texlive[57759] Master: numerica (15feb21)
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numerica (15feb21)
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+numerica: a package to numerically evaluate mathematical
+expressions in the LaTeX form in which they are typeset.
+
+Andrew Parsloe (ajparsloe at gmail.com) 05 February 2021
+
+This work may be distributed and/or modified under the conditions
+of the LaTeX Project Public License, either version 1.3c of this
+license or any later version; see
+http://www.latex-project.org/lppl.txt
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+This is version 1.0.0 of the numerica bundle
+The LaTeX3 bundles l3kernel and l3packages
+are required; also amsmath and mathtools.
+
+numerica-basics.pdf explains more.
+
+Manifest
+%%%%%%%%
+README.txt this document
+
+numerica.sty LaTeX .sty file
+numerica-plus.def optionally loaded files for use
+numerica-tables.def with numerica.sty
+numerica-lyx.def
+
+numerica-basics.pdf documentation for numerica.sty &
+ numerica-lyx.def
+numerica-plus.pdf documentation for numerica-plus.def
+numerica-tables.pdf documentation for numerica-tables.def
+
+numerica-basics.tex LaTeX documentation source files
+numerica-plus.tex
+numerica-tables.tex
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+%% LyX 2.4.0-alpha1 created this file. For more info, see https://www.lyx.org/.
+%% Do not edit unless you really know what you are doing.
+\documentclass[english,tableposition=top]{report}
+\usepackage{lmodern}
+\renewcommand{\sfdefault}{lmss}
+\renewcommand{\ttdefault}{lmtt}
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\usepackage[utf8]{inputenc}
+\setcounter{secnumdepth}{3}
+\usepackage{color}
+\definecolor{shadecolor}{rgb}{0.667969, 1, 1}
+\usepackage{babel}
+\usepackage{array}
+\usepackage{cprotect}
+\usepackage{wrapfig}
+\usepackage{booktabs}
+\usepackage{framed}
+\usepackage{url}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[unicode=true,pdfusetitle,
+ bookmarks=true,bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=2,
+ breaklinks=true,pdfborder={0 0 1},backref=section,colorlinks=true]
+ {hyperref}
+
+\makeatletter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+\providecommand{\LyX}{\texorpdfstring{\ensureascii{%
+ L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@}}{LyX}}
+\DeclareRobustCommand*{\lyxarrow}{%
+\@ifstar
+{\leavevmode\,$\triangleleft$\,\allowbreak}
+{\leavevmode\,$\triangleright$\,\allowbreak}}
+%% Because html converters don't know tabularnewline
+\providecommand{\tabularnewline}{\\}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
+\newenvironment{centred}%
+ {\begin{center}\baselineskip=13pt\parskip=1pt}{\end{center}}
+\newenvironment{lyxcode}
+ {\par\begin{list}{}{
+ \setlength{\rightmargin}{\leftmargin}
+ \setlength{\listparindent}{0pt}% needed for AMS classes
+ \raggedright
+ \setlength{\itemsep}{0pt}
+ \setlength{\parsep}{0pt}
+ \normalfont\ttfamily}%
+ \item[]}
+ {\end{list}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
+\usepackage{numerica}
+
+\newcommand\rel{\,\varrho\;}
+\DeclareMathOperator{\erf}{erf}
+\DeclareMathOperator{\gd}{gd}
+
+\reuse
+
+\usepackage{upquote}
+
+\makeatother
+
+\begin{document}
+\title{\texttt{numerica-basics}}
+\author{Andrew Parsloe\\
+(\url{ajparsloe at gmail.com})\\
+}
+\maketitle
+\begin{abstract}
+The \verb`numerica` package defines a command to wrap around a mathematical
+expression in its LaTeX form and, once values are assigned to variables,
+numerically evaluate it. The intent is to avoid the need to modify
+the LaTeX form of the expression being evaluated. For programs with
+a preview facility like LyX, or compile-as-you-go systems, interactive
+back-of-envelope calculations and numerical exploration are possible
+within the document being worked on. The package requires the bundles
+\verb`l3kernel` and \verb`l3packages`, and the \verb`amsmath` and
+\verb`mathtools` packages. Additional modules define commands to
+iterate and find fixed points of functions of a single variable, to
+find the zeros or extrema of such functions, to calculate the terms
+of recurrence relations, and to create multi-column tables of function
+values (which requires the \verb`booktabs` package).\\
+\\
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\paragraph*{Note:}
+\begin{itemize}
+\item {\normalsize This document applies to version 1.0.0 of }{\normalsize\texttt{numerica.sty}}{\normalsize .}{\small\par}
+\item {\normalsize Reasonably recent versions of the \LaTeX 3 bundles }{\normalsize\texttt{l3kernel}}{\normalsize{}
+and }{\normalsize\texttt{l3packages}}{\normalsize{} are required.}{\small\par}
+\item {\normalsize The package requires }{\normalsize\texttt{amsmath}}{\normalsize{}
+and }{\normalsize\texttt{mathtools}}{\normalsize .}{\small\par}
+\item {\normalsize I refer many times in this document (especially §\ref{sec:Argument-parsing})
+to }{\normalsize\emph{Handbook of Mathematical Functions}}{\normalsize ,
+edited by Milton Abramowitz and Irene A. Segun, Dover, 1965. This
+is abbreviated to }{\normalsize\emph{HMF}}{\normalsize , often followed
+by a number like 1.2.3 to locate the actual expression referenced.}{\small\par}
+\end{itemize}
+\end{shaded}%
+\end{minipage}
+\end{abstract}
+\begin{center}
+\tableofcontents{}
+\par\end{center}
+
+\chapter{Introduction}
+
+\texttt{numerica} is a \LaTeX{} package offering the ability to numerically
+evaluate mathematical expressions in the \LaTeX{} form in which they
+are typeset.\footnote{\texttt{numerica} evolved from the author's \texttt{calculyx} package
+that was designed for use with the document processor \LyX{} and available
+for download from a link on the \LyX{} wiki website (but not from CTAN). }
+
+There are a number of packages which can do calculations in \LaTeX ,\footnote{A simple search finds the venerable \texttt{calc} in the \LaTeX{} base,
+\texttt{calculator }(including an associated \texttt{calculus} package),
+\texttt{fltpoint}, \texttt{fp} (\emph{fixed} rather than floating
+point), \texttt{spreadtab} (using either \texttt{fp} or \texttt{l3fp}
+as its calculational engine) if you want simple spreadsheeting with
+your calculations, the elaborate \texttt{xint}, \texttt{pst-calculate}
+(a limited interface to \texttt{l3fp}), \texttt{l3fp} in the \LaTeX 3
+kernel, and \texttt{xfp}, the \LaTeX 3 interface to \texttt{l3fp}.
+Other packages include a calculational element but are restricted
+in their scope. (\texttt{longdivision} for instance is elegant, but
+limited only to long division.) } but those I am aware of all require the mathematical expressions
+they operate on to be changed to an appropriate syntax. Of these packages
+\texttt{xfp} comes closest to my objective with \texttt{numerica}.
+For instance, given a formula
+\begin{centred}
+\verb`\frac{\sin (3.5)}{2} + 2\cdot 10^{-3}`
+\end{centred}
+(in a math environment), this can be evaluated using \texttt{xfp}
+by transforming the expression to \verb`sin(3.5)/2 + 2e-3` and wrapping
+this in the command \verb`\fpeval`. In \texttt{numerica} you don't
+need to transform the formula, just wrap it in an \verb`\eval` command
+(for the acutal calculation see §\ref{subsec:introSimple-examples}):
+\begin{centred}
+\verb`\eval{ \frac{\sin (3.5)}{2} + 2\cdot 10^{-3} }`.
+\end{centred}
+\texttt{numerica}, like \texttt{xfp} and a number of other packages,
+uses \texttt{l3fp} (the \LaTeX 3 floating point module in \texttt{l3kernel})
+as its calculational engine. To some extent the main command, \verb`\nmcEvaluate`,
+short-name form \verb`\eval`, is a pre-processor to \texttt{l3fp},
+converting mathematical expressions written in the \LaTeX\texttt{
+}form in which they will be typeset into an `fp-ified' form that
+is digestible by \texttt{l3fp}. The aim is to make the command act
+as a wrapper around such formulas. Ideally, one should not have to
+make \emph{any} adjustment to them, although any text on Fourier series
+suggests that hope in full generality is delusional. Surprisingly
+often however it \emph{is} possible. We will see shortly that even
+complicated formulas like
+
+\[
+\cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)\frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi},
+\]
+and
+\[
+\left(\frac{1-4\sin^{2}\tfrac{m}{3n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi}\right)\sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi,
+\]
+
+\noindent can be evaluated `as is' (see below, §\ref{subsec:introReassurance}).
+There is no need to shift the position of the superscript $2$ on
+the sines, no need to parenthesize the arguments of $\sin$ and $\cos$,
+no need to insert asterisks to indicate multiplication, no need to
+change the \verb`\frac` and \verb`\tfrac`-s to slashes, \texttt{/},
+no need to delete the \verb`\left` and \verb`\right` that qualify
+the big parentheses (in the underlying \LaTeX ) in the second expression.
+Of course, if there are variables in an expression, as in these examples,
+they will need to be assigned values. And how the result of the evaluation
+is presented also requires specifying, but the aim is always: to evaluate
+mathematical expressions in \LaTeX{} with as little adjustment as possible
+to the form in which they are typeset.
+
+\texttt{numerica} is written in \texttt{expl3}, the programming language
+of the \LaTeX 3 project. It uses the \LaTeX 3 module \texttt{l3fp}
+(part of \texttt{l3kernel}) as its calculational engine. This enables
+floating point operations to 16 significant figures, with exponents
+ranging between $-10000$ and $+10000$. Many functions and operations
+are built-in to \texttt{l3fp} – arithmetic operations, trigonometric,
+exponential and logarithm functions, factorials, absolute value, max
+and min. Others have been constructed for \texttt{numerica }from \texttt{l3fp}
+ingredients – binomial coefficients, hyperbolic functions, sums and
+products – but to the user there should be no discernible difference.
+
+Associated modules provide for additional operations: iteration, finding
+zeros, recurrence relations, mathematical table building. Further
+modules are planned (e.g. calculus).
+
+\section{How to use \texttt{numerica}}
+
+The package is invoked in the usual way:\footnote{I use the angle-bracket notation to indicate optional user input.
+Of course what is input does not include the angle brackets.} put
+\begin{lyxcode}
+\textbackslash usepackage{[}<options>{]}\{numerica\}
+\end{lyxcode}
+\noindent in the \LaTeX{} preamble. \texttt{numerica} requires the
+\texttt{amsmath} and \texttt{mathtools} packages and loads these automatically.
+\texttt{numerica} will also accept use of some relational symbols
+from the \texttt{amssymb} package provided that package is loaded;
+see §\ref{subsec:evalBoolean-output}.
+
+\subsection{Packages and package options}
+
+\label{subsec:introPackagesOptions}Version 1.0.0 of \texttt{numerica}
+has three package options.
+\begin{description}
+\item [{\texttt{plus}}] By calling \texttt{numerica} with the \texttt{plus}
+package option,\\
+{\ttfamily{}%
+\noindent\begin{minipage}[t]{1\columnwidth}%
+~~~~\textbackslash usepackage{[}plus{]}\{numerica\}%
+\end{minipage}}\texttt{}~\\
+the file \texttt{numerica-plus.def }is loaded where a number of additional
+commands: \verb`\nmcIterate`, \verb`\nmcSolve`, \verb`\nmcRecur`
+are defined. These enable the iteration of functions of a single variable\footnote{At this stage!},
+including finding fixed points; the solving of equations of the form
+$f(x)=0$ (or the location of local maxima or minima); and the calculation
+of terms in recurrence relations (like the Fibonacci series or othogonal
+polynomials). See the associated document \texttt{numerica-plus.pdf}.
+\item [{\texttt{tables}}] By calling \texttt{numerica} with the \texttt{tables}
+package option\\
+{\ttfamily{}%
+\noindent\begin{minipage}[t]{1\columnwidth}%
+~~~~\textbackslash usepackage{[}tables{]}\{numerica\}%
+\end{minipage}}\texttt{}~\\
+the file \texttt{numerica-tables.def} is loaded with with the command
+\verb`\nmcTabulate` enabling the creation of multi-column tables
+of function values with a wide variety of formatting options (most
+of those employed in \emph{HMF} in fact). See the associated document
+\texttt{numerica-tables.pdf}.
+\item [{\texttt{lyx}}] By calling \texttt{numerica} with the \texttt{lyx}
+package option,\\
+{\ttfamily{}%
+\noindent\begin{minipage}[t]{1\columnwidth}%
+~~~~\textbackslash usepackage{[}lyx{]}\{numerica\}%
+\end{minipage}}\texttt{}~\\
+the file \texttt{numerica-lyx.def }is loaded with code enabling the
+full use of the \verb`\nmcReuse` command in the document processor
+\LyX{} (along with all other commands of the \texttt{numerica} package).
+Use of \texttt{numerica} in \LyX{} exploits the mini-\LaTeX{} runs of
+the instant preview facility of that program to give immediate feedback
+on calculations without requiring the whole document to be compiled.
+See Chapter~\ref{chap:LyX}.
+\end{description}
+More than one option can be used at a time by separating the options
+with a comma; e.g. \texttt{\textbackslash usepackage{[}plus,tables{]}\{numerica\}}.
+However, apart from Chapter~\ref{chap:LyX}, the present document
+focuses on \texttt{numerica} when called with no options: \texttt{\textbackslash usepackage\{numerica\}}.
+
+\subsection{Simple examples of use}
+
+\label{subsec:introSimple-examples}A simple example of use is provided
+by the document
+\begin{verbatim}
+ \documentclass{minimal}
+ \usepackage{numerica}
+ \begin{document}
+
+ \eval{$ mc^2 $}[m=70,c=299 792 458][8x]
+
+ \end{document}
+
+\end{verbatim}
+\noindent We have a formula between math delimiters: \verb`$ mc^2 $`.
+We have wrapped a command \verb`\eval` around the lot, added an optional
+argument in parentheses specifying numericaal values for the quantities
+\texttt{m} and \texttt{c}, and concluded it all with a trailing optional
+argument specifying that the result should be presented to $8$ places
+of decimals and in scientific notation (the \verb`x`). Running \texttt{pdflatex}
+on this document generates a pdf displaying
+\begin{centred}
+\eval{$ mc^2 $}[m=70,c=299 792 458][8x]
+\end{centred}
+\noindent where the formula ($mc^{2})$ is equated to the numerical
+value resulting from substituting the given values of $m$ and $c$.
+Those values are displayed in a list following the result. The calculation
+is presented to $8$ decimal places in scientific notation. (According
+to Einstein's famous equation $E=mc^{2}$ this is the enormous energy
+content, in joules, of what was once considered an average adult Caucasian
+male. Only a minute fraction is ever available.)
+
+A second example is provided by the formula in earlier remarks:
+\begin{verbatim}
+ \documentclass{minimal}
+ \usepackage{numerica}
+ \begin{document}
+
+ \eval{\[ \frac{\sin(3.5)}{2} + 2\cdot 10^{-3} \]}
+
+ \end{document}
+
+\end{verbatim}
+Running \texttt{pdflatex} on this document produces the result
+
+\eval{\[ \frac{\sin(3.5)}{2} + 2\cdot 10^{-3} \]}
+
+The \verb`\eval` command used in these examples is the main command
+of the \texttt{numerica} package. It is discussed in full in the next
+chapter, but first some preliminaries.
+
+\subsection{Display of the result}
+
+\label{subsec:introResultDisplay}In what follows I shall write things
+like (but generally more complicated than)
+\begin{centred}
+\verb`$ \eval{ 1+1 } $` $\Longrightarrow \eval{ 1 + 1 } $
+\end{centred}
+to mean: run \texttt{pdflatex} on a document containing \verb`\eval{1+1}`
+in the document body to generate a pdf containing the calculated result
+($2$ in this instance). In this case the \verb`\eval` command is
+used \emph{within} a math environment (delimited by the dollar signs).
+It is not limited to this behaviour. The command can also wrap \emph{around}
+the math delimiters (as we saw in the previous examples):
+\begin{centred}
+\verb`\eval{$ 1+1 $}` $\Longrightarrow$ \eval{$ 1+1 $}.
+\end{centred}
+As you can see, the display that results is different.
+\begin{itemize}
+\item When the \verb`\eval` command is used\emph{ within} a math environment,
+only the \emph{result,} followed possibly by the \emph{variable~=~value
+list} (see §\ref{subsec:evalVv-list}) is displayed.
+\end{itemize}
+Environments may include the various AMS environments as well as the
+standard \LaTeX{} inline ( \verb`$ $` ), \texttt{equation} ( \verb`\[ \]`
+) and \texttt{eqnarray} environments. For an example of \verb`\eval`
+within an \verb`align*` environment see §\ref{subsec:introExploring}
+below.
+\begin{itemize}
+\item When the \verb`\eval` command is wrapped\emph{ around} a math environment,
+the result is displayed in the form, \emph{formula = result} (followed
+possibly by the \emph{variable~=~value list}) within that environment,
+\begin{itemize}
+\item If the formula is long or contains many variables then it may be desirable
+to split the display over two lines; see §\ref{subsec:evalChanging-display-format}
+and §\ref{subsec:settings New-line-display},
+\end{itemize}
+\end{itemize}
+the whole presented as an inline expression if \verb`$` delimiters
+are used, or as a display-style expression otherwise. (See the $mc^{2}$
+example for an illustration.)
+
+It is not clear to me that wrapping \verb`\eval` \emph{around} the
+AMS environments, except for \texttt{multline}, makes much sense,
+although it can be done. Here is an example of \verb`\eval` wrapped
+around a \texttt{multline{*}} environment (the phantom is there so
+that the hanging $+$ sign spaces correctly),\medskip{}
+
+\begin{minipage}[t]{0.8\columnwidth}%
+\begin{verbatim}
+\eval{ \begin{multline*}
+ 1+2+3+4+5+6+7+8+9+10+\phantom{0}\\
+ 11+12+13+14+15+16+17+18+19
+ \end{multline*} }
+\end{verbatim}
+%
+\end{minipage}
+
+\medskip{}$\Longrightarrow$ %
+\begin{minipage}[t]{0.8\columnwidth}%
+\vspace{-2ex}\eval{ \begin{multline*}
+ 1+2+3+4+5+6+7+8+9+10+\phantom{0}\\ 11+12+13+14+15+16+17+18+19
+\end{multline*} }%
+\end{minipage}
+\begin{itemize}
+\item It is also possible to dispense with math delimiters entirely, neither
+wrapped within nor wrapped around the \verb`\eval` command, but in
+that case \texttt{numerica} acts as if \verb`\eval` had been used
+within \verb`\[` and \verb`\]` and displays the result accordingly.
+\end{itemize}
+
+\subsection{Exploring}
+
+\label{subsec:introExploring}When working on \texttt{numerica}'s
+predecessor package, I constantly tested it against known results
+to check for coding errors. One test was to ensure that
+\[
+\left(1+\frac{1}{n}\right)^{n}
+\]
+did indeed converge to the number $e$ as $n$ increased.\texttt{ }Let's
+do that here. Try first $n=10$:
+\begin{center}
+\verb`\eval{$ e-(1+1/n)^n $}[n=10][x]` $\Longrightarrow$ \eval{$ e-(1+1/n)^n $}[n=10][x].
+\par\end{center}
+
+\noindent (The default number of decimal places displayed is $6$.)
+The difference between $e$ and $(1+1/n)^{n}$ is about an eighth
+($0.125$) when $n=10$, which is encouraging but hardly decisive.
+The obvious thing to do is increase the value of $n$. I'll use an
+\verb`align*` environment to `prettify' the presentation of the
+results:
+\begin{verbatim}
+ \begin{align*}
+ e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^5][*x],\\
+ e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^6][*x],\\
+ e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^7][*x],\\
+ e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^8][*x].
+ \end{align*}
+\end{verbatim}
+(most of which was written using copy and paste) which produces
+
+\begin{align*}
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{5}][x],\\
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{6}][*x],\\
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{7}][x],\\
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{8}][x].
+\end{align*}
+Clearly $(1+1/n)^{n}$ converges to $e$, the difference between them
+being of order $1/n$, but that is not what catches the eye. There
+is an unanticipated regularity here. 1.35914? Double the number: \verb`$\eval{2\times 1.35914}[5]$`\texttt{
+}$\Longrightarrow\eval{2\times1.35914}[5]$ which is close enough
+to $e$ to suggest a relationship, namely,
+\[
+\lim_{n\to\infty}n\left(e-\left(1+\frac{1}{n}\right)^{n}\right)=\tfrac{1}{2}e.
+\]
+
+This was new to me. Is it true? From the familiar expansion of the
+logarithm
+
+\begin{align*}
+\ln\left(1+\frac{1}{n}\right)^{n} & =n\ln\left(1+\frac{1}{n}\right)\\
+ & =n\left(\frac{1}{n}-\frac{1}{2}\frac{1}{n^{2}}+\frac{1}{3}\frac{1}{n^{3}}-\ldots\right)\\
+ & =1-\frac{1}{2n}\left(1-\frac{2}{3}\frac{1}{n}+\frac{2}{4}\frac{1}{n^{2}}-\right)\\
+ & \equiv1-\frac{1}{2n}E_{n},
+\end{align*}
+say. Since $E_{n}$ is an alternating series and the magnitudes of
+the terms of the series tend to $0$ monotonically, $1>E_{n}>1-2/3n$.
+From this and the inequalities $1/(1-x)>e^{x}>1+x$ when $x<1$ it
+proved a straightforward matter to verify the proposed limit.
+
+\subsection{Reassurance}
+
+\label{subsec:introReassurance}In the course of some hobbyist investigations
+in plane hyperbolic geometry I derived the formula
+\[
+\Phi_{1}(m,n)=\cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)\frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi},
+\]
+for $m=2,3,\ldots$ and integral $n\ge2m+1$. A key concern was: when
+is $\Phi_{1}$ positive? After an embarrassingly laborious struggle,
+I managed to work this expression into the form
+\[
+\Phi_{2}(m,n)=\left(\frac{1-4\sin^{2}\tfrac{m}{3n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi}\right)\sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi,
+\]
+in which the conditions for positivity are clear: with $n\ge2m+1$,
+so that $m\pi/3n<\pi/6$, the first factor is always positive, the
+second is positive for $m\ge2$, and the third is positive for $m\ge4$.
+All well and good, but given the struggle to derive $\Phi_{2}$, was
+I confident that $\Phi_{1}$ and $\Phi_{2}$ really\emph{ }are equal?
+It felt all too likely that I had made a mistake.
+
+The simplest way to check was to see if the two expressions gave the
+same numericaal answers for a number of $m,\thinspace n$ values.
+I wrote \verb`\eval{\[ \]}[m=2,n=5]` twice and between the delimiters
+pasted the already composed expressions for $\Phi_{1}$ and $\Phi_{2}$,
+namely:
+\begin{verbatim}
+ \eval{\[
+ \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
+ \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \]}[m=2,n=5]
+ \eval{\[
+ \left(
+ \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \right)
+ \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
+ \]}[m=2,n=5]
+\end{verbatim}
+I have added some formatting – indenting, line breaks – to make the
+formulas more readable for the present document but otherwise left
+them unaltered. The \verb`\eval` command can be used for even quite
+complicated expressions without needing to tinker with their \LaTeX{}
+form, but you may wish – as here – to adjust white space to clarify
+the component parts of the formula. Running \texttt{pdflatex} on these
+expressions, the results were
+
+\eval{\[
+ \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
+ \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \]}[m=2,n=5]
+
+\eval{\[
+ \left(
+ \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \right)
+ \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
+ \]}[m=2,n=5]
+
+\noindent which was reassuring. Doing it again but with different
+values of $m$ and $n$, again the results coincided:
+
+\eval{\[
+ \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
+ \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \]}[m=5,n=13]
+
+\eval{\[
+ \left(
+ \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \right)
+ \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
+ \]}[m=5,n=13]
+
+\noindent Thus reassured that there was \emph{not }an error in my
+laborious derivation of $\Phi_{2}$ from $\Phi_{1}$, it was not difficult
+to work back from $\Phi_{2}$ to $\Phi_{1}$ then reverse the argument
+to find a straightforward derivation.
+
+\chapter{\texttt{\textbackslash nmcEvaluate} (\texttt{\textbackslash eval)}}
+
+The main calculational command in \texttt{numerica} is \verb`\nmcEvaluate`.
+Unlike some other commands which are loaded optionally, \verb`\nmcEvaluate`
+is \emph{always} loaded, and therefore always available. Because \verb`\nmcEvaluate`
+would be tiresome to write too frequently,\texttt{ }particularly for
+back-of-envelope calculations, there is an equivalent short-name form,
+\verb`\eval`, used almost exclusively in the following. But note:
+wherever you see the command \verb`\eval`, you can substitute \verb`\nmcEvaluate`
+and obtain the same result.
+
+\verb`\eval` (like other short-name forms of other commands in the
+\texttt{numerica} suite) is defined using \verb`\ProvideDocumentCommand`
+from the \texttt{xparse} package. Hence if \verb`\eval` has already
+been defined in some other package already loaded, it will not be
+redefined by \texttt{numerica}. It will retain its meaning in the
+other package. Its consequent absence from \texttt{numerica} may be
+an irritant, but only that; \verb`\nmcEvaluate` is defined using
+\texttt{xparse}'s \verb`\DeclareDocumentCommand` which would override
+any (freakishly unlikely) previous definition of \verb`\nmcEvaluate`
+in another package and would therefore still be available.
+
+\section{Syntax of \texttt{\textbackslash nmcEvaluate (\textbackslash eval)}}
+
+There are five arguments to the \verb`\nmcEvaluate` (or \verb`\eval`)
+command, of which only one, the third, is mandatory. All others are
+optional. If all are deployed the command looks like
+\begin{centred}
+\noindent \verb`\eval*[settings]{expr.}[vv-list][num. format]`
+\end{centred}
+I discuss the various arguments in the referenced sections.
+\begin{enumerate}
+\item \verb`*` optional switch; if present ensures display of only the
+numerical result (suppresses display of the formula and vv-list);
+see §\ref{subsec:evalVvSuppresList}
+\item \verb`[settings]` optional comma-separated list of \emph{key=value
+}settings for this particular calculation; see §\ref{sec:settingsOption}
+\item \verb`{expr.}` the only mandatory argument; the mathematical expression/formula
+in \LaTeX{} form that is to be evaluated
+\item \verb`[vv-list]` optional comma-separated list of \emph{variable=value
+}items; see §\ref{subsec:evalVv-list}
+\item \verb`[num. format]` optional format specification for presentation
+of the numerical result (rounding, padding with zeros, scientific
+notation, boolean output); see~§\ref{subsec:evalRoundingEtc}
+\end{enumerate}
+Note that arguments 4 and 5 are both square-bracket delimited optional
+arguments. Should only one such argument be used, \texttt{numerica}
+determines which is intended by looking for an equals sign within
+the argument. Its presence indicates the argument is the vv-list;
+its absence indicates the argument is the number format specification.
+
+The vv-list and number-format specification are \emph{trailing} optional
+arguments. There is a possibility that the \verb`\eval` command could
+be followed by a square-bracketed mathematical expression which \texttt{numerica}
+might therefore confuse with one of its trailing arguments. Experience
+using \texttt{numerica} suggests that this will be a (very) rare occurrence
+and is easily prevented by inserting an empty brace pair (\verb`{}`)
+before the offending square-bracketed expression. Allowing spaces
+between the arguments enables complicated expressions and large vv-lists
+to be formatted, in the interests of clarity, with new lines and white
+space – without requiring the insertion of comment characters (\verb`%`).
+
+Recommended practice is to minimise the number of optional arguments
+used in \LaTeX{} commands by consolidating such arguments into a single
+\emph{key=value} list. Although \texttt{numerica} uses such an argument,
+the vv-list does not fit naturally into that scheme. And practice
+suggests that separating out the elements of the number format specification
+(rounding value, padding with zeros, scientific notation, boolean
+output) and placing them in a trailing argument feels natural for
+the kind of back-of-envelope calculations envisaged for \texttt{numerica}.
+
+\section{The variable=value list}
+
+\label{subsec:evalVv-list}To evaluate algebraic, trigonometric and
+other formulas that involve \emph{variables} we need to give those
+variables values. This is done in the \emph{variable=value list} –
+or \emph{vv-list} for short. This is the fourth argument of the \texttt{\textbackslash nmcEvaluate}
+command and is a square-bracket delimited optional argument (optional
+because an expression may depend only on constants and numbers).
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+I was sorely tempted to use parentheses to delimit this argument,
+since then both the placement and delimiters of the vv-list would
+anticipate the way it is displayed in the evaluated result (see the
+$mc^{2}$ example in §\ref{subsec:introSimple-examples} above). But
+there is good reason not to. Parentheses will often occur in expressions
+in the vv-list. With parentheses nested within parentheses it is all
+too easy to get a pairing wrong, which would cause a \LaTeX{} error
+and halt compilation. As it is, using the standard square bracket
+delimiters, unbalanced parentheses cause a \texttt{numerica} error
+(see §\ref{subsec:errorsMismatched-brackets}), which does not halt
+compilation. (Of course unbalanced \emph{square} brackets now will
+cause a \LaTeX{} error, but such brackets are used less often in mathematical
+expressions and are rarely nested within other square-bracketed expressions.)\end{shaded}%
+\end{minipage}
+
+\subsection{Variable names}
+
+\label{subsec:evalVariableNames}In mathematical practice, variable
+names are generally single letters of the Roman or Greek alphabets,
+sometimes also from other alphabets, in a variety of fonts, and often
+with subscripts or primes or other decorations. In \texttt{numerica}
+a variable name is \emph{what lies to the left of the equals sign
+in an item} of the vv-list. Thus variables can be multi-token affairs:
+$x',x'',x^{iv},x_{n},x'_{n},x''_{mn}$, $^{k}C_{n},var,\mathrm{var},Fred,\mathbf{Fred},\mathcal{FRED}\ldots$
+Although variable names start and end with non-space tokens, a variable
+name may contain spaces – for instance \verb`x x` should not cause
+a \verb`numerica` error, but such names are not part of mathematical
+practice. Usually, for the kind of back-of-envelope calculations envisaged
+for \verb`numerica`, and for ease of typing, most variables will
+be single letters from the Roman or Greek alphabets.
+
+Because equals signs and commas give structure to the vv-list, it
+should also be clear that a variable name should not contain a \emph{naked}
+equals sign or a \emph{naked} comma. They can be incorporated in a
+variable name but only when decently wrapped in braces, like \verb`R_{=}`
+displaying as $R_{=}$ or \verb`X_{,i}` displaying as $X_{,i}$.
+
+Note that $x$ and $\mathrm{x}$ will be treated by \verb`numerica`
+as \emph{different} variables since, in the underlying \LaTeX , one
+is \texttt{x} and the other \verb`\mathrm{x}`. Even names that look
+identical in the pdf may well be distinct in \LaTeX . This is true
+particularly of superscripts and subscripts: \verb`x_0` and \verb`x_{0}`
+appear identical in the pdf but in the underlying \LaTeX{} they are
+distinct, and will be treated as distinct variables by \verb`numerica`.
+
+Although multi-token variables are perfectly acceptable, \emph{internally}
+\verb`numerica` expects variables to be single tokens. Hence a necessary
+initial step for the package is to convert all multi-token variable
+names in the vv-list and the formula to single tokens. \verb`numerica`
+does this by turning the multi-token variable names into control sequences
+with names in the sequence \verb`\_nmca`, \verb`\_nmcb`, \verb`\_nmcc`,
+etc., then searches through the vv-list and the formula for every
+occurrence of the multi-token names and replaces them with the relevant
+control sequences. It does this in order of decreasing size of name,
+working from the names that contain most tokens down to names containing
+only two tokens.
+
+The conversion process uses computer resources. Even if there are
+no multi-token variables present, \verb`numerica` still needs to
+check that this is so – unless the user alerts the program to the
+fact. This can be done by making a brief entry \texttt{xx=0 }in the
+settings option (the second optional argument of \verb`\nmcEvaluate`);
+see §\ref{subsec:settingsMultitokSwitch}. If the user never (or hardly
+ever) uses multi-token variables, then a more permanent solution is
+to create a file \verb`numerica.cfg`\texttt{ }with the line \texttt{multitoken-variables
+= false}; see §\ref{sec:settingsDefaults} for this.
+
+\subsection{The vv-list and its use}
+
+A vv-list is a comma-separated list where each item is of the form
+\emph{variable=value}. It might be something simple like
+\begin{lyxcode}
+{[}g=9.81,t=2{]}
+\end{lyxcode}
+or something more complicated like
+\begin{lyxcode}
+{[}V\_S=\textbackslash tfrac43\textbackslash pi~r\textasciicircum 3,V\_C=2\textbackslash pi~r\textasciicircum 2h,h=3/2,r=2{]}.
+\end{lyxcode}
+Spaces around the equals signs or the commas are stripped away during
+processing so that
+\begin{lyxcode}
+{[}g=9.81,t=2{]}~\textrm{and}~{[}~g~=~9.81~,~t~=~2{]}
+\end{lyxcode}
+are the \emph{same} variable=value list.
+
+\subsubsection{Evaluation from right to left}
+
+In these examples, with variables depending on other variables, there
+is an implication: that the list is evaluated \emph{from the right}.
+Recall how a function of a function is evaluated, say\emph{ }$y=f(g(h(x)))$.
+To evaluate $y$, first $x$ is assigned a value then $h(x)$ is calculated,
+then $g(h(x))$ then $f(g(h(x)))=y$. We work from right to left,
+from the innermost to the outermost element. Or consider an example
+like calculating the area of a triangle by means of the formula
+\[
+A=\sqrt{s(s-a)(s-b)(s-c)}.
+\]
+First we write the formula; then we state how $s$ depends on $a,b,c$,
+namely $s=\frac{1}{2}(a+b+c)$, then we give values to $a,b,c$. In
+\texttt{numerica} this is mirrored in the layout of the \verb`\eval`
+command:
+\begin{verbatim}
+ \eval{$ \sqrt{s(s-a)(s-b)(s-c)} $}
+ [s=\tfrac12(a+b+c),a=3,b=4,c=5]
+\end{verbatim}
+The formula in a sense is the leftmost extension of the vv-list. The
+entire evaluation occurs from right to left.
+
+This means that the rightmost variable in the vv-list can depend only
+on (mathematical) constants and numbers – although it may be a complicated
+expression of those elements. Other variables in the vv-list can depend
+on variables \emph{to their right} but not to their left.
+
+\subsubsection{Expressions in the variable=value list}
+
+Suppose our expression is $\tfrac{4}{3}\pi r^{3}$, the volume $V_{S}$
+of a sphere in terms of its radius $r$, and we want to calculate
+the volume for different values of $r$ to get a sense of how rapidly
+volume increases with radius.
+\begin{centred}
+\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=1] $` $\Longrightarrow$ $ V_S=\eval{ \tfrac43\pi r^3 }[r=1] $.
+\end{centred}
+Having set up this calculation it is now an easy matter to change
+the value of $r$ in the vv-list:
+\begin{centred}
+\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=1.5] $` $\Longrightarrow$
+$ V_S= \eval{ \tfrac43\pi r^3 }[r=1.5] $.
+
+\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=2] $` $ \Longrightarrow $ $V_S= \eval{ \tfrac43\pi r^3 }[r=2] $.
+\end{centred}
+To compute the volume $V_{C}=\pi r^{2}h$ of a cylinder, we have two
+variables to assign values to:
+\begin{centred}
+\verb`$ V_C=\eval{ \pi r^2h }[h=4/3,r=1] $` $\Longrightarrow$ $ V_C=\eval{ \pi r^2h }[h=4/3,r=1] $.
+\end{centred}
+Although values in the vv-list are generally either numbers or simple
+expressions (like \texttt{4/3}), that is not essential. A little more
+complicated is
+\begin{centred}
+\verb`$ V_C=\eval{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
+$ V_C=\eval{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $.
+\end{centred}
+where calculation of the volume of the cylinder has been split into
+two: first calculate the area $A_{C}$ of its circular base and then,
+once that has been effected, calculate the volume.
+
+A second example is provided by Brahmagupta's formula for the area
+of a triangle in terms of its semi-perimeter. In a triangle ABC, the
+sides are $a=3$, $b=4$ and $c=5$. (Of course we know this is a
+right-angled triangle with area $\tfrac{1}{2}ab=6$.) The semi-perimeter
+$s=\tfrac{1}{2}(a+b+c)$ and the area of ABC is \medskip{}
+\begin{verbatim}
+ \eval{$ \sqrt{s(s-a)(s-b)(s-c) $}
+ [s=\tfrac12(a+b+c),a=3,b=4,c=5]
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \sqrt{s(s-a)(s-b)(s-c)} $}
+ [s=\tfrac12(a+b+c),a=3,b=4,c=5].
+
+\subsubsection{Constants}
+
+There are five constants built-in to \texttt{numerica}: \verb`\pi`,
+the ratio of circumference to diameter of a circle, \verb`e`, the
+base of natural logarithms, \verb`\gamma`, the limit of $\left(\sum_{1}^{N}1/n\right)-\ln N$
+as $N\to\infty$, \verb`\phi`, the golden ratio, equal to $\tfrac{1}{2}(1+\surd5)$,
+and the utilitarian constant \verb`\deg`, the number of radians in
+a degree.
+\begin{centred}
+\verb`\eval{$ \pi $}` $\Longrightarrow$ \eval{$ \pi $},
+
+\verb`\eval{$ e $}` $\Longrightarrow$ \eval{$ e $},
+
+\verb`\eval{$ \gamma $}` $\Longrightarrow$ \eval{$ \gamma $},
+
+\verb`\eval{$ \phi $}` $\Longrightarrow$ \eval{$ \phi $},
+
+\verb`\eval{$ \deg $}` $\Longrightarrow$ \eval{$ \deg $},
+\end{centred}
+so that \verb`\eval{$ 180\deg $}` $\Longrightarrow$ \eval{$ 180\deg $}
+(as it should).
+
+Let's combine two of these in a formula:
+\begin{centred}
+\verb`\eval{$ e^\pi-\pi^e $}` $\Longrightarrow$ \eval{$ e^\pi-\pi^e $},
+\end{centred}
+which is close-ish to $\tfrac{1}{4}e$: \verb`\eval{$ \tfrac14e $}`
+$\Longrightarrow$ \eval{$ \tfrac14e $}.
+
+\subsubsection{Use of \texttt{\textbackslash pi},\texttt{ e},\texttt{ \textbackslash gamma},\texttt{
+\textbackslash phi} as variables}
+
+In some contexts it may feel natural to use any or all of \verb`\pi`,
+\verb`e`, \verb`\gamma` and \verb`\phi` as variables by assigning
+values to them in the vv-list. \texttt{numerica} does not object.
+The values assigned in this way override the constants' values. For
+example, if the triangle we labelled ABC previously was instead labelled
+CDE then it has sides $c=3,d=4$ and (note!) $e=5$. It's area therefore
+is\medskip{}
+\begin{verbatim}
+ \eval{$ \sqrt{s(s-c)(s-d)(s-e)} $}
+ [s=\tfrac12(c+d+e),c=3,d=4,e=5]
+\end{verbatim}
+$\Longrightarrow$
+
+\eval{$ \sqrt{s(s-c)(s-d)(s-e)} $}
+ [s=\tfrac12(c+d+e),c=3,d=4,e=5].\medskip{}
+
+\noindent Since this is the correct area we see that \texttt{e} has
+been treated as a variable with the assigned value $5$, not as the
+constant. But if \texttt{e} (or \verb`\pi` or \verb`\gamma` or \verb`\phi`)
+is not assigned a value in the vv-list then it has, by default, the
+value of the constant.
+
+In the case of \texttt{e}, if you wish to use it as a variable, the
+constant is always available as \verb`\exp(1)`. There is no similar
+alternative available for \verb`\pi`, \verb`\gamma` or \verb`\phi`
+although you can always do something like \verb`[\pi=<new value>,\pi_0=\pi]`
+in the vv-list, so that \verb`\pi_0` now has the constant's value.
+
+\subsection{Display of the vv-list}
+
+By default, the vv-list is displayed with (in fact following) the
+numerical result. That and the format of the display can both be changed.
+
+\subsubsection{Star option: suppressing display of the vv-list}
+
+\label{subsec:evalVvSuppresList}If display of the vv-list is not
+wanted at all, only the numerical result, it suffices to attach an
+asterisk (star) to the \texttt{\textbackslash eval} command:
+\begin{centred}
+\verb`$ V_C=\eval*{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
+$ V_C=\eval*{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $,
+\end{centred}
+or simply the naked result:
+\begin{centred}
+\verb`\eval*{$ hA_C $}[A_C=\pi r^2,h=4/3,r=1]` $\Longrightarrow$
+\eval*{$ hA_C $}[A_C=\pi r^2,h=4/3,r=1].
+\end{centred}
+In the latter case, note that a negative result will display with
+a hyphen for the minus sign unless you, the user, explicitly write
+math delimiters around the \verb`\eval*` command:
+\begin{centred}
+\verb`\eval*{$ y $}[y=ax+b,x=2,a=-2,b=2]` $\Longrightarrow$ \eval*{$ y $}[y=ax+b,x=2,a=-2,b=2]
+\end{centred}
+The \verb`$` signs that \verb`\eval*` wraps around are ignored.
+The star option delivers a number, pure and simple, with no accompaniments.
+
+\subsubsection{Suppressing display of items}
+
+\label{subsec:evalVvSuppressVars}You may wish to retain some variables
+in the vv-list display, but not all. For those variables you wish
+omitted from the display, wrap each variable (but not the equals sign
+or value) in braces. When calculating the volume of a cylinder in
+the previous examples, the base area $A_{C}$ has a different status
+from the `fundamental' variables $r$ and $h$. It is an intermediate
+value, one that we pass through on the way to the final result. To
+suppress it from display enclose the variable in braces:
+\begin{centred}
+\verb`$ V_C=\eval{ hA_C }[{A_C}=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
+$ V_C=\eval{ hA_C }[{A_C}=\pi r^2,h=4/3,r=1] $.
+\end{centred}
+As you can see, $A_{C}$ no longer appears in the displayed vv-list.
+Of course the name and its value are still recorded `behind the scenes'
+and can still be used in calculations.
+
+\subsubsection{Changing the display format}
+
+\label{subsec:evalChanging-display-format}In two examples above,
+we have calculated the area of a triangle using Brahmagupta's formula.
+Display of the result is crowded. Two remedies have just been suggested,
+but a third one and preferable in this case would be to force display
+of the vv-list and result to a new line. This can be done through
+the settings option to the \verb`\eval` command, discussed in §\ref{subsec:settings New-line-display}.
+However, if \verb`\eval` is wrapped around an \emph{appropriate}
+environment (like \verb`multline`, but not \verb`equation`) it can
+also be done simply by including \texttt{\textbackslash\textbackslash}
+at the end of the formula.
+
+In the following example I use Brahmagupta's formula for calculating
+the area of a cyclic quadrilateral (of which his formula for a triangle
+is a special case). The cyclic quadrilateral in the example is formed
+by a 45-45-90 triangle of hypotenuse 2 joined along the hypotenuse
+to a 30-60-90 triangle. The sides are therefore $\surd2,\surd2,\surd3,1$.
+Adding the areas of the two triangles, the area of the quadrilateral
+is $A=1+\tfrac{1}{2}\surd3$, or in decimal form, \verb`$\eval{1+\tfrac12\surd3}$`
+$\Longrightarrow$ $\eval{1+\tfrac12\surd3}$. Let's check with Brahmagupta's
+formula:
+\begin{verbatim}
+ \eval{
+ \begin{multline*}
+ \sqrt{(s-a)(s-b)(s-c)(s-d)}\\
+ \end{multline*}
+ }[s=\tfrac12(a+b+c+d),
+ a=\surd2,b=\surd2,c=\surd3,d=1]
+\end{verbatim}
+$\Longrightarrow$ \eval{
+ \begin{multline*}
+ \sqrt{(s-a)(s-b)(s-c)(s-d)}\\
+ \end{multline*}
+ }[s=\tfrac12(a+b+c+d),
+ a=\surd2,b=\surd2,c=\surd3,d=1]
+
+\section{Formatting the numerical result}
+
+\label{subsec:evalRoundingEtc} A result of a calculation is displayed,
+by default, to $6$ decimal places. All our results so far have been
+rounded to this figure, although not all digits are displayed, for
+instance if the sixth one is $0$, or the result is an integer. Like
+other elements of the display, both rounding value and the (dis)appearance
+of trailing zeros can be customized, in this case by means of an optional
+argument following the vv-list (or the formula if there is no vv-list).
+This optional argument may contain up to four juxtaposed items from
+seven possibilities:
+\begin{itemize}
+\item a question mark ?, which gives boolean output, or
+\item an integer, the \emph{rounding value}, positive, negative or zero,
+specifying how many decimal places to display the result to, or
+\item an asterisk {*}, which pads the result with zeros should it not have
+as many decimal places as the rounding value specifies, or
+\item the character \texttt{x} (lower case!) which presents the result in
+`proper' scientific notation (a form like $1.234\times10^{5}$ for
+123450), or
+\item the character \texttt{t} (lower case!) which presents the result in
+a bastardized form of scientific notation useful in tables (a form
+like $(5)1.234$ for 123450), or
+\item a character other than \texttt{?}, \texttt{{*}}, \texttt{x}, \texttt{t}
+or a digit, usually (but not necessarily) one of the letters\texttt{
+e d} \texttt{E D}, which presents the result in scientific notation
+using that character as the exponent mark (a form like $1.234\text{e}5$
+for $123450$), or
+\item a prime
+\begin{itemize}
+\item attached to the character specifying scientific notation, which extends
+that notation to numbers in the interval \texttt{{[}1,10)}, or
+\item attached to a question mark, which changes the format of boolean output.
+\end{itemize}
+\end{itemize}
+If you use \texttt{?} in the same specification as some other text
+character, the \texttt{?} prevails; if you use \texttt{x} in the same
+specification as some other text character except for \texttt{?},
+the \texttt{x} prevails; if you use \texttt{t} in the same specification
+as some other text character except for \texttt{?} or \texttt{x},
+the \texttt{t} prevails.
+
+\subsection{Rounding value }
+
+\label{subsec:evalRounding-value}The rounding value specifies the
+number of decimal places displayed:
+\begin{centred}
+\verb`$ \eval{ 1/3 }[4] $` $\Longrightarrow$ $ \eval{ 1/3 }[4] $
+\end{centred}
+The default rounding value is $6$:
+\begin{centred}
+\verb`$ \eval{ 35/3 } $` $\Longrightarrow$ $ \eval{ 35/3 } $
+\end{centred}
+Following the default behaviour in \texttt{l3fp}, `ties' are rounded
+to the nearest \emph{even} digit. Thus a number ending $55$ has a
+`choice' of rounding to $5$ or $6$ and rounds up to the even digit
+$6$, and a number ending $65$ with a `choice' of rounding to $6$
+or $7$ rounds down to the even digit $6$:
+\begin{centred}
+\verb`$ \eval{ 0.1234555 } $` $\Longrightarrow\eval{0.1234555}$
+
+\verb`$ \eval{ 0.1234565 } $` $\Longrightarrow\eval{0.1234565}$
+\end{centred}
+The calculational engine which \texttt{numerica} uses, \texttt{l3fp},
+works to 16 significant figures and never displays more than that
+number (and often less).
+\begin{itemize}
+\item In the first of the following although I have specified a rounding
+value of $19$ only $16$ decimal places are displayed, with the final
+digit rounded up to $7$;
+\item in the second I have added $10$ zeros after the decimal point, meaning
+that all $19$ decimal places specified by the rounding value can
+be displayed since the 10 initial zeros do not contribute to the significant
+figures;
+\item in the third I have changed the figure \emph{before} the decimal point
+to $1$ so that the $10$ added zeros are now included among the significant
+figures;
+\item and in the fourth, I have added $9$ digits before the decimal point:
+\end{itemize}
+\begin{centred}
+\verb`$ \eval{ 0.1234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 0.1234567890123456789 }[19]$
+
+\verb`$ \eval{ 0.00000000001234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 0.00000000001234567890123456789 }[19]$
+
+\verb`$ \eval{ 1.00000000001234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 1.00000000001234567890123456789 }[19]$
+
+\verb`$ \eval{ 987654321.1234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 987654321.1234567890123456789 }[19]$
+\end{centred}
+In all cases, no more than $16$ \emph{significant} figures are displayed,
+although the number of decimal places displayed may exceed $16$ as
+in the second example.
+
+It is possible to use \emph{negative} rounding values. Such a value
+zeroes the specified number of digits \emph{before} the decimal point.
+\begin{centred}
+\verb`$ \eval{ 987654321.123456789 }[-4] $` $\Longrightarrow$ $\eval{ 987654321.123456789 }[-4]$
+\end{centred}
+A rounding value of $0$ rounds to the nearest integer:
+\begin{centred}
+\verb`$ \eval{ 987654321.123456789 }[0] $` $\Longrightarrow$ $\eval{ 987654321.123456789 }[0]$
+\end{centred}
+If you wish to change the \emph{default} rounding value from $6$
+to some other value, this can be done by creating or editing a file
+\texttt{numerica.cfg} in a text editor; see §\ref{sec:settingsDefaults}.
+
+\subsection{Padding with zeros }
+
+\label{subsec:evalPadding-with-zeros}A result may contain fewer decimal
+places than the rounding value specifies, the trailing zeros being
+suppressed by default (this is how \texttt{l3fp} does it). Sometimes,
+perhaps for reasons of presentation like aligning columns of figures,
+it may be desirable to pad results with zeros. This is achieved by
+inserting an asterisk, {*}, into the final optional argument of the
+\verb`\eval` command:
+\begin{centred}
+\verb`$ \eval{ 1/4 }[4] $` $\Longrightarrow$ $ \eval{ 1/4 }[4] $,
+
+\verb`$ \eval{ 1/4 }[4*] $` $\Longrightarrow$ $ \eval{ 1/4 }[4*] $.
+\end{centred}
+
+\subsection{Scientific notation }
+
+\label{subsec:evalScientificNotation} The \texttt{l3fp} package can
+output numbers in scientific notation. For example, $1234$ is rendered
+as $\eval{1234}[e]$, denoting $1.234\times10^{3}$ , and $0.008$
+as $\eval{0.008}[e]$, denoting $8\times10^{-3}$. The `e' here,
+the \emph{exponent mark}, separates the \emph{significand} ($1.234$)
+from the \emph{exponent} ($3$). To switch on output in scientific
+notation in \texttt{numerica} enter \texttt{e} in the trailing optional
+argument:
+\begin{centred}
+\verb`$ \eval{ 123456789 }[e] $` $\Longrightarrow$ $ \eval{ 123456789 }[e] $.
+\end{centred}
+The default rounding value $6$ is in play here. In \texttt{numerica},
+when scientific notation is selected rounding takes a different meaning:
+it is the \emph{significand }which is rounded (not the number as a
+whole). One digit precedes the decimal point, at most $15$ follow
+it.
+
+Negative rounding values are pointless for scientific notation. A
+zero might on occasion be relevant:
+\begin{centred}
+\verb`$ \eval{ 987654321 }[0e] $` $\Longrightarrow$ $ \eval{ 987654321 }[0e] $.
+\end{centred}
+Sometimes letters other than `e' are used to indicate scientific
+notation, like `E' or `d' or `D'. With a few exceptions, \texttt{numerica}
+allows any letter or text character to be used as the exponent marker:
+\begin{centred}
+\verb`\eval{$ 1/23456789 $}[4d]`\texttt{ $\Longrightarrow$} \eval{$ 1/23456789 $}[4d].
+\end{centred}
+But when \texttt{x} is inserted in the trailing optional argument,
+the output is in the form $d_{0}.d_{1}\ldots d_{m}\times10^{n}$ (except
+when $n=0$), where each $d_{i}$ denotes a digit.
+\begin{centred}
+\verb`\eval{$ 1/23456789 $}[4x]`\texttt{ $\Longrightarrow$ }\eval{$ 1/23456789 $}[4x] .
+\end{centred}
+The requirements of tables leads to another form of scientific notation.
+Placing \texttt{t} in the trailing argument turns on this table-ready
+form of notation:
+\begin{centred}
+\verb`\eval{$ 1/23456789 $}[4t]`\texttt{ $\Longrightarrow$ }\eval{$ 1/23456789 $}[4t].
+\end{centred}
+This is discussed more fully in the associated document \texttt{numerica-tables.pdf}.
+
+In the next example three options are used in the trailing argument.
+The order in which the items are entered does not matter:
+\begin{centred}
+\verb`\eval{$ 1/125 $}[*e4]` $\Longrightarrow$ \eval{$ 1/125 $}[*e4].
+\end{centred}
+Finally, to illustrate that `any' text character\footnote{Be sensible! An equals sign for instance might confuse \texttt{numerica}
+into thinking the number-format option is the vv-list, and will certainly
+confuse the reader.} save for \texttt{x} or \texttt{t} can be used to distinguish the
+exponent, I use an @ character:
+\begin{centred}
+\verb`\eval{$ 1/123 $}[@4]` $\Longrightarrow$ \eval{$ 1/123 $}[@4].
+\end{centred}
+
+\subsubsection{Numbers in \texttt{{[}1,10)}}
+
+Usually when scientific notation is being used, numbers with magnitude
+in the interval $[1,10)$ are rendered in their normal decimal form,
+$3.14159$ and the like. Occasionally it may be desired to present
+numbers in this range in scientific notation (this can be the case
+in tables where the alignment of a column of figures might be affected).
+\texttt{numerica} offers a means of extending scientific notation
+to numbers in this range by adding a prime to the letter chosen as
+the exponent mark in the trailing optional argument.
+\begin{centred}
+\verb`\eval{$ \pi $}[4t']` $\Longrightarrow$ \eval{$ \pi $}[4t']
+\end{centred}
+
+\subsubsection{\textbackslash eval{*} and scientific notation}
+
+Scientific notation can be used for the numerical result output by
+\verb`\eval*`:
+\begin{centred}
+\verb`\eval*{$ \pi $}[e']` $\Longrightarrow$ \eval*{$ \pi $}[e']
+\end{centred}
+There is one catch: if you substitute \texttt{x} for \texttt{e} here,
+\LaTeX{} will complain about a missing \verb`$`. An \texttt{x} in
+the number-format option produces a \verb`\times` in the output which
+requires a math environment. It is up to you, as the user, to provide
+the necessary delimiters outside the \verb`\eval*` command. (This
+applies even when \verb`\eval*` wraps around math delimiters.)
+
+(Because of the way \texttt{numerica} parses the number-format option,
+entering a prime with neither exponent character nor question mark
+specified will result in scientific output using \texttt{e} as the
+exponent mark. The last example could have been written \verb`\eval*{$ \pi $}[']`.)
+
+\subsection{Boolean output}
+
+\label{subsec:evalBoolean-output}\texttt{l3fp} can evaluate comparisons,
+outputting $0$ if the comparison is false, $1$ if it is true. By
+entering a question mark, \texttt{?}, in the trailing optional argument,
+you can force \texttt{numerica} to do the same depending as the result
+of a calculation is zero or not. The expression being evaluated does
+not need to be a comparison, \verb`$ \eval{\pi}[?] $` $\Longrightarrow$ $ \eval{\pi}[?]$,
+but comparisons are what this is designed for.
+
+Possible comparison relations are \verb`=`, \verb`<`, \verb`>`,
+\verb`\ne`, \verb`\neq`, \verb`\ge`, \verb`\geq`, \verb`\le`,
+\verb`\leq`. Although programming languages use combinations like
+\verb`<=` or \verb`>=`, \texttt{numerica} does \emph{not} accept
+these (they are not part of standard \emph{mathematical} usage) and
+will generate an error. An example where the relation is equality
+exhibits a numerological curiosity:\footnote{The \texttt{{[}p=.{]}} of this and the next example ensures a full
+stop appears in the correct place; see §\ref{subsec:settingsPunctuation}.}
+\begin{centred}
+\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[5?]` $\Longrightarrow$
+\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[5?]
+\end{centred}
+Notice the $5$ alongside the question mark in the trailing argument.
+That is critical. Change the $5$ to a $6$ (or omit it since the
+default rounding value is $6$) and the outcome is different:
+\begin{centred}
+\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?]` $\Longrightarrow$
+\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?]
+\end{centred}
+Now the relation is false. Evaluating the fraction to more than $6$
+places, say to $9$, we can see what is going on:
+\begin{centred}
+\verb`\eval{$ 1/0.0123456789 $}[9]` $\Longrightarrow$ \eval{$ 1/0.0123456789 $}[9].
+\end{centred}
+
+\subsubsection{Outputting \texttt{T} or \texttt{F}}
+
+To my eye, outputting $0$ or $1$ in response to a `question' like
+$1/0.0123456789=81$ is confusing. It is easy to change the boolean
+output from $0,1$ to a more appropriate $F,T$, or \texttt{$\texttt{F,\texttt{T}}$}
+by adding a prime or two primes respectively to the question mark
+in the number-format option.
+\begin{centred}
+\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?'']` $\Longrightarrow$
+\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?'']
+\end{centred}
+The default boolean output format is chosen to be $0,1$ in case an
+\verb`\eval*` command is used within another \verb`\eval` command
+(`nesting'– see Chapter~\ref{chap:Nesting}~). The inner command
+needs to output a \emph{numerical} answer.
+
+\subsubsection{Rounding error tolerance}
+
+\label{subsec:evalToleranceRounding}If at least one of the terms
+in a comparison is the result of a calculation, then it's value is
+likely to contain rounding errors. What level of rounding error can
+we tolerate before such errors interfere with the comparison being
+made? \texttt{l3fp} tolerates none. It decides the truth or falsity
+of a comparison to all $16$ significant figures: 1.000 0000 0000
+0000 and 1.000 0000 0000 0001 are \emph{not }equal in \texttt{l3fp}.
+But for most purposes this will be far too severe a criterion.
+
+Suppose our comparison relation is $\varrho$, denoting one of =,
+<, >, \verb`\le`, etc. If $X\rel Y$ then $X-Y\rel Y-Y$, i.e. $X-Y\rel0$.
+This is what \texttt{numerica} does. It takes the right-hand side
+of the relation from the left-hand side and then compares the \emph{rounded}
+difference under $\varrho$ to $0$. The rounding value used is the
+number specified with the question mark in the trailing argument of
+the \verb`\eval` command or, if no number is present, the default
+rounding value (`out of the box' this is $6$). Thus, in a recent
+example, $1/0.0123456789-81$ when rounded to $5$ decimal places
+is \@.$0.00000$, indistinguishable from zero at this rounding value;
+hence the equality $1/0.0123456789=81$ is true. But when rounded
+to $6$ places it is $0.000001$ which \emph{is} distinguishable from
+zero and so the equality is false. Truth or falsity depends on the
+rounding value.
+
+When dealing with numbers generated purely mathematically, rounding
+values of $5$ or $6$ are likely to be too small. More useful would
+be rounding values closer to \texttt{l3fp}'s $16$ – perhaps $14$?
+– depending on how severe the calculations are that generate the numbers.
+However if the numbers we are dealing with come from outside mathematics,
+from practical experiments perhaps, then even a rounding value of
+$5$ or $6$ may be too large.
+
+\subsubsection{Rationale}
+
+Mathematically, the claim that $X=Y$ at a rounding value $n$ is
+the claim that
+\[
+\abs{X-Y}\le5\times10^{-(n+1)}.
+\]
+since this rounds \emph{down} to zero at $n$ places of decimals.
+This gives a more accurate test of equality than doing things in the
+opposite order – rounding each number first and then taking the difference.
+One might, for instance, have numbers like $X=0.12345$, $Y=0.12335$.
+Rounding to $n=4$ places, both round to $0.1234$ and yet the difference
+between them is $0.0001$ – they are distinguishable numbers to $4$
+places of decimals. This is why \texttt{numerica} forms the difference
+\emph{before }doing the rounding.
+
+\subsubsection{And, Or, Not}
+
+For logical And \LaTeX{} provides the symbols \verb`\wedge` and \verb`\land`,
+both displaying as $\land$, but \texttt{numerica} adds thin spaces
+( \verb`\,` ) around the symbol for \verb`\land` (copying the package
+\texttt{gn-logic14.sty}). For logical Or \LaTeX{} provides the symbols
+\verb`\vee` and \verb`\lor`, both displaying as $\lor$, but again
+\texttt{numerica} adds thin spaces around the symbol for \verb`\lor`.
+\begin{centred}
+\verb`\eval{$ 1<2 \wedge 2<3 $}[?'']` $\Longrightarrow$ \eval{$ 1<2 \wedge 2<3 $}[?''],
+
+\verb`\eval{$ 1<2 \land 2<3 $}[?'']` $\Longrightarrow$ \eval{$ 1<2 \land 2<3 $}[?''].
+\end{centred}
+To my eye the second of these with its smidgen more space around the
+wedge symbol displays the meaning of the overall expression better
+than the first. Both And and Or have equal precedence; in cases of
+ambiguity the user needs to parenthesize as necessary to clarify what
+is intended.
+
+\LaTeX{} provides two commands for logical Not, \verb`\neg` and \verb`\lnot`,
+both displaying as $\lnot$ . Not binds tightly to its argument:
+\begin{centred}
+\verb`\eval{$ \lnot A \land B $}[A=0,B=0]` $\Longrightarrow$ \eval{$ \lnot A \land B $}[A=0,B=0].
+\end{centred}
+Here \verb`\lnot` acts only on the $A$; if it had acted on $A\land B$
+as a whole the result would have been $1$.
+
+For a little flourish, I evaluate a more complicated logical statement:\footnote{Quoting from an article in \emph{Quanta Magazine} (August 2020) by
+Kevin Hartnett: `Let’s say you and two friends are planning a party.
+The three of you are trying to put together the guest list, but you
+have somewhat competing interests. Maybe you want to either invite
+Avery or exclude Kemba. One of your co-planners wants to invite Kemba
+or Brad or both of them. Your other co-planner, with an ax to grind,
+wants to leave off Avery or Brad or both of them. Given these constraints,
+you could ask: Is there a guest list that satisfies all three party
+planners?' I have written $C$ for Kemba, $A$ and $B$ for Avery
+and Brad.}
+\begin{verbatim}
+ \eval{$(A\lor\lnot C)\land(C\lor B)\land
+ (\lnot A\lor\lnot B)$}[A=1,B=0,C=1][?'']
+\end{verbatim}
+$\Longrightarrow$ \eval{$(A\lor\lnot C)\land(C\lor B)\land
+ (\lnot A\lor\lnot B)$}[A=1,B=0,C=1][?'']
+
+\subsubsection{Chains of comparisons}
+
+\texttt{numerica} can handle chains of comparisons like $1<2<1+2<5-1$.
+`Behind the scenes' it inserts logical And-s into the chain, $1<2\land2<1+2\land1+2<5-1$,
+and evaluates the modified expression:
+\begin{centred}
+\verb`\eval{$ 1<2<1+2<5-1 $}[?'']` $\Longrightarrow$ \eval{$ 1<2<1+2<5-1 $}[?''].
+\end{centred}
+
+\subsubsection{\texttt{amssymb} comparison symbols}
+
+\label{subsec:evalAmssymb-comparisons}\texttt{numerica} accepts some
+alternative symbols for the basic comparison relations from the \texttt{amssymb}
+package provided that package is loaded, i.e. the preamble of your
+document includes the statement
+\begin{lyxcode}
+\textbackslash usepackage\{amssymb\}
+\end{lyxcode}
+The variants from this package are: \verb`\leqq` ( $\leqq$ ), \verb`\leqslant`
+( $\leqslant$ ), \verb`\geqq` (~$\geqq$~), and \verb`\geqslant`
+( $\geqslant$ ).\footnote{No, that is not \texttt{eggplant}.} There
+are also negations: \verb`\nless` ( $\nless$ ), \verb`\nleq` (~$\nleq$~),
+\verb`\nleqq` ( $\nleqq$ ), \verb`\nleqslant` ( $\nleqslant$ ),
+\verb`\ngtr` ( $\ngtr$ ), \verb`\ngeq` ( $\ngeq$ ), \verb`\ngeqq`
+( $\ngeqq$ ), \verb`\ngeqslant` ( $\ngeqslant$ ).
+
+\section{Calculational details}
+
+\subsection{Arithmetic}
+
+\label{subsec:evalArithmetic}Addition, subtraction, multiplication,
+division, square roots, \emph{$n$}-th roots, and exponentiating (raising
+to a power) are all available.
+
+Multiplication can be rendered explicitly with an asterisk,
+\begin{centred}
+\verb`\eval{$ 9*9 $}` $\Longrightarrow$ \eval{$ 9*9 $},
+\end{centred}
+but that's ugly. More elegant is to use \verb`\times`:
+\begin{centred}
+\verb`\eval{$ 9\times9 $}` $\Longrightarrow$ \eval{$ 9\times9 $}.
+\end{centred}
+\verb`\cdot` is also available and in many cases juxtaposition alone
+suffices:
+\begin{centred}
+\verb`\eval{$ \surd2\surd2 $}` $\Longrightarrow$ \eval{$ \surd2\surd2 $},
+
+\verb`\eval{$ ab $}[a=123,b=1/123]` $\Longrightarrow$ \eval{$ ab $}[a=123,b=1/123].
+\end{centred}
+Division can be rendered in multiple ways too:
+\begin{centred}
+\verb`\eval{$ 42/6 $}` $\Longrightarrow$ \eval{$ 42/6 $},
+
+\verb`\eval{$ 42\div6 $}` $\Longrightarrow$ \eval{$ 42\div6 $},
+\end{centred}
+or by using \verb`\frac` or \verb`\tfrac` or \verb`\dfrac` as in
+\begin{centred}
+\verb`\eval{$ \frac{42}6 $}` $\Longrightarrow$ \eval{$ \frac{42}6 $}.
+\end{centred}
+But note that since juxtaposition means multiplication, it is also
+true that $42\tfrac{1}{6}$\texttt{ }evaluates to $7$ inside an \verb`\eval`
+command rather than denoting `forty two and a sixth'. Hence if you
+want to use `two and a half' and similar values in \texttt{numerica},
+they need to be entered as improper fractions like $\tfrac{5}{2}$
+or in decimal form, $2.5$ (as one does automatically in mathematical
+expressions anyway because of the ambiguity in a form like $2\tfrac{1}{2}$).
+
+\subsubsection{Square roots and $n$-th roots}
+
+\label{subsec:evalSquareRootsEtc}Let us check that 3, 4, 5 and 5,
+12, 13 really are Pythagorean triples (I use \verb`\sqrt` in the
+first, \verb`\surd` in the second):
+\begin{centred}
+\verb`\eval{$ \sqrt{3^2+4^2} $}` $\Longrightarrow$ \eval{$\sqrt{3^{2}+4^{2}}$},
+
+\verb`\eval{$ \surd(5^2+12^2) $}` $\Longrightarrow$ \eval{$ \surd(5^2+12^2) $}.
+\end{centred}
+The \verb`\sqrt` command has an optional argument which can be used
+for extracting $n$-th roots of a number. This notation is generally
+used when $n$ is a small positive integer like $3$ or $4$. This
+practice is followed in \texttt{numerica}: $n$ must be a (not necessarily
+small) \emph{positive integer}:
+\begin{centred}
+\verb`\eval{$ \sqrt[4]{81} $}` $\Longrightarrow$ \eval{$ \sqrt[4]{81} $},
+
+\verb`\eval{$ \sqrt[n]{125} $}[n=\floor{\pi}]` $\Longrightarrow$
+\eval{$ \sqrt[n]{125} $}[n=\floor{\pi}].
+\end{centred}
+If $n$ should not be a positive integer, an error message is generated;
+see §\ref{sec:evalErrors}.
+
+For display-style expressions, the \verb`\sqrt` command grows to
+accommodate the extra vertical height; the surd doesn't. Here is an
+example which anticipates a number of matters not discussed yet. It
+shows \verb`\eval` wrapping around a square root containing various
+formatting commands (negative spaces, \verb`\left` and \verb`\right`
+nested within \verb`\bigg` commands), all digested without complaint
+(see §\ref{subsec:evalFormatting-commands}; and see §\ref{subsec:settingsPunctuation}
+for the \verb`[p=.]`): \medskip{}
+\begin{verbatim}
+ \eval[p=.]
+ {\[
+ \sqrt[3]
+ {\!\biggl(\!\left.\frac AD\right/\!\frac BC\biggr)}
+ \]}[A=729,B=81,C=9,D=3]
+\end{verbatim}
+$\Longrightarrow$\eval[p=.]
+ {\[
+ \sqrt[3]
+ {\!\biggl(\!\left.\frac AD\right/\!\frac BC\biggr)}
+ \]}[A=729,B=81,C=9,D=3]
+
+As implemented in \texttt{numerica}, $n$-th roots found using \verb`\sqrt[n]`
+are \verb`n=<integer>`\emph{ }roots. This raises an interesting question:
+if the `$n$' of an $n$-th root is the result of a calculation,
+what happens with rounding errors? The calculation may not produce
+an \emph{exact} integer. (This problem also arises with factorials;
+see §\ref{subsec:evalFactorialsBinom}.) The solution employed in
+\texttt{numerica} is to make what is considered an integer depend
+on a rounding value. Most calculations will produce rounding errors
+in distant decimal places. For `int-ifying' calculations, \texttt{numerica}
+uses a rounding value of $14$: a calculation produces an integer
+if, when rounded to $14$ figures, the result is an integer. Since
+\texttt{l3fp} works to $16$ significant figures, a rounding value
+of $14$ allows ample `elbowroom' for rounding errors to be accommodated
+when judging what is an integer and what is not. As a practical matter
+problems should not arise.
+
+\subsubsection{\emph{n}-th roots of negative numbers}
+
+Odd (in the sense of `not even') integral roots of \emph{negative}
+numbers are available with \verb`\sqrt`,
+\begin{centred}
+\verb`\eval{$ \sqrt[3]{-125} $}` $\Longrightarrow$ \eval{$ \sqrt[3]{-125} $},
+
+\verb`\eval{$ \sqrt[3]{-1.25} $}` $\Longrightarrow$ \eval{$ \sqrt[3]{-0.125} $}.
+\end{centred}
+
+\subsubsection{Inverse integer powers }
+
+Of course to find an $n$-th root we can also raise to the inverse
+power,
+\begin{centred}
+\verb`\eval{$ 81^{1/4} $}` $\Longrightarrow$ \eval{$ 81^{1/4} $}.
+\end{centred}
+However, raising a \emph{negative} number to an inverse power generates
+an error even when, mathematically, it should not. This matter is
+discussed below in §\ref{subsec:errorsInverse-powers}.
+
+\subsection{Precedence, parentheses}
+
+The usual precedence rules apply: multiplication and division bind
+equally strongly and more strongly than addition and subtraction which
+bind equally stongly. Exponentiating binds most strongly. Evaluation
+occurs from the left.
+\begin{centred}
+\verb`\eval{$ 4+5\times6+3 $}` $\Longrightarrow$ \eval{$ 4+5\times6+3 $},
+
+\verb`\eval{$ 6\times10^3/2\times10^2 $}` $\Longrightarrow$ \eval{$ 6\times10^3/2\times10^2 $},
+\end{centred}
+which may not be what was intended. Parentheses (or brackets or braces)
+retrieve the situation:
+\begin{centred}
+\verb`\eval{$ (4+5)(6+3) $}` $\Longrightarrow$ \eval{$ (4+5)(6+3) $},
+
+\verb`\eval{$ (6\times10^3)/(2\times10^2) $}` $\Longrightarrow$
+\eval{$ (6\times10^3)/(2\times10^2) $}.
+\end{centred}
+Because exponentiating binds most strongly, negative values must be
+parenthesized when raised to a power. If not,
+\begin{centred}
+\verb`\eval{$ -4^2 $}` $\Longrightarrow$ \eval{$ -4^2 $},
+\end{centred}
+which is clearly not $(-4)^{2}$. But
+\begin{centred}
+\verb`\eval{$ (-4)^2 $}` $\Longrightarrow$ \eval{$ (-4)^2 $}.
+\end{centred}
+
+\subsubsection{Command-form brackets}
+
+\label{subsec:evalCommandBrackets}Note that brackets of all three
+kinds are available also in command form: \verb`\lparen \rparen`
+(from \verb`mathtools`) for \verb`( )`, \verb`\lbrack \rbrack`
+for \verb`[ ]`, and \verb`\lbrace \rbrace` for \verb`\{ \}`.
+
+\subsection{Modifiers\texttt{ (\textbackslash left \textbackslash right}, etc.)}
+
+The \verb`\left` and \texttt{\textbackslash right} modifiers and
+also the series of \verb`\big...` modifiers\texttt{ }(\texttt{\textbackslash}\verb`bigl \bigr`,
+\verb`\Bigl \Bigr`, \verb`\biggl \biggr`, \verb`\Biggl \Biggr`)
+are available for use with all brackets (parentheses, square brackets,
+braces):
+\begin{verbatim}
+ \eval[p=.]{\[ \exp\left(
+ \dfrac{\ln2}{4}+\dfrac{\ln8}{4}
+ \right) \]}
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]{\[ \exp\left( \dfrac{\ln2}{4}+\dfrac{\ln8}{4} \right) \]}
+
+\texttt{numerica} also accepts their use with \texttt{.} (dot) and
+with \texttt{/} (as noted earlier, the \verb`[p]` and \verb`[p=.]`
+are explained at §\ref{subsec:settingsPunctuation}):
+\begin{centred}
+\verb`\eval[p]{\[ \left.\dfrac{3+4}{2+1}\right/\!\dfrac{1+2}{4+5} \]}`
+$\Longrightarrow$ \eval[p=.]{\[ \left. \dfrac{3+4}{2+1} \right/\!\dfrac{1+2}{4+5} \]}
+\end{centred}
+They can be nested.
+
+\subsection{Other formatting commands}
+
+\label{subsec:evalFormatting-commands}There are many formatting commands
+which change the layout of a formula on the page but do not alter
+its content. These include various spacing commands like \verb`\!`,
+\verb`\quad`, etc., phantoms (\verb`\phantom` etc.), \verb`\mathstrut`
+from \TeX{} and its \verb`mathtools` cousin \verb`\xmathstrut`.
+
+Consider the same package's \verb`\splitfrac` and \verb`\splitdfrac`.
+The \verb`mathtools` documentation gives an example to illustrate
+the use of these last two. I've mangled it to produce a ridiculous
+illustration of their use, and of the modifiers \verb`\left \right`,
+and of the command-form alternatives to parentheses \verb`\lparen \rparen`;
+also the use of \verb`\dfrac`. A little mental arithmetic will convince
+that we are evaluating the square root of $(9\times7)^{2}$ which
+indeed is what we get:\footnote{For the \texttt{{[}p=.,vvd={]}} see §\ref{subsec:settingsPunctuation}
+and §\ref{subsec:settingsVvDisplayChangeLocal}. The first puts the
+concluding full stop in the right place; the second suppresses the
+vv-list.} \medskip{}
+\begin{verbatim}
+ \eval[p=.,vvd=]{\[
+ \sqrt{\left\lparen
+ \frac{ \splitfrac{xy + xy + xy + xy + xy}
+ {+ xy + xy + xy + xy}
+ }
+ { \dfrac z7}
+ \right\rparen \left\lparen
+ \frac{ \splitdfrac{xy + xy + xy + xy + xy}
+ {+ xy + xy + xy + xy}
+ }
+ {\dfrac z7}\right\rparen}
+ \]}[x=2,y=5,z=10]
+\end{verbatim}
+$\Longrightarrow$\eval[p=.,vvd=]{\[
+ \sqrt{\left\lparen
+ \frac{ \splitfrac{xy + xy + xy + xy + xy}
+ {+ xy + xy + xy + xy}
+ }
+ { \dfrac z7}
+ \right\rparen \left\lparen
+ \frac{ \splitdfrac{xy + xy + xy + xy + xy}
+ {+ xy + xy + xy + xy}
+ }
+ {\dfrac z7}\right\rparen}
+ \]}[x=2,y=5,z=10]\texttt{numerica} essentially ignores formatting commands (the ones
+it knows of). They do not alter the mathematical content of a formula,
+only how it looks. But there will undoubtedly be formatting commands
+it does not recognize which will probably trigger an `Unknown token'
+message. Please contact the author in that case.\footnote{ajparsloe at gmail.com}
+
+\subsection{Trigonometric \& hyperbolic functions}
+
+\LaTeX{} provides all six trignometric functions, \verb`\sin`, \verb`\cos`,
+\verb`\tan`, \verb`\csc`, \verb`\sec`, \verb`\cot` and the three
+principal inverses \verb`\arcsin`, \verb`\arccos`, \verb`\arctan`.
+It also provides four of the six hyperbolic functions: \verb`\sinh`,
+\verb`\cosh`, \verb`\tanh`, \verb`\coth`, and \emph{no} inverses.
+\texttt{numerica} provides the missing hyperbolic functions, \verb`\csch`
+and \verb`\sech`, and all missing inverses, the three trigonometric
+and all six hyperbolic: \verb`\arccsc`, \verb`\arcsec`, \verb`\arccot`,
+and \verb`\asinh`, \verb`\acosh`, \verb`\atanh`, \verb`\acsch`,
+\verb`\asech`, \verb`\acoth`. (\emph{HMF} writes $\text{arcsinh}$,
+$\text{arccosh}$, etc. and ISO recommends $\text{arsinh}$, $\text{arcosh}$,
+etc. The first seems ill-advised, the second not widely adopted. At
+present neither is catered for in \texttt{numerica}.)\emph{ }
+\begin{centred}
+\verb`\eval{$ \arctan1/1\deg $}` $\Longrightarrow$ \eval{$ \arctan 1/1\deg $} ,
+
+\verb`\eval{$ \atanh\tanh3 $}` $\Longrightarrow$ \eval{$ \atanh\tanh3 $} .
+\end{centred}
+Inverses can also be constructed using the `$-1$' superscript notation.
+Thus
+\begin{centred}
+\verb`\eval{$ \sin^{-1}(1/\surd2)/1\deg $}` $\Longrightarrow$ \eval{$ \sin^{-1}(1/\surd2)/1\deg $} ,
+
+\verb`\eval{$ \tanh\tanh^{-1}0.5 $}` $\Longrightarrow$ \eval{$ \tanh\tanh^{-1}0.5 $} .
+\end{centred}
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\subsubsection*{Hyperbolic functions}
+
+Please note that \texttt{l3fp} does not (as yet) provide \emph{any}
+hyperbolic functions natively. The values \texttt{numerica} provides
+for these functions are \emph{calculated} values using familiar formulas
+involving exponentials (for the direct functions) and natural logarithms
+and square roots for the inverses. Rounding errors mean the values
+calculated may not have $16$-figure accuracy. The worst `offenders'
+are likely to be the least used, \verb`\acsch` and \verb`\asech`.
+For instance,
+\[
+\acsch x=\ln\left[\frac{1}{x}+\left(\frac{1}{x^{2}}+1\right)^{1/2}\right],
+\]
+
+\begin{centred}
+\verb`\eval{$ \csch \acsch 7 $}[16]` $\Longrightarrow$ \eval{$ \csch \acsch 7 $}[16].
+\end{centred}
+\end{shaded}%
+\end{minipage}
+
+\subsection{Logarithms}
+
+The natural logarithm \verb`\ln`, base $10$ logarithm \verb`\lg`,
+and binary or base $2$ logarithm \verb`\lb` are all recognized,
+as is \verb`\log`, preferably with a subscripted base:
+\begin{centred}
+\verb`\eval{$ \log_{12}1728 $}` $\Longrightarrow$ \eval{$ \log_{12}1728 $}
+\end{centred}
+If there is no base indicated, base $10$ is assumed. (The notations
+\verb`\ln`, \verb`\lg`, and \verb`\lb` follow ISO 80000-2 recommendation,
+which frowns upon the use of the unsubscripted \verb`\log` although
+only \verb`\ln` appears widely used.) The base need not be explicitly
+entered as a number. It could be entered as an expression or be specified
+in the vv-list:
+\begin{centred}
+\verb`\eval*{$ \log_b c $}[b=2,c=1024]` $\Longrightarrow$ \eval*{$ \log_b c $}[b=2,c=1024],
+\end{centred}
+the log to base $2$ in this case. It is possible to use the unadorned
+\verb`\log` with a base different from $10$; if you wish to do this
+only for a particular calculation see §\ref{subsec:settingsLogBase},
+or see §\ref{sec:settingsDefaults} if you want to make this default
+behaviour.
+
+\subsection{Other unary functions}
+
+Other unary functions supported are the exponential function \verb`\exp`
+and signature function \verb`\sgn` (equal to $-1$, $0$, or $1$
+depending as its argument is $<0$, $=0$, or $>0$).
+
+\subsection{Squaring, cubing, \ldots unary functions}
+
+\texttt{numerica} has no difficulty reading a familiar but `incorrectly
+formed' expression like
+\[
+\sin^{2}1.234+\cos^{2}1.234.
+\]
+You do not have to render it $(\sin1.234)^{2}+(\cos1.234)^{2}$ or
+(heaven forbid) $(\sin(1.234))^{2}+(\cos(1.234))^{2}$. The everyday
+usage is fine:
+\begin{centred}
+\verb`\eval{$ \sin^2\theta+\cos^2\theta $}[\theta=1.234]` $\Longrightarrow$
+\eval{$ \sin^2\theta+\cos^2\theta $}[\theta=1.234] .
+\end{centred}
+Equally \texttt{numerica} has no difficulty reading the `correct'
+but pedantic form
+\begin{centred}
+\verb`\eval{$ (\sin(\theta))^2+(\cos(\theta))^2 $}[\theta=1.234]`
+$\Longrightarrow$ \eval{$ (\sin(\theta))^2+(\cos(\theta))^2 $}[\theta=1.234] .
+\end{centred}
+A hyperbolic identity is confirmed in this example:
+\begin{centred}
+\verb`\eval{$ \sinh 3x $}[x=1]` $\Longrightarrow$ \eval{$ \sinh 3x $}[x=1],\medskip{}
+
+\verb`\eval{$ 3\sinh x+4\sinh^3x $}[x=1]` $\Longrightarrow$ \eval{$ 3\sinh x+4\sinh^3x $}[x=1].
+\end{centred}
+In fact all named unary functions in \texttt{numerica} can be squared,
+cubed, etc., in this `incorrect' but familiar way, although the
+practice outside the trigonometric and hyperbolic context seems (vanishingly?)
+rare.
+
+When the argument of the function is parenthesized and raised to a
+power – like $\sin(\pi)^{2}$ – it is read by \texttt{numerica} as
+the `sine of the square of pi', $\sin(\pi^{2})$, and \emph{not
+}as the `square of the sine of pi', $(\sin\pi)^{2}$:
+\begin{centred}
+\verb`\eval{$ \sin(\pi)^2 $}` $\Longrightarrow$ \eval{$ \sin(\pi)^2 $} .
+\end{centred}
+Things are done like this in \texttt{numerica} above all to handle
+the logarithm in a natural way. Surely $\ln x^{n}=n\ln x=\ln(x^{n})$
+rather than $(\ln x)^{n}$? And if we wish to write (as we do) $\ln(1+1/n)^{n}=n\ln(1+1/n)=1-1/2n+1/3n^{2}-\ldots$
+to study the limiting behaviour of $(1+1/n)^{n}$, then we cannot
+avoid $\ln(x)^{n}=n\ln(x)=\ln(x^{n})$.
+
+\subsection{\emph{n}-ary functions}
+
+The functions of more than one variable ($n$-ary functions) that
+\texttt{numerica} supports are \verb`\max`, \verb`\min` and \verb`\gcd`,
+greatest common divisor. The comma list of arguments to \verb`\max`,
+\verb`\min` or \verb`\gcd` can be of arbitrary length. The arguments
+themselves can be expressions or numbers. For \verb`\gcd`, non-integer
+arguments are truncated to integers. Hence both $y$ and $3y$ are
+independently truncated in the following example – to $81$ and $243$
+respectively:
+\begin{centred}
+\verb`\eval{$ \gcd(12,10x^2,3y,y,63) $}[y=1/0.0123456789,x=3]` $\Longrightarrow$
+\eval{$ \gcd(12,10x^2,3y,y,63) $}[y=1/0.0123456789,x=3] .
+\end{centred}
+(The truncation occurs in the argument of \verb`\gcd`, not in the
+vv-list.)
+
+For $n$-ary functions, squaring, cubing, etc. follows a different
+pattern from that for unary functions. For \verb`\max`, \verb`\min`,
+\verb`\gcd` the argument of the function is a comma list. Squaring
+the argument makes no sense. We understand the superscript as applying
+to the function as a whole. (Consistency is not the point here; it
+is what mathematicians do that \texttt{numerica} tries to accommodate.)
+\begin{centred}
+\verb`\eval{$ \gcd(3x,x,\arcsin 1/\deg)^2 $}[x=24]` $\Longrightarrow$
+\eval{$ \gcd(3x,x,\arcsin 1/\deg)^2 $}[x=24] .
+\end{centred}
+
+\subsection{Delimiting arguments with brackets \& modifiers }
+
+Arguments of unary and $n$-ary functions can be delimited not only
+with parentheses, but also with square brackets and braces, both in
+explicit character form and also in the command form of §\ref{subsec:evalCommandBrackets}.
+The brackets, of whatever kind, can be qualified with \verb`\left \right`,
+\verb`\bigl \bigr`, etc.\footnote{See §\ref{subsec:settingsPunctuation} for the \texttt{{[}p=.{]}}
+(which ensures the concluding full stop appears in the correct place.}
+\begin{centred}
+\verb`\eval[p=.]{\[ \sin\left\lbrack \dfrac\pi{1+2+3}\right\rbrack \]}`
+$\Longrightarrow$\eval[p=.]{\[ \sin\left\lbrack\dfrac\pi{1+2+3}\right\rbrack \]}
+\end{centred}
+
+\subsection{Absolute value, floor \& ceiling functions}
+
+It is tempting to use the \texttt{|} key on the keyboard for inserting
+an absolute value sign. \texttt{numerica} accepts this usage, but
+it is deprecated. The spacing is incorrect – compare $|-l|$ using
+\texttt{|} against $\lvert-l\rvert$ using \verb`\lvert \rvert`.
+Also, the identity of the left and right delimiters makes nested absolute
+values difficult to parse. \texttt{numerica} does not attempt to do
+so. Placing an absolute value constructed with \texttt{|} within another
+absolute value constructed in the same way is likely to produce a
+compilation error or a spurious result. \verb`\lvert \rvert` are
+better in every way except ease of writing.\texttt{ }To aid such ease
+\texttt{numerica }provides the \verb`\abs` function (using the \texttt{\textbackslash DeclarePairedDelimiter}
+command of the \texttt{mathtools} package). This takes a mutually
+exclusive star (asterisk) or square bracketed optional argument, and
+a mandatory braced argument. The starred form expands to \verb`\left\lvert #1 \right\rvert`
+where \verb`#1` is the mandatory argument:
+\begin{centred}
+\verb`\eval[p=.]{\[ 3\abs*{\frac{\abs{n}}{21}-1} \]}[n=-7]` $\Longrightarrow$
+\eval[p=.]{\[ 3\abs*{\frac{\abs{n}}{21}-1} \]}[n=-7]
+\end{centred}
+The optional argument provides access to the \verb`\big...` modifiers:
+\begin{verbatim}
+ \eval[p=.]{\[
+ \abs[\Big]{\abs{a-c}-\abs[\big]{A-C}}
+ \]}[A=12,a=-10,C=7,c=-5]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]{\[
+ \abs[\Big]{\abs{a-c}-\abs[\big]{A-C}}
+ \]}[A=12,a=-10,C=7,c=-5]
+
+The form without either star or square bracket option dispenses with
+the modifiers altogether:
+\begin{centred}
+\verb`\eval{$ \tfrac12(x+y)+\tfrac12\abs{x-y} $}[x=-3,y=7].` $\Longrightarrow$
+\eval{$ \tfrac12(x+y)+\tfrac12\abs{x-y} $}[x=-3,y=7].
+\end{centred}
+As noted, the star and square bracketed option are mutually exclusive
+arguments.
+
+\texttt{numerica} also provides the functions \verb`\floor` and \verb`\ceil`,
+defined in the same way, taking a mutually exclusive star or square
+bracketed optional argument and for the starred forms expanding to
+\verb`\left\lfloor #1 \right\rfloor` and \verb`\left\lceil #1 \right\rceil`
+where \verb`#1` is the mandatory argument, and for the square bracket
+option forms replacing the \verb`\left` and \verb`\right` with the
+corresponding \verb`\big` commands. The form without star or square-bracket
+option dispenses with any modifier at all.
+\begin{centred}
+\verb`\eval{$ \floor{-\pi} $}` $\Longrightarrow$ \eval{$ \floor{-\pi} $},
+
+\verb`\eval{$ \ceil{\pi} $}` $\Longrightarrow$ \eval{$ \ceil{\pi} $}.
+\end{centred}
+The floor function, $\lfloor x\rfloor$, is the greatest integer $\le x$;
+the ceiling function, $\lceil x\rceil$ is the smallest integer $\ge x$.
+Like the absolute value, the floor and ceiling functions, can be nested:
+\begin{centred}
+\verb`\eval{$ \floor{-\pi+\ceil{e}} $}` $\Longrightarrow$ \eval{$ \floor{-\pi+\ceil{e}} $}.
+\end{centred}
+
+\subsubsection{Squaring, cubing, \ldots{} absolute values, etc.}
+
+These three functions can be raised to a power \emph{without} extra
+parentheses:
+\begin{centred}
+\verb`\eval{$ \ceil{e}^2 $},` $\Longrightarrow$ \eval{$ \ceil{e}^2 $},
+
+\verb`\eval{$ \abs{-4}^2 $}.` $\Longrightarrow$ \eval{$ \abs{-4}^2 $}.
+\end{centred}
+
+\subsection{Factorials, binomial coefficients}
+
+\label{subsec:evalFactorialsBinom}Factorials use the familiar trailing
+\texttt{!} notation:
+\begin{centred}
+\verb`\eval{$ 7! $}` $\Longrightarrow$ \eval{$ 7! $},
+
+\verb`\eval{$ (\alpha+\beta)!-\alpha!-\beta! $}[\alpha=2,\beta=3]`
+$\Longrightarrow$ \eval{$ (\alpha+\beta)!-\alpha!-\beta! $}[\alpha=2,\beta=3].
+\end{centred}
+The examples illustrate how \texttt{numerica} interprets the argument
+of the factorial symbol:\texttt{ }it `digests'
+\begin{enumerate}
+\item a preceding (possibly multi-digit) integer, or
+\item a preceding variable token, or
+\item a bracketed expression, or
+\item a bracket-like expression – an absolute value, floor or ceiling function,
+\end{enumerate}
+since they delimit arguments in a bracket-like way:
+\begin{centred}
+\verb`\eval{$ \abs{-4}!+\floor{\pi}!+\ceil{e}! $}` $\Longrightarrow$
+\eval{$ \abs{-4}!+\floor{\pi}!+\ceil{e}! $}.
+\end{centred}
+The result of feeding the factorial an expression different in kind
+from one of these four cases may give an error message or an unexpected
+result. Use parentheses around such an expression; for example write
+$(3^{2})!$, rather than $3^{2}!$.
+
+Nesting of brackets for factorials is accepted:
+\begin{centred}
+\verb`\eval{$ ((5-2)!+1)! $}` $\Longrightarrow$ \eval{$ ((5-2)!+1)! $}.
+\end{centred}
+The factorials of negative integers or of non-integers are not defined
+in \texttt{numerica}. Again there is the problem met in relation to
+$n$-th roots of what happens if the argument of a factorial is the
+result of a calculation and rounding errors mean it is not an exact
+integer. This problem is unlikely to be of practical concern since
+\texttt{numerica} rounds the result of such a calculation by default
+to $14$ significant figures before offering it to the factorial.
+Since \texttt{l3fp} works to $16$ significant figures, there is ample
+`elbowroom' to accommodate rounding errors before the result of
+a calculation ceases to round to an integer.
+
+\subsubsection{Double factorials}
+
+The double factorial, written $n!!,$ is the product $n(n-2)(n-4)\ldots\times4\times2$
+when $n$ is even and the product $n(n-2)(n-4)\ldots\times3\times1$
+when $n$ is odd.
+\begin{centred}
+\verb`\eval{$ 6!! $}` $\Longrightarrow$ \eval{$ 6!! $},
+
+\verb`\eval{$ n!! $}[n=\sqrt{49}]` $\Longrightarrow$ \eval{$ n!! $}[n=\sqrt{49}],
+\end{centred}
+Since $n!=n!!(n-1)!!$ it follows that
+\[
+n!!=\frac{n!}{(n-1)!!}=\frac{(n+1)!}{(n+1)!!}.
+\]
+Putting $n=0$ in the outer equality shows that $0!!=1$. Now putting
+$n=0$ in the left equality gives $(-1)!!=1$. Double factorials therefore
+are defined for integers $\ge-1$.
+
+\subsubsection{Binomial coefficients}
+
+Binomial coefficients are entered in \LaTeX{} with the \verb`\binom`\textbf{
+}command. It takes two arguments and has a text-style version \verb`\tbinom`
+and a display-style version \verb`\dbinom`. As implemented in \texttt{numerica},
+these are \emph{generalised} binomial coefficients:
+\[
+\binom{x}{k}=\frac{x(x-1)\dots(x-k+1)}{k(k-1)\dots1},\quad(x\in\mathbb{R},~k\in\mathbb{N}),
+\]
+where $x$ need not be a non-negative integer, and where $\binom{x}{0}=1$
+by definition. Although the first (or upper) argument can be any real
+number, the lower argument \emph{must} be a non-negative integer.
+Thus, \verb`\eval{$ \tbinom53 $}` $\Longrightarrow$ \eval{$\tbinom53$},
+\verb`\eval{$ \tbinom70 $}` $\Longrightarrow$ \eval{$\tbinom70$},
+\verb`\eval{$ \tbinom{4.2}3 $}` $\Longrightarrow$ \eval{$\tbinom{4.2}3$},
+but if the second (or lower) argument of \verb`\binom` is \emph{not}
+a non-negative integer, \texttt{numerica} displays a message; see
+§\ref{subsec:errorsIntegerArgs}.
+
+\subsection{Sums and products}
+
+\texttt{numerica} recognizes sums (\verb`\sum` displaying as $\sum$)
+and products (\verb`\prod` displaying as $\prod$), and expects both
+symbols to have lower and upper summation/product limits specified.
+The lower limit must be given in the form \emph{sum/prod variable
+= initial value}; the upper limit requires only the final value to
+be specified (although it can also be given in the form \emph{sum/prod
+variable = final value}). The values may be expressions depending
+on other variables and values but must evaluate to integers (or infinity
+– see §\ref{sec:settingsInfiniteSumsProds}). Evaluating to an integer
+means that they \emph{round} to an integer, using a rounding value
+that is set by default to $14$; (recall that \texttt{l3fp} works
+to $16$ significant figures). If a limit evaluates to a non-integer
+at this `int-ifying' rounding value, an error message results. (To
+change this `int-ifying' rounding value, see §\ref{subsec:defaultsIntifyingRounding}.)
+
+As an example of expressions in the limits, this example uses the
+floor and ceiling functions to convert combinations of constants to
+integers (the \verb`[p]` is explained in §\ref{subsec:settingsPunctuation}),
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{n=\floor{\pi/e}}^{\ceil{\pi e}}n \]}` $\Longrightarrow$\eval[p]{\[ \sum_{n=\floor{\pi/e}}^{\ceil{\pi e}}n \]}
+\end{centred}
+\noindent (which is $\sum_{n=1}^{9}n$).\emph{ }If the upper limit
+is less than the lower limit the result is zero. Notice that there
+is no vv-list. The summation variable does not need to be included
+there unless there are other variables that depend on it. However,
+in the case
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{k=1}^N\frac1{k^3} \]}[N=100][4]` $\Longrightarrow$
+\eval[p]{\[ \sum_{k=1}^N\frac1{k^3} \]}[N=100][4]
+\end{centred}
+the upper limit $N$ is necessarily assigned a value in the vv-list.
+
+To the author it seems natural to enter the lower limit first, immediately
+after the \verb`\sum` command (the sum is \emph{from }something \emph{to}
+something), but no problem will accrue if the upper limit is placed
+first (after all, the appearance of the formula in the pdf is the
+same):
+\begin{centred}
+\verb`\eval[p=.]{\[ \sum^N_{k=1}\frac1{k^3} \]}[N=100][4]` $\Longrightarrow$
+\eval[p=.]{\[ \sum^N_{k=1}\frac1{k^3} \]}[N=100][4]
+\end{centred}
+Another example of a sum, using binomial coefficients this time, is
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{m=0}^5\binom{5}{m}x^m y^{5-m} \]}[x=0.75,y=2.25]`
+$\Longrightarrow$ \eval[p]{\[ \sum_{m=0}^5\binom{5}{m}x^m y^{5-m} \]}[x=0.75,y=2.25]
+\end{centred}
+which is just \verb`\eval{$(x+y)^5$}[x=0.75,y=2.25]` $\Longrightarrow$
+\eval{$ (x+y)^5 $}[x=0.75,y=2.25], or $3^{5}$.
+
+Now let's calculate a product:
+\begin{verbatim}
+ \eval[p]{\[
+ \prod_{k=1}^{100}
+ \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+ \]}[x=1][3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[\prod_{k=1}^{100} \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)\]}[x=1][3]
+
+\noindent to be compared with \verb`\eval{$ \sinh 1 $}[3]` $\Longrightarrow$
+\eval{$ \sinh1 $}[3]. Obviously more terms than $100$ are required
+in the product to achieve 3-figure accuracy.
+
+\subsubsection{Infinite sums and products}
+
+There is a strong urge to use $\infty$ in the upper limit of this
+product. Let's do so:
+\begin{verbatim}
+ \eval[p=.]{\[
+ \prod_{k=1}^{\infty}
+ \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+ \]}[x=1][3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]
+ {\[
+ \prod_{k=1}^{\infty}
+ \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+ \]}[x=1][3]
+
+\noindent Disappointingly, we still get the same result, deficient
+by $1$ in the third decimal place. Obviously \texttt{numerica} has
+not multiplied an infinite number of terms and, just as obviously,
+the finite number of terms it \emph{has} multiplied are too few. How
+\texttt{numerica} decides when to stop evaluating additional terms
+in an infinite sum or product is discussed later, §\ref{sec:settingsInfiniteSumsProds}.
+
+For this particular product the problem is that it converges slowly.
+Any criterion for when to stop multiplying terms or, for an infinite
+sum adding terms, seems bound to fail for some product or series.
+Presumably any stopping criterion must measure smallness in some way.
+But terms of the divergent harmonic series, $\sum(1/n)$ can always
+be found smaller than any value we care to specify. It is not surprising
+that a sufficiently slowly converging product or series falls foul
+of a given criterion.
+
+The default criterion however can be changed. Because this involves
+values assigned in the settings option of the \verb`\eval` command,
+I discuss infinite sums and products in the section discussing that
+optional argument; see §\ref{sec:settingsInfiniteSumsProds}.
+
+Other infinite sums converge more rapidly, and the default settings
+work admirably. For example \verb`\eval{$ (1+0.1234)^{4.321} $}`
+$\Longrightarrow$ \eval{$ (1+0.1234)^{4.321} $}. Using binomial
+coefficients we can express this as an infinite sum:\medskip{}
+\begin{verbatim}
+ \eval[p=.]{\[
+ \sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n}
+ \]}[\alpha=4.321,x=0.1234]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]
+ {\[
+ \sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n}
+ \]}[\alpha=4.321,x=0.1234]
+
+\section{Error messages }
+
+\label{sec:evalErrors}There are two kinds of error in \texttt{numerica}:
+those in the underlying \LaTeX{} which are reported in the \LaTeX{}
+log, shown on the terminal, and generally halt compilation, and specifically
+\texttt{numerica}-related errors which do not halt compilation and
+produce messages displayed in the pdf where one would expect the result
+of the calculation to be. The original reason for doing things this
+way was to enable \texttt{numerica} to be used effectively with the
+instant preview facility of the document processor \LyX . More philosophically,
+one might view such errors as similar to grammatical errors or spelling
+mistakes in text. It is not clear that they should halt compilation.
+Hence strictly \texttt{numerica}-related errors leave brief messages
+in the pdf at the offending places.
+
+Before discussing specific error messages, note that there is a debug
+facility (of a sort) discussed below in §\ref{subsec:settingsDebug}.
+
+Error messages are in two parts: a \emph{what} part and a \emph{where}
+part.
+
+\subsection{Mismatched brackets}
+
+\label{subsec:errorsMismatched-brackets}An unmatched left parenthesis
+or other left bracket (in this case a missing right parenthesis) usually
+results in a \texttt{numerica} error:
+\begin{centred}
+\verb`$\eval{\sin(\pi/(1+x)}[x=1]$` $\Longrightarrow$ $\eval{\sin(\pi/(1+x)}[x=1]$
+\end{centred}
+For the same error in the vv-list, the what-part remains unchanged
+but the where-part is altered:
+\begin{centred}
+\verb`$\eval{ 1+y }[x=1,y=\sin(\pi/(1+x)]$` $\Longrightarrow$ $\eval{ 1+y }[y=\sin(\pi/(1+x),x=1]$
+\end{centred}
+The \emph{what} message is the same; the \emph{where} is different.
+
+An unmatched right parenthesis or other right bracket (in this case
+a missing \emph{left} parenthesis) usually results in a similar \texttt{numerica}
+error:
+\begin{centred}
+\verb`$\eval{2((x+y)/(y+z)))^2}[x=1,y=2,z=3]$` $\Longrightarrow$
+\eval{2((x+y)/(y+z)))^{2}}[x=1,y=2,z=3]
+\end{centred}
+But note that an unmatched modifier like \verb`\left` or \verb`\right`
+is a \LaTeX{} error and is caught by \LaTeX{} before \texttt{numerica}
+can respond and so results in a terminal and logfile message.
+
+\subsection{Unknown tokens}
+
+An `Unknown token' message can arise in a number of ways. If an
+expression involves a number of variables, some of which depend on
+others, their order in the vv-list matters:
+\noindent \begin{center}
+\verb`$\eval{\tfrac12 vt}[t=2,v=gt,g=9.8]$` $\Longrightarrow$ \eval{\tfrac{1}{2}vt}[t=2,v=gt,g=9.8]
+\par\end{center}
+
+The vv-list is evaluated from the \emph{right} so that in this example
+the variable \texttt{v} depends on a quantity \texttt{t} that is not
+yet defined. Hence the message. The remedy is to move \texttt{t} to
+the right of \texttt{v} in the vv-list.
+
+Similarly, if we use a variable in the formula that has not been assigned
+a value in the vv-list, we again get the `Unknown token' message,
+but this time the location is the formula:
+\begin{centred}
+\verb`$\eval{\pi r^2h}[r=3]$` $\Longrightarrow$ \eval{\pi r^{2}h}[r=3]
+\end{centred}
+The remedy obviously is to assign a value to \texttt{h} in the vv-list\texttt{.}
+
+The same message will result if a mathematical operation or function
+is used that has not been implemented in \texttt{numerica}:
+\begin{centred}
+\verb`$\eval{u \bmod v }[v=7,u=3]$` $\Longrightarrow$ \eval{u\bmod v}[v=7,u=3]
+\end{centred}
+A missing comma in the vv-list will generally result in an unknown
+token message:
+\begin{centred}
+\verb`$\eval{axy}[a=3 y=2,x=1]$` $\Longrightarrow$ \eval{axy}[a=3y=2,x=1]
+\end{centred}
+Because of the missing comma, \texttt{numerica} assumes \texttt{a}
+has the `value' \texttt{3y=2} and has no knowledge of \texttt{y}
+as a variable.
+
+The presence of multi-token variables can also cause this error message
+if the check for such variables is turned off; see §\ref{subsec:settingsMultitokSwitch}.
+
+\subsection{Other vv-list errors}
+
+Other errors that can occur with the vv-list are overlooked value
+assignments to variables, or missing commas. For the first, it is
+essential that we do actually give a value to all variables occuring
+in the vv-list:
+\begin{centred}
+\verb`$\eval{axy}[a=3,y=,x=1]$` $\Longrightarrow$ \eval{axy}[a=3,y=,x=1]
+\end{centred}
+The remedy is obvious – assign a value to \texttt{$y$}.
+
+\emph{Extra} commas in the vv-list should cause no problems:
+\begin{centred}
+\verb`$\eval{axy}[,a=3,,y=2,x=1,]$` $\Longrightarrow$ $\eval{axy}[,a=3,,y=2,x=1,]$
+\end{centred}
+
+\subsection{Inverse powers of negative numbers}
+
+\label{subsec:errorsInverse-powers}Inverse integer powers of positive
+numbers should always be possible, but raising a \emph{negative} number
+to an inverse power generates an error even when, mathematically,
+it should not:
+\begin{centred}
+\verb`\eval{$ (-125)^{1/3} $}` $\Longrightarrow$ \eval{$ (-125)^{1/3} $}
+\end{centred}
+This is a feature of floating point arithmetic. When a number is raised
+to a rational power, say $p/q$ where $p$ and $q$ are non-zero integers,
+then the result is the $p$-th power of the $q$-th root of the number.
+Can a $q$-th root be taken? If our floating point system used (for
+ease of illustration) only $4$ significant digits, $p/q=1/3$ would
+be the fraction $3333/10^{4}$, an odd numerator over an even denominator.
+But a negative number does not possess an even ($10^{4}$-th) root.
+
+\subsection{Integer argument errors}
+
+\label{subsec:errorsIntegerArgs}Some functions require integer arguments
+– factorials, the second argument of a binomial coefficient, and (in
+\texttt{numerica}) $n$-th roots using the optional argument of \texttt{\textbackslash sqrt};
+also summation and product variables. If integers are explicitly entered
+for these arguments there is no problem, but if the value of the argument
+is the result of a calculation, rounding errors require thinking about.
+What accumulation of rounding errors is \emph{too} much so that the
+result of the calculation \emph{cannot} be considered an integer?
+\texttt{numerica} is generous: in the default setup, if a calculation
+rounds to an integer at rounding value $14$ the result of the calculation
+is considered an integer (obviously, the integer resulting from the
+rounding). Since \texttt{l3fp} works to $16$ significant figures
+that gives ample room for rounding errors to `get lost in' and be
+ignored, while still ruling out such things as (recall the example
+in §\ref{subsec:evalBoolean-output}),
+\begin{centred}
+\verb`\eval{\[ \sum_{n=1}^N n \]}[N=1/0.0123456789]` $\Longrightarrow$
+\eval{\[ \sum_{n=1}^N n \]}[N=1/0.0123456789]
+\end{centred}
+where $N$ differs from $81$ not until the seventh decimal place.
+
+The default rounding value of $14$ for `int-ifying' calculations
+can be changed: see §\ref{subsec:defaultsIntifyingRounding}.
+
+\subsection{Comparison errors}
+
+Should a user try to make a comparison using a combination like \verb`>=`
+rather than \verb`\geq`, \texttt{numerica} admonishes like this:
+\begin{centred}
+\verb`$\eval{ \pi^e >= e^\pi }[?]$` $\Longrightarrow$ $\eval{ \pi^e >= e^\pi }[?]$
+\end{centred}
+(The relation is false by the way.)
+
+\subsection{Invalid base for \texttt{\textbackslash log}}
+
+ISO recommends using \verb`\log` only with a subscripted base specified.
+Otherwise how is one to know whether the base is \verb`e` or $10$
+or $2$ or whatever? Nonetheless \texttt{numerica} assumes that when
+\verb`\log` is used unsubscripted, the base is 10. Suppose you want
+to make $12$ the base, but forget to put braces around the $12$:
+\begin{centred}
+\verb`$\eval{ \log_12 1728 }$` $\Longrightarrow$ $\eval{ \log_12 1728 }$
+\end{centred}
+Here, \texttt{numerica} has taken \texttt{1} as the base (and $21728$
+as the argument) of the logarithm and responds accordingly.
+
+\subsection{\texttt{l3fp} errors}
+
+Some errors arising at the \texttt{l3fp} level are trapped and a message
+displayed.
+\begin{itemize}
+\item Dividing by zero
+\end{itemize}
+\begin{centred}
+\verb`$\eval{1/\sin x}[x=0]$` $\Longrightarrow$ \eval{1/\sin x}[x=0]
+\end{centred}
+Note however that \verb`$\eval{1/\sin x}[x=\pi]$` $\Longrightarrow\,\eval{1/\sin x}[x=\pi]$,
+because of rounding errors in distant decimal places. No doubt this
+is true for other functions as well.
+\begin{itemize}
+\item Invalid operation
+\end{itemize}
+\begin{centred}
+\verb`$\eval{\arccos x}[x=2]$` $\Longrightarrow$ \eval{\arccos x}[x=2]
+\end{centred}
+In this case the inverse cosine has been fed a value of $x$ outside
+its domain of definition, the interval $[-1,1]$. Trying to evaluate
+an expression that resolves to $0/0$ also produces this message:
+\begin{centred}
+\verb`$\eval{\frac{1-y}{x-2}}[x=2,y=1]$`$\Longrightarrow$ \eval{\frac{1-y}{x-2}}[x=2,y=1]
+\end{centred}
+\begin{itemize}
+\item Overflow/underflow
+\end{itemize}
+The factorial (discussed in §\ref{subsec:evalFactorialsBinom}) provides
+an example of overflow:
+\begin{centred}
+\verb`$\eval{3249!}$`\texttt{ }$\Longrightarrow$ \eval{3249!}
+\end{centred}
+This is hardly surprising since
+\begin{centred}
+\verb`$\eval{3248!}[x]$` $\Longrightarrow$ $\eval{3248!}[x]$.
+\end{centred}
+There is a limit on the size of exponents that \texttt{l3fp} can handle.
+A number in the form $a\times10^{b}$ must have $-10001\le b<10000$.
+If this is not the case an overflow or underflow condition occurs.
+As the examples show, an overflow condition generates a \texttt{numerica}
+error. For underflow, where the number is closer to $0$ than $10^{-10001}$,
+\texttt{l3fp} assigns a zero value to the quantity. \texttt{numerica}
+accepts the zero value.
+
+\chapter{Settings}
+
+\label{chap:Settings}A calculation is effected against a background
+of default values for various quantities. For a particular calculation,
+these values may not be appropriate; or you may have different preferences.
+The way to change settings for a particular calculation is through
+the settings option of \verb`\nmcEvaluate` discussed next. The way
+to change a \emph{default} setting is by creating a configuration
+file \texttt{numerica.cfg} discussed in §\ref{sec:settingsDefaults}.
+
+\section{Settings option}
+
+\label{sec:settingsOption}The second argument of the \verb`\nmcEvaluate`
+command is the settings option, delimited by square brackets. This
+option is a \emph{key=value} list, hence comma-separated. \emph{Key=value
+}lists tend to be wordy. For back-of-envelope calculations one wants
+to be able to `dash off' the calculation, hence the short, cryptic
+nsture of the keys. Most settings are generic, applicable not only
+to \verb`\nmcEvaluate` but also to other commands that are available
+if \texttt{numerica} is loaded with the \texttt{plus} option; see
+the associated document \texttt{numerica-plus.pdf}.
+
+\subsection{\textquoteleft Debug\textquoteright{} facility}
+
+\label{subsec:settingsDebug}It is rather grandiose to call this a
+debug facility, but if a calculation goes wrong or produces a surprising
+result, \texttt{numerica} offers a means of examining some intermediate
+stages on the way to the final result. To use the facility, enter
+\begin{lyxcode}
+dbg~=~<integer>
+\end{lyxcode}
+into the settings option. (White space around the equals sign is optional.)
+\begin{itemize}
+\item \texttt{dbg=0 }turns off the debug function, displays the result or
+error message (this is the default);
+\item \texttt{dbg=1 }equivalent to \texttt{dbg=2{*}3{*}5{*}7};
+\end{itemize}
+The `magic' integers are the following primes and their products:
+
+\begin{table}[t]
+\centering
+\noindent \centering{}\caption{Settings options}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{reuse}} & {\small int} & {\small form of result saved with }{\small{\small\verb`\nmcReuse`}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{xx}} & {\small int (0/1)} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{()}} & {\small int (0/1/2)} & {\small trig. arg. parsing} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{o}} & & {\small degree switch for trig. funcions} & \tabularnewline
+{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{vvmode}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvd}} & {\small token(s)} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{vvi}} & {\small token(s)} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{*} & & {\small suppress equation numbering if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
+in }{\small\texttt{vvd}} & \tabularnewline
+{\small\texttt{p}} & token(s) & {\small punctuation (esp. in display-style)} & {\small\texttt{,}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+
+\begin{itemize}
+\item \texttt{dbg=2} displays the vv-list after multi-token variables have
+been converted to their single token form, \texttt{\textbackslash\_nmca},
+\texttt{\textbackslash\_nmcb}, etc.;
+\item \texttt{dbg=3} displays the formula after multi-token variables have
+been converted to their single token form;
+\item \texttt{dbg=5} displays the stored variables and their evaluated values
+(\texttt{dbg=2} lists the values as expressions; here they have been
+evaluated); note that any saved values (Chapter~\ref{chap:Saving-and-retrieving})
+that have been loaded will also feature in this list which might lead
+to a messy display depending on the nature of those values;
+\item \texttt{dbg=7} displays the formula after it has been fp-ified (but
+before it has been fed to \texttt{l3fp} to evaluate);
+\begin{itemize}
+\item should the formula successfully evaluate, the result of the evaluation
+is also displayed.
+\end{itemize}
+\end{itemize}
+\vspace{-1ex}
+
+\begin{wraptable}{o}{0.5\columnwidth}%
+\centering{}\caption{Magic integers}\label{tab:settingsMagic-integers}
+\begin{center}
+\begin{tabular}{rrl}
+\toprule
+integer & & \multicolumn{1}{c}{factors}\tabularnewline
+\midrule
+6 & & 2,3\tabularnewline
+10 & & 2,5\tabularnewline
+14 & & 2,7\tabularnewline
+15 & & 3,5\tabularnewline
+21 & & 3,7\tabularnewline
+30 & & 2,3,5\tabularnewline
+35 & & 5,7\tabularnewline
+42 & & 2,3,7\tabularnewline
+70 & & 2,5,7\tabularnewline
+105 & & 3,5,7\tabularnewline
+210 & & 2,3,5,7\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}\end{wraptable}%
+To display two or more of these elements simultaneously, use the product
+of their debug numbers for the magic integer. This can be entered
+either as the multiplied-out product, or as the `waiting to be evaluated'
+product with asterisks (stars) between the factors. Thus \texttt{dbg=6}
+or \verb`dbg=2*3` display both the vv-list and formula after multi-token
+variables have been converted to single token form; \texttt{dbg=10}
+or \verb`dbg=2*5` display both the vv-list after multi-token variables
+have been converted to single token form and the recorded variables
+with their evaluated values. And similarly for the other magic integers
+listed. For other integers, if they are divisible by $2$ or $3$
+or $5$ or $7$, they will display the corresponding component. Both
+\texttt{dbg=210} and \verb`dbg=2*3*5*7` display all four elements,
+but rather than remembering this product, it suffices to put \texttt{dbg=1}.
+This is equivalent and displays all elements.
+
+The debug option uses an \texttt{aligned }or \texttt{align{*}} environment
+to display its wares, depending on the presence or absence of math
+delimiters around the \texttt{\textbackslash eval} command. In the
+next example I have used multi-token variables to illustrate the different
+elements in the debug display, and a chain of comparisons to show
+how \texttt{numerica} treats these (§\ref{subsec:evalBoolean-output}).
+\begin{verbatim}
+ \eval[dbg=1]{ a_1<2a_2<3a_3<\pi+e }
+ [a_1=\pi,a_2=\phi,a_3=e\gamma][6?'']
+\end{verbatim}
+$\Longrightarrow$ \eval[dbg=1]{ a_1<2a_2<3a_3<\pi+e }
+ [a_1=\pi,a_2=\phi,a_3=e\gamma][6?'']
+
+Note that the four elements are displayed in temporal order: first
+comes the vv-list after conversion of multi-token to single-token
+variables, then the formula in the single-token variables; these are
+created essentially at the same time. The vv-list is presented in
+left-to-right order because that is the direction of evaluation \emph{internally}
+in \texttt{numerica}. Next the stored values of the variables are
+displayed. These are the values \emph{after }evaluation. The fourth
+element both in the display and chronologically is the fp-ified formula;
+this is often a thicket of parentheses. The final element of the display
+and chronologically is the result of evaluating the formula. This
+is displayed only if $7$ is a factor of the \texttt{dbg} integer,
+and there is no error.
+
+When interpreting the fp-form, one should be aware of differences
+between \texttt{numerica} and \texttt{l3fp}. In particular be aware
+that in \texttt{l3fp} function calls bind most tightly so that, for
+example, \verb`sin 2pi` evaluates not to zero but to $(\sin2)\times\pi$
+and \verb`sin x^2` evaluates to $(\sin x)^{2}$. This should not
+be of any concern to the user except as here in debug mode when interpreting
+fp-forms.
+
+Finally, note that those mathematical operations that have no direct
+representation in \texttt{l3fp} contribute only their value to the
+fp-form. This applies to sums and products, double factorials and
+partly to binomial coefficients as illustrated in the followng (ridiculous)
+example:
+\begin{verbatim}
+ \eval[dbg=1]{\[
+ \sum_{n=1}^5 n + \binom{10}{m}
+ + \prod_{n=2}^5 (1-1/n) + m!! \][m=6]
+\end{verbatim}
+$\Longrightarrow$ \eval[dbg=1]{\[
+ \sum_{n=1}^5 n+\binom{10}{m}
+ +\prod_{n=2}^5(1-1/n) +m!! \]}[m=6]
+
+\noindent The various contributions to the overall result are displayed
+simply as numbers because \texttt{l3fp} does not (at least as yet)
+handle these elements natively.
+
+\subsubsection{Negative \texttt{dbg} values}
+
+Negative \texttt{dbg} values are possible: \texttt{dbg=-2}, \texttt{dbg=-3},
+etc. (and \texttt{dbg=-1} meaning \texttt{dbg=-210}) have exactly
+the same effects as the corresponding positive values except for some
+details of display. The display for positive \texttt{dbg} values is
+the one evident in the examples above. Lines wrap, the left margin
+is not indented and the display occupies the page width. For negative
+\texttt{dbg} values, lines do not wrap, the left margin is indented
+and the display occupies the text width. An example is presented in
+§\ref{subsec:nestDebugging} below where the display for a nested
+\verb`\eval` is significantly improved with a negative \texttt{dbg}
+value.
+
+\subsection{Reuse setting}
+
+This setting determines whether the entire display or only the numerical
+result is saved to file with the \verb`\nmcReuse` command. See below,
+Chapter~\ref{chap:Saving-and-retrieving}, §\ref{subsec:reuse=00005Ceval's-reuse-setting}.
+
+\subsection{Inputting numbers in scientific notation}
+
+\label{subsec:settingsInputtingSciNotation}\emph{Outputting} numbers
+in scientific notation is controlled by the final trailing argument
+of the \texttt{\textbackslash eval} command. That is turned off by
+default and needs to be explicitly ordered. Similarly, \emph{inputting}
+numbers in scientific notation is turned off by default and needs
+to be explicitly ordered. To turn it on, write
+\begin{lyxcode}
+\textasciicircum ~=~<char>
+\end{lyxcode}
+\noindent in the settings option, where \texttt{<char>} is any single
+character, usually \texttt{e} or \texttt{d} or their upper-casings,
+but not restricted to them: \texttt{\textasciicircum =@} for instance
+is perfectly possible, and has the advantage over \texttt{e} or \texttt{d}
+that it doesn't conflict with the use of the character as a variable
+or constant.
+\begin{centred}
+\verb`$ \eval[^=@]{ 1.23 at -1 } $` $\Longrightarrow$ $ \eval[^=@]{ 1.23 at -1 } $.
+\end{centred}
+With letters for the exponent mark – say \verb`d` or \verb`e` –
+the problem is interpreting forms like \texttt{8d-3} or \texttt{2e-1}.
+Does such a form denote a number in scientific notation or an algebraic
+expression? In \texttt{numerica}, if the settings option shows \texttt{\textasciicircum =d},
+then a form like \texttt{8d-3} is treated as a number in scientific
+notation. Similarly for \texttt{e} or any other letter used as the
+exponent marker for the input of scientific numbers. (But only one
+character can be so used at a time.) Note that the number \emph{must}
+start with a digit: \verb`e-1` for instance does not and will be
+treated as an algebraic expression involving the exponential constant
+(unless \verb`e` is assigned a different value in the vv-list).
+\begin{centred}
+\verb`$ \eval[^=e]{ x+e-1 }[x=2e-1] $` $\Longrightarrow$ $ \eval[^=e]{ x+e-1 }[x=2e-1] $.
+\end{centred}
+The problem here is that \verb`2e-1` is treated as a number in scientific
+notation but displays in the vv-list as if it were an algebraic expression.
+The solution is to put \verb`2e-1` into an \verb`\mbox` or \verb`\text`
+command in the vv-list:
+\begin{centred}
+\verb`$ \eval[^=e]{ x+e-1 }[x=\text{2e-1}] $` $\Longrightarrow$
+$ \eval[^=e]{ x+e-1 }[x=\text{2e-1}] $.
+\end{centred}
+If you use a particular character as the exponent marker for inputting
+numbers in scientific notation, it is good practice \emph{not} to
+use that character as a variable, not because it will cause an error
+but because it makes expressions harder to read.
+
+\subsection{Multi-token variables}
+
+\label{subsec:settingsMultitokSwitch}Variables need not consist of
+a single character or token (like $x$ or $\alpha$). Multi-token
+symbols like $x'$ or $t_{i}$ or $var$ are perfectly acceptable.
+For its internal operations, \texttt{numerica} converts such multi-token
+names to single tokens (as discussed in §\ref{subsec:evalVariableNames}).
+This conversion takes time. Even if there are no multi-token variables
+used at all, \texttt{numerica} still needs to check that that is so.
+There is a setting that allows a user to turn off or turn on the check
+for such variables by entering
+\begin{lyxcode}
+xx~=~<integer>
+\end{lyxcode}
+into the settings option. If \texttt{<integer>} is \texttt{0}, the
+check for (and conversion of) multi-token variables is turned off;
+if \texttt{<integer>} is \texttt{1} (or any other\emph{ non-zero}
+integer), the check, and conversion if needed, goes ahead. By default,
+checking for multi-token variables and converting them if found is
+turned \emph{on}. (The name for the key, \texttt{xx}, is chosen because
+\texttt{x} is the most familiar variable of all, introduced in elementary
+algebra, and doubling it like this suggests multi-token-ness.)
+
+If checking is turned off when a multi-token variable is present,
+an error results. We don't need to enter \texttt{xx=1} in the first
+of the following examples because the check for multi-token variables
+is on by default. Explicitly turning it off in the second produces
+an error.
+\begin{centred}
+\verb`\eval{$ x_0^{\,2} $}[x_0=5]` $\Longrightarrow$ \eval{$ x_0^{\,2} $}[x_0=5],\medskip{}
+\verb`\eval[xx=0]{$ x_0^{\,2} $}[x_0=5]` $\Longrightarrow$ \eval[xx=0]{$ x_0^{\,2} $}[x_0=5]
+\end{centred}
+
+\subsection{Parsing arguments of trigonometric functions}
+
+This setting allows a wider range of arguments to trigonometric functions
+to be used (think Fourier series) without needing to insert extra
+parentheses in order for them to be read correctly by \verb`\eval`;
+see §\ref{subsec:parseTrigFns}.
+
+\subsection{Using degrees rather than radians}
+
+\label{subsec:settingsDegrees}You may find it more convenient to
+use degrees rather than radians with trigonometric functions.This
+can be switched on simply by entering a lowercase \texttt{o} in the
+settings option. (The author's fond hope is that the charitable eye
+might see a degree symbol in the \texttt{o}.) Thus
+\begin{centred}
+\verb`\eval[o]{$ \sin 30 $}` $\Longrightarrow$ \eval[o]{$ \sin 30 $},
+
+\verb`\eval[o]{$ \arcsin 0.5 $}` $\Longrightarrow$ \eval[o]{$ \arcsin 0.5 $}.
+\end{centred}
+
+\subsection{Specifying a logarithm base}
+
+\label{subsec:settingsLogBase}If you wish to use \verb`\log` without
+a subscripted base in a particular calculation, then add an entry
+like
+\begin{lyxcode}
+log~=~<positive~number${}\,\mathtt{\ne1}$>
+\end{lyxcode}
+to the settings option of the \verb`\eval` command. The \verb`<positive number>`
+does not need to be an integer. It could be \verb`e` (if you object
+to writing \verb`\ln`) but is more likely to be $2$ or another small
+integer; $10$ is the default. If you want to use this changed base
+not for one but most calculations, then add an entry with your choice
+of base to a configuration file; see §\ref{sec:settingsDefaults}.
+
+\subsection{Calculation mode}
+
+\label{subsec:settingsRecalcMode}A variable may change in the course
+of a calculation. This is certainly true of sums and products. If
+a parameter in the vv-list depends on the variable then that parameter
+will need to be recalculated, perhaps repeatedly, in the course of
+a calculation. By entering \label{subsec:settingsCalculationMode}
+\begin{lyxcode}
+vvmode~=~<integer>
+\end{lyxcode}
+in the settings option it is possible to turn on or off the ability
+to repeatedly evaluate the vv-list; \verb`<integer>` here takes two
+possible values, \texttt{0} or \texttt{1}. \texttt{vvmode=0} means
+the vv-list is evaluated once at the start of the calculation; \texttt{vvmode=1}
+means the vv-list is recalculated every time the relevant variable
+changes.
+
+For example, it may be desirable to place the summand, or some part
+of it, in the vv-list. Since the summation variable obviously changes
+during the course of the calculation, we need to enter \texttt{vvmode=1}
+in the settings option. Repeating an earlier sum (the seting \verb`p=.`
+is discussed in §\ref{subsec:settingsPunctuation}) \medskip{}
+\begin{verbatim}
+ \eval[p=.,vvmode=1]{\[ \sum_{k=1}^N f(k) \]}
+ [N=100,f(k)=1/k^3,{k}=1][4]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.,vvmode=1]{\[ \sum_{k=1}^N f(k) \]}
+ [N=100,f(k)=1/k^3,{k}=1][4]
+
+As you can see, the summand \texttt{f(k)} has been given explicit
+form in the vv-list – equated to \texttt{1/k\textasciicircum 3}.
+That means we need to give a preceding value to \texttt{k} in the
+vv-list; hence the rightmost entry. But we don't want \texttt{k=1}
+appearing in the final display, so we wrap \texttt{k} in braces (see
+§\ref{subsec:evalVvSuppressVars}). Since the value \texttt{k=1} applies
+only to the first term in the sum, to ensure it is not used for all
+terms, we enter \texttt{vvmode=1} in the settings option. This turns
+vv-recalculation mode on and ensures \texttt{k=1} is overwritten by
+\texttt{k=2}, \texttt{k=3} and so on, and the vv-list recalculated
+each time. The final result is the same as before, although recalculating
+the vv-list at each step is a more resource-hungry process. The difference
+may not be marked for this example; with more complicated expressions
+it noticeably takes longer.
+
+Because it is necessary to activate this switch when using \emph{implicit}
+notations – like $f(k)$ in the example – rather than the explicit
+form of the function in the main argumet it seems natural to call
+\texttt{vvmode=1} \emph{implicit }mode and \texttt{vvmode=0} (the
+default) \emph{explicit }mode. Most calculations are explicit mode
+– the vv-list is evaluated only once.\emph{ }
+
+\subsection{Changing the vv-list display format}
+
+\label{subsec:settingsVvDisplayChangeLocal}In previous formulas with
+variables the vv-list has been displayed following the result. It
+is wrapped in parentheses following a comma followed by a space. These
+formatting elements – comma, space, parentheses – can all be changed
+with the settings option.
+
+The default format specification is
+\begin{lyxcode}
+\{,\}\textbackslash mskip~12mu~plus~6mu~minus~9mu(vv)
+\end{lyxcode}
+for a text-style display (an inline formula) and
+\begin{lyxcode}
+\{,\}\textbackslash mskip~36mu~minus~24mu(vv)
+\end{lyxcode}
+in a display-style context. The commas are wrapped in braces because
+these are items in a comma-separated list. Both entries exhibit the
+elements: punctuation (comma), preceding a variable space, preceding
+the parenthesized vv-list (the \texttt{vv} placeholder). No full stop
+is inserted after the closing parentheses because the \texttt{\textbackslash eval}
+command may occur in the middle of a sentence (even in display style).
+For inline use, the elasticity of the space becomes relevant when
+\TeX{} is adjusting individual lines to fit sentences into paragraphs
+and paragraphs into pages. The largest spacing that can be stretched
+to is a quad, $18$~mu (mu~=~math unit), and the smallest that
+can be shrunk to is a thin space, $3$~mu. In display style,\texttt{
+}the largest spacing specified is the double quad, in line with the
+recommendation in \emph{The \TeX{} Book}, Chapter~18, but this can
+shrink to a single quad, for instance if the vv-list is heavily populated
+with variables so that the evaluated result is pushed well to the
+left by the vv-list. (But see below, §\ref{subsec:settings New-line-display}.)
+
+If you want to change these defaults, enter in the settings option
+\begin{lyxcode}
+vvi~=~<new~specification>
+\end{lyxcode}
+to change the inline display and
+\begin{lyxcode}
+vvd~=~<new~specification>
+\end{lyxcode}
+to change the display-style display For example the settings
+\begin{lyxcode}
+vvi~=~\{,\}\textbackslash quad(vv)
+
+vvd~=~\{,\}\textbackslash qquad(vv)
+\end{lyxcode}
+would give a comma (in braces since the settings option is a comma-separated
+list) and a fixed space (of one or two quads) between the result and
+the parenthesized vv-list.
+
+The vv-list itself in the display specification is represented by
+the placeholder \texttt{vv}. If the \texttt{vv} is omitted from the
+specification, then the vv-list will not appear at all:
+\begin{centred}
+\verb`\eval[vvi=?!]{$ \pi $}[\pi=3]` $\Longrightarrow$ \eval[vvi=?!]{$ \pi $}[\pi=3]
+\end{centred}
+More relevantly, it may well be the case that all variables in the
+vv-list are suppressed (wrapped in braces). In that case the display
+would look something like \verb`, ()`. To prevent this enter \verb`vvi=`
+in the vv-list, in the inline case, or \verb`vvd=` in the display-style
+case, i.e. enter an empty value. (Alternatively, use the star option
+of the \verb`\eval` command.)
+
+Another minor wrinkle occurs if you want to change parentheses around
+the vv-list to square brackets. Because the settings option is a square-bracket
+delimited argument, the square brackets in the specification will,
+like commas, need to be hidden in braces, although you can get away
+with braces around the whole spec.:
+\begin{lyxcode}
+vvi=\{,\textbackslash mskip~12mu~plus~6mu~minus~9mu~{[}vv{]}\}
+\end{lyxcode}
+
+\subsection{Displaying the vv-list on a new line }
+
+\label{subsec:settings New-line-display}Display of a long formula
+with many variables, hence a full vv-list, may not fit comfortably
+on a line. In an earlier example I used Brahmagupta's formula to calculate
+the area of a triangle. It squeezed onto a line. I shall now use his
+formula for the area of a cyclic quadrilateral:
+\[
+A=\sqrt{(s-a)(s-b)(s-c)(s-d)}.
+\]
+The extra side (quadrilateral as against triangle) means there is
+a further variable to accommodate, not only in the formula but also
+in the vv-list. In the following example, the cyclic quadrilateral
+is formed by a 45-45-90 triangle of hypotenuse 2 joined along the
+hypotenuse to a 30-60-90 triangle. The sides are therefore $\surd2,\surd2,\surd3,1$.
+Adding the areas of the two triangles, the area of the quadrilateral
+is $A=1+\tfrac{1}{2}\surd3$, or in decimal form, \verb`$\eval{1+\tfrac12\surd3}$`
+$\Longrightarrow$ $\eval{1+\tfrac12\surd3}$. Let's check with Brahmagupta's
+formula:
+\begin{verbatim}
+ \eval[p=.,vvd={,}\\(vv),*]
+ {\[ \sqrt{(s-a)(s-b)(s-c)(s-d)} \]}
+ [s=\tfrac12(a+b+c+d),
+ a=\surd2,b=\surd2,c=\surd3,d=1]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.,vvd={,}\\(vv),*]
+ {\[ \sqrt{(s-a)(s-b)(s-c)(s-d)} \]}
+ [s=\tfrac12(a+b+c+d),
+ a=\surd2,b=\surd2,c=\surd3,d=1]
+
+\noindent The values agree. The point to note here is the\texttt{
+vvd=\{,\}\textbackslash\textbackslash (vv)} and the \texttt{{*}}
+in the settings option. The \texttt{\textbackslash\textbackslash}
+in a specification for \texttt{vvd} acts as a trigger for \texttt{numerica}
+to replace whatever math delimiters are enclosed by the \verb`\eval`
+command with a \verb`multline` environment. As you can see, the specification
+inserts a comma after the formula and places the parenthesized vv-list
+on a new line. The star \texttt{{*}} if present suppresses equation
+numbering by turning the \verb`multline` into a \verb`multline*`
+environment.
+
+Things to note in the use of\texttt{ \textbackslash\textbackslash}
+in a \texttt{vvd} specification are that
+\begin{itemize}
+\item it applies only to the \texttt{vvd} specification, not the \texttt{vvi}
+spec.;
+\item it applies only when\emph{ }\verb`\eval`\emph{ wraps around }a math
+environment of some kind;
+\item it has no effect when the \verb`\eval` command is used \emph{within}
+a math environment when the presentation of the result is of the form
+\emph{result, vv-list}. The formula is not displayed and so the pressure
+on space is less and the `ordinary' vv-list specification is used.
+\end{itemize}
+
+\subsection{Punctuation}
+
+\label{subsec:settingsPunctuation}The \verb`\eval` command can be
+used within mathematical delimiters or it can be wrapped around mathematical
+delimiters. The latter gives a \emph{formula=result} style of display
+automatically, which is convenient. One doesn't need to write the
+\emph{formula=} part of the expression, but it causes a problem when
+\verb`\eval` wraps around a display-style or similar environment:
+how to display a following punctuation mark? For an inline display
+we can simply follow the \verb`\eval` command with the appropriate
+punctuation, for instance: \verb`\eval{$ 1+1 $}.` $\Longrightarrow$
+\eval{$ 1+1 $}. But with \verb`\[ \]` delimiters used \emph{within}
+the \verb`\eval` command – \verb`\eval{\[ 1+1 \]}.` – the fullstop
+slides off to the start of the next line, since it is beyond the closing
+delimiter. We want it to display as if it were the last element \emph{before}
+the closing delimiter.
+
+Explicitly putting it there – \verb`\eval{\[ 1+1. \]}` – means the
+punctuation mark becomes part of the formula. Potentially \texttt{numerica}
+then needs to check not just for a fullstop but also other possible
+punctuation marks like comma, semicolon, perhaps even exclamation
+and question marks. All these marks have roles in mathematics or \texttt{l3fp}.
+Including them in the formula means distinguishing their punctuation
+role from their mathematical role and can only cause difficulties
+(and code bloat).
+
+Instead, \texttt{numerica} uses the setting
+\begin{lyxcode}
+p~=~<char(s)>~
+\end{lyxcode}
+to place the \texttt{char(s)} after the result but within the environment
+delimiters. The default punctuation mark is the comma so that simply
+entering \texttt{p} will produce a comma in the appropriate place.
+This saves having to write \texttt{p=\{,\}} as would otherwise be
+required, since the settings option is a \emph{comma}-separated list.
+
+Nor is one limited to a single punctuation mark:
+\begin{centred}
+\verb`\eval[p=\ (but no 8!)]{\[ \frac{1}{81} \]}[9]` $\Longrightarrow$
+\eval[p=\ \text{(but no 8!)}]{\[ \frac{1}{81} \]}[9]
+\end{centred}
+
+\section{Infinite sums and products}
+
+\label{sec:settingsInfiniteSumsProds}There are ways of tweaking various
+default settings to nudge infinite sums and products to a correct
+limit. These tweaks are applied via the settings option of the \verb`\eval`
+command.
+
+\begin{table}
+\centering
+\noindent \centering{}\caption{Settings for infinite sums \& products}\label{tab:settingsSumsProducts}
+\noindent \begin{center}
+{\small{}%
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+key & type & meaning & default\tabularnewline
+\midrule
+\texttt{S+} & int & extra rounding for stopping criterion & \texttt{2}\tabularnewline
+\texttt{S?} & $\text{int}\ge0$ & stopping criterion query terms for sums & \texttt{0}\tabularnewline
+\texttt{P+} & int & extra rounding for stopping criterion & \texttt{2}\tabularnewline
+\texttt{P?} & $\text{int}\ge0$ & stopping criterion query terms for products & \texttt{0}\tabularnewline
+\bottomrule
+\end{tabular}}
+\par\end{center}
+\end{table}
+The normal convergence criterion used by \texttt{numerica} to determine
+when to stop adding/multiplying terms in an infinite sum/product is
+\emph{when the next term added/multiplied leaves the total unaltered
+when rounded to 2 more digits than the specified rounding value.}
+Suppose $T_{k}$ is the sum/product after the inclusion of $k$ terms,
+and $r$ is the rounding value. Denote $T_{k}$ rounded to $r$ figures
+by $\left(T_{k}\right)_{r}$. \emph{The infinite sum or product stops
+at the $(k+1)$-th term (and the value is attained at the $k$-th
+term) when }$\left(T_{k+1}\right)_{r+2}=\left(T_{k}\right)_{r+2}$.
+The hope is that if this is true at rounding value $r+2$ then at
+rounding value $r$ the series or product will have attained a stable
+value at that level of rounding.
+
+For a series of monotonic terms converging quickly to a limit, this
+stopping criterion works well, less so if convergence is slower, as
+seen earlier with the infinite product for $\sinh1$. The criterion
+can fail completely when terms behave in a non-monotonic manner. Distant
+terms of a Fourier series, for example, may take zero values; the
+criterion is necessarily satisfied but the series may still be far
+from its limit. In a product the equivalent would be a distant term
+taking unit value. Such series or products may also have initial `irregular'
+terms including zero/unit terms. A summation/product might stop after
+only one or two additions/multiplications if the criterion were applied
+to them.
+
+To cope with these possibilities, \texttt{numerica} offers two settings
+for sums, two for products, summarized in Table~\ref{tab:settingsSumsProducts}.
+These are entered in the settings option of the \verb`\eval` command.
+\begin{itemize}
+\item \texttt{S+=<integer> }or \texttt{P+=<integer>} additional rounding
+on top of the specified (or default) rounding for the calculation;
+default = $2$
+\begin{itemize}
+\item the larger the additional \texttt{<integer>} is, the more likely that
+sum or product has attained a stable value at the specified rounding
+$r$
+\end{itemize}
+\item \texttt{S?=<integer${}\,\mathtt{\ge0}$> }or \texttt{P?=<integer${}\,\mathtt{\ge0}$>}
+the number of final terms to query after the stopping criterion has
+been achieved to confirm that it is not an `accident' of particular
+values; default = $0$
+\begin{itemize}
+\item a final few terms to be summed/multiplied and the rounded result after
+each such operation to be compared with the rounded result at the
+time the stopping criterion was achieved. Suppose the additional rounding
+(\texttt{S+} or \texttt{P+}) is $n$ on top of the specified rounding
+$r$ and let the number of final checking terms be $m$. Suppose $T_{k_{0}}$
+is the first term at which the stopping criterion is achieved: $\left(T_{k_{0}}\right)_{r+n}=\left(T_{k_{0}+1}\right)_{r+n}$.
+What we require of the final query terms is that $\left(T_{k_{0}}\right)_{r+n}=\left(T_{k_{0}+1+j}\right)_{r+n}$
+for $j=0,1,\ldots,m$.
+\end{itemize}
+\end{itemize}
+Previously we found that the infinite product for $\sinh1$ with the
+default settings gave the wrong value, $0.174$, deficient by $1$
+in the last digit. We now have the means to tweak the stopping criterion
+by increasing the additional rounding:
+\begin{verbatim}
+ \eval[p,P+=3]{\[
+ \prod_{k=1}^{\infty}
+ \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+ \]}[x=1][3] \nmcInfo{prod}.
+\end{verbatim}
+\noindent $\Longrightarrow$ \noindent \eval[p,P+=3]{\[
+ \prod_{k=1}^{\infty}
+ \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+\]}[x=1][3] \nmcInfo{prod}.
+
+\noindent To obtain that last item of information (350 factors), I've
+anticipated a little and used the command \verb`\nmcInfo` with the
+argument \verb`prod`; see Chapter~\ref{sec:info}. The product now
+produces the correct three-figure value, but it takes $350$ factors
+to do so.
+
+Knowing how many terms or factors have been needed helps assess how
+trustworthy the result from an infinite sum or product is. For example,
+for the exponential series,
+\begin{verbatim}
+ \eval[p]{\[
+ \sum_{k=0}^\infty \frac1{k!}
+ \]}[9] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[\sum_{k=0}^\infty \frac1{k!} \]}[9] \nmcInfo{sum}.
+
+To $9$ places of decimals, using the default value \texttt{S+=2},
+the exponential series arrives at the right sum after only $15$ terms.
+Convergence is rapid. We can trust this result (and it is in fact
+the correct nine-figure value). By contrast, if we didn't know the
+value of $\sinh1$ beforehand, noting the number of factors required
+would make us justly cautious about accepting the result of the infinite
+product calculation.
+
+One way to gain confidence in a result is to choose a possibly unrealistic
+rounding value – say, the default $6$ for the infinite product \-\-
+then use \emph{negative} values for the extra rounding, \texttt{S+=-5},
+\texttt{S+=-4}, \ldots{} , so that the stopping criterion applies at
+rounding values $s$ of $6+(-5)=1$, $6+(-4)=2$, and so on, but the
+result is always presented to $6$ decimal places. One can then see
+how the $6$-figure results behave relative to the number of terms
+it takes to meet the stopping criterion. A little experimenting shows
+that for our infinite product for $\sinh1$ the number of factors
+$N_{s}$ at a stopping rounding value $s$ increases in geometric
+proportion with a scale factor of about $3$: $N_{s+1}/N_{s}\approx3$.
+For the exponential series on the other hand $N_{s}=4+s$, the number
+of terms increasing in direct proportion to the stopping rounding
+value.
+
+A similar calculation for the sum of inverse fourth powers of the
+integers $\zeta(4)=\sum_{n=1}^{\infty}\frac{1}{n^{4}}$, inverse third
+powers, $\zeta(3)$, and inverse squares, $\zeta(2)$, using \verb`\nmcInfo`
+to find how many terms are required at each stopping rounding value,
+shows that at least over the rounding value range $1$ to $8$, for
+inverse fourth powers $N_{s+1}/N_{s}\approx1.7$, for inverse third
+powers $N_{s+1}/N_{s}\gtrsim2$ and for inverse squares $N_{s+1}/N_{s}\approx3$.
+All are geometric rather than arithmetic progressions, but for inverse
+fourth powers the scale factor ($\approx1.7$) is sufficiently small
+that for these low values of $s$ the number of terms required does
+not grow too quickly. It is a standard result (Euler) that the series
+sums to $\pi^{4}/90$: \verb`$ \eval{ \pi^4/90 } $` $\Longrightarrow$
+$ \eval{ \pi^4/90 } $ to six places, and indeed, with the default
+\texttt{S+=2},
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{k=1}^\infty \frac1{k^4} \]}` $\Longrightarrow$
+\eval[p]{\[ \sum_{k=1}^\infty \frac1{k^4} \]}
+\end{centred}
+there is complete agreement.
+
+For inverse third powers, the number of terms required to reach the
+stopping criterion grows rapidly for rounding values from $7$ onwards
+($2^{7}=128$, $2^{8}=256$, \ldots ). This suggests trying for a
+five-figure result (with the default setting \texttt{S+=2} the stopping
+rounding value is $7$). Doing this gives a result $1.20205$ to five
+decimal places. \emph{HMF} Table 23.3 has this quantity tabulated
+to $20$ places and shows our result is too small by $1$ in the final
+figure.
+
+For inverse second powers, the number of terms required to reach the
+stopping criterion increase even more quickly: $3^{4}=81$, $3^{5}=243$,
+and so on. A three figure answer (with the default setting \texttt{S+=2}
+the stopping rounding value is $5$) seems the best we can hope for.
+Doing the evaluation gives $1.642$ whereas we know that $\zeta(2)=\pi^{2}/6$
+(Euler's famous result), evaluating to $1.645$. Even with \texttt{S+=3},
+the sum is still too small, $1.644$ after 1007 terms. Increasing
+the additional rounding to $4$, \texttt{S+=4}, does finally give
+the correct three-figure result, $1.645$, but only after summing
+$3180$ terms.
+
+\subsection{Premature ending of infinite sums}
+
+All the series considered so far have been monotonic. Trigonometric
+series will generally not be so, nor even single-signed.
+
+Trigonometric sums are computationally intensive and so, for the following
+example, I have specified a rounding value of 2. The series
+\[
+\sum_{n=1}^{\infty}\frac{4}{n^{2}\pi^{2}}(1-\cos n\pi)\cos2\pi nt
+\]
+is the Fourier series for the triangular wave function /\!\textbackslash\!/\!\textbackslash\!/\!\textbackslash\!/\!\textbackslash{} \ldots{}
+of period 1, symmetric about the origin where it takes its maximum
+value 1, crossing the axis at $t=0.25$ and descending to its minimum
+$-1$ at $t=0.5$, before ascending to a second maximum at $t=1$
+(and so on). In the interval $[0,0.5)$ the series should sum to $1-4t$.
+The problem is that the summand $\frac{4}{n^{2}\pi^{2}}(1-\cos n\pi)\cos2\pi nt$
+vanishes both when $n$ is even and when $4nt$ is an odd integer.
+If $t=0.1$ then $4nt$ is never an odd integer so the summand vanishes
+only for $n$ even, every second term. We expect the result to be
+$1-4\times0.1=0.6$.
+\begin{verbatim}
+ \eval[p]{\[
+ \sum_{n=1}^{\infty}
+ \frac{4}{n^{2}\pi^{2}}
+ (1-\cos n\pi)\cos2\pi nt
+ \]}[t=0.1][2] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[
+ \sum_{n=1}^{\infty}
+ \frac{4}{n^{2}\pi^{2}}
+ (1-\cos n\pi)\cos2\pi nt
+\]}[t=0.1][2] \info{sum}.
+
+\noindent Only one term? Of course – since the second term $n$ is
+even; the term vanishes and the stopping criterion is satisfied. The
+way around this problem is to query terms \emph{beyond} the one where
+the stopping criterion is achieved, i.e., to set \texttt{S?} to a
+nonzero value. We try \texttt{S?=1}:
+\begin{verbatim}
+ \eval[p,S?=1]{\[
+ \sum_{n=1}^{\infty}
+ \frac{4}{n^{2}\pi^{2}}
+ (1-\cos n\pi)\cos2\pi nt
+ \]}[t=0.1][2] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p,S?=1]{\[
+ \sum_{n=1}^{\infty}
+ \frac{4}{n^{2}\pi^{2}}
+ (1-\cos n\pi)\cos2\pi nt
+\]}[t=0.1][2] \info{sum}.
+
+Table~\ref{tab:settingsFinite-sums} lists the results of evaluating
+the \emph{finite }sums from $n=1$ to $N$ for values of $N$ around
+$65$. Since we have specified a rounding value of $2$ for the calculation,
+the stopping criterion\emph{ }applies at a rounding value of $2$
+more than that, $4$. Since $N=64$ is even, the summand for the $64$th
+term is zero and the sum takes the same value as for $N=63$. The
+$65$th term is the query term and the sum differs, so the summation
+continues. The $66$th term vanishes, so the stopping criterion is
+met. This time for the query term, the $67$th, the sum retains the
+same $4$-figure value, and the summation stops. The result was attained
+at the $65$th term.
+
+\begin{wraptable}{o}{0.5\columnwidth}%
+\centering{}\vspace{-4ex}
+\caption{Finite sums}\label{tab:settingsFinite-sums}
+\setlength{\abovetopsep}{1ex}%
+\begin{tabular}{cc}
+\toprule
+$N$ & $\Sigma$\tabularnewline
+\midrule
+$63$ & $0.6001$\tabularnewline
+$64$ & $0.6001$\tabularnewline
+$65$ & $0.5999$\tabularnewline
+$66$ & $0.5999$\tabularnewline
+$67$ & $0.5999$\tabularnewline
+\bottomrule
+\end{tabular}\end{wraptable}%
+Should we be confident in the result? Increase the number of query
+terms to $3$ (there is no point in increasing \texttt{S?} to $2$
+because of the vanishing of the even terms), the sum stops after $113$
+terms, with the same $0.6$ result. Indeed, increasing \texttt{S?}
+to $5,7,\ldots$ makes no difference. It still takes $113$ terms
+to reach the stable two-figure result $0.6$.
+
+For a final example, consider the error function
+\[
+\erf z=\dfrac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt
+\]
+which can also be rendered as an infinite sum (\emph{HMF }7.1.5):
+\[
+\erf z=\sum_{n=0}^{\infty}(-1)^{n}\frac{z^{2n+1}}{n!(2n+1)}.
+\]
+(\verb`\erf` expanding to \verb`erf` has been defined in the preamble
+to this document using \verb`\DeclareMathOperator`.) We calculate
+this sum for $z=2$ to $10$ places of decimals. Although this is
+an alternating series, it is obvious that the summand never vanishes
+when $z\ne0$ as here. Hence there seems no need to change the default
+value \texttt{S?=0}.
+\begin{verbatim}
+ \eval[p]{\[
+ \frac2{\sqrt{\pi}}
+ \sum_{n=0}^\infty(-1)^n
+ \frac{z^{2n+1}}{n!(2n+1)}
+ \]}[z=2][10*] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[
+ \frac2{\sqrt{\pi}}
+ \sum_{n=0}^\infty(-1)^n
+ \frac{z^{2n+1}}{n!(2n+1)}
+\]}[z=2][10*] \nmcInfo{sum}.
+
+According to \emph{HMF }Table 7.1, this calculated value of $\erf2$
+is correct to all $10$ places. But beyond $z=2$ errors will begin
+to interfere with the result. Note that $26$ terms means $n=26$
+was the last value of $n$ for which the summand was evaluated. (The
+sum stops at the $26$th term, $n=25$, but the next term $n=26$
+needs to be calculated for the stopping criterion.) Fortuitously,
+$2^{2\times26+1}=2^{53}$ is the greatest power of $2$ that can be
+\emph{exactly} rendered to the $16$ significant figures that \texttt{l3fp}
+uses. But $n!$ exceeds the $16$-significant figure limit of \texttt{l3fp}
+when $n>21$, so despite the 10-figure result, errors have already
+begun to occur in the denominator of the summand and accrue in the
+sum when $z=2$. For larger $z$ values the errors can only get worse
+and at some point will render the calculated value worthless at any
+meaningful rounding value. For example, when $z=7$ the sum apparently
+`evaluates' to over $929$ whereas we know that
+\[
+\erf z<\dfrac{2}{\sqrt{\pi}}\int_{0}^{\infty}e^{-t^{2}}dt=1.
+\]
+
+
+\subsection{Double sums or products}
+
+Sums or products can be iterated. For instance, the exponential function
+can be calculated this way:
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{k=0}^{\infty}\prod_{m=1}^{k}\frac{x}{m} \]}[x=2]`
+$\Longrightarrow$ \eval[p]{\[ \sum_{k=0}^{\infty}\prod_{m=1}^{k}\frac{x}{m} \]}[x=2]
+\end{centred}
+\noindent which is \verb`\eval{$ e^2 $}` $\Longrightarrow\eval{\ensuremath{e^{2}}}$.
+
+A second example is afforded by Euler's transformation of series (\emph{HMF}
+3.6.27). To calculate $e^{-1}$ we use
+\begin{verbatim}
+ \eval[p={,}\quad \mbox{\nmcInfo{sum}}.]
+ {\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p={,}\quad \mbox{\nmcInfo{sum}}.]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+
+(Note the placement of the information command as the value of the
+punctuation key. This keeps it \emph{within }the \texttt{\textbackslash{[}
+\textbackslash{]}} delimiters.) Following Euler, this series can
+be transformed to the form
+\begin{verbatim}
+ \eval[p,S?=1]{\[
+ \sum_{k=0}^\infty \frac{(-1)^k}{2^{k+1}}
+ \sum_{n=0}^k(-1)^n\binom kn \frac1{(k-n)!}
+ \]}[3] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p,S?=1]{\[ \sum_{k=0}^\infty \frac{(-1)^k}{2^{k+1}}\sum_{n=0}^k(-1)^n\binom kn \frac1{(k-n)!} \]}[3] \nmcInfo{sum}.
+
+\noindent Note the setting \verb`S?=1`. Without it, the summation
+stops after $1$ term, the $k=0$ term, because the $k=1$ term vanishes.
+With \verb`S?=1` it takes $16$ terms of the \emph{outer }sum to
+reach the stopping criterion. Since that sum starts at $0$, that
+means that changing the upper limit from $\infty$ to $15$ should
+give the same result – which it does – but it takes $\tfrac{1}{2}\times16\times17=136$
+terms in total to get there, to be compared with the $9$ terms of
+the earlier simpler sum, and the terms are more complicated. Obviously
+such double sums are computationally intensive.
+
+\section{Changing default values}
+
+\label{sec:settingsDefaults}The settings option enables various settings
+to be changed for an individual calculation. You may find yourself
+wanting to make such changes sufficiently often that a change of default
+value is a better plan than encumbering each calculation with a list
+of settings.
+
+\begin{table}[h]
+\centering
+\noindent \centering{}\caption{Default values, \texttt{\textbackslash eval} command}\label{tab:settingsDefaults}
+\noindent \begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small value}\tabularnewline
+\midrule
+{\small rounding} & {\small\texttt{6}}\tabularnewline
+{\small pad} & {\small\texttt{0}}\tabularnewline
+{\small output-sci-notation } & {\small\texttt{0}}\tabularnewline
+{\small output-exponent-char} & {\small\texttt{e}}\tabularnewline
+{\small\%} & \tabularnewline
+{\small input-sci-notation } & {\small\texttt{0}}\tabularnewline
+{\small input-exponent-char} & {\small\texttt{e}}\tabularnewline
+{\small multitoken-variables } & {\small\texttt{1}}\tabularnewline
+logarithm-base & {\small\texttt{10}}\tabularnewline
+{\small vv-display} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{\small vv-inline} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\%} & \tabularnewline
+{\small intify-rounding} & {\small\texttt{14}}\tabularnewline
+\% & \tabularnewline
+{\small sum-extra-rounding} & {\small\texttt{2}}\tabularnewline
+{\small sum-query-terms} & {\small\texttt{0}}\tabularnewline
+{\small prod-extra-rounding} & {\small\texttt{2}}\tabularnewline
+{\small prod-query-terms} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+The way to do that is to create a \emph{configuration file }with the
+name \texttt{numerica.cfg} in a text editor. Its entries, one per
+line, are of the form \emph{key=value} followed by a comma, and for
+clarity preferably one entry per line (although this is not essential).The
+key names are noticeably more verbose than the corresponding keys
+of the settings option.\emph{ }The possible keys are listed in Table~\ref{tab:settingsDefaults},
+together with their current default values.
+
+Keys taking one of two possible values, \verb`0` (for \verb`false/off`)
+or \verb`1` (for \verb`true/on`), are \verb`pad` (the result with
+zeros), \verb`output-sci-notation`, \verb`input-sci-notation`, and
+(check for) \verb`multitoken-variables`.
+
+The table is divided into four parts.
+\begin{itemize}
+\item The top four rows concern elements that can be changed for individual
+calculations with the trailing optional argument of \verb`\eval`:
+rounding, padding with zeros, and outputting in scientific notation;
+see §\ref{subsec:evalRoundingEtc}.
+\begin{itemize}
+\item Note that to output the result always in scientific notation requires
+two settings, first setting \texttt{output-sci-notation} to \texttt{1},
+and then choosing a character to act as the exponent marker. Because
+\texttt{l3fp} uses \texttt{e} for this character, \texttt{numerica}
+has made \texttt{e} its default. But this option is turned off by
+default (hence the \texttt{0} against this key).
+\end{itemize}
+\item The next block of rows concern general elements that can be changed
+for individual calculations with the settings option of \verb`\eval`;
+see §\ref{sec:settingsOption}. Obviously the key names are more expansive
+in the present context but the effect is the same.
+\begin{itemize}
+\item But note that to input numbers in scientific notation requires two
+settings, first setting \texttt{input-sci-notation} to \texttt{1},
+and then choosing a character to act as the exponent marker. Because
+\texttt{l3fp} uses \texttt{e} for this character, \texttt{numerica}
+has made \texttt{e} its default. The option is turned off by default
+(hence the \texttt{0} against this key).
+\end{itemize}
+\item The third block is a single row specifying at what rounding value
+a floating point should be considered an integer; see §\ref{subsec:defaultsIntifyingRounding}
+below.
+\item The last four rows concern default settings for infinite sums and
+products. These correspond to the keys \texttt{S+}, \texttt{S?} and
+\texttt{P+}, \texttt{P?} of the settings option that can be used to
+tweak the behaviour of the stopping criterion for such sums or products;
+see §\ref{sec:settingsInfiniteSumsProds}.
+\end{itemize}
+If you are dissatisfied with any of the default values listed, then
+in a text editor create a new file called \texttt{numerica.cfg} and
+assign \emph{your} values to the relevant keys. For instance, if you
+find yourself working to $4$ figures, that rounding to $6$ is too
+many, then make the entry \texttt{rounding=}4. If also you want results
+always presented in proper scientific notation,\emph{ $d.d_{1}d_{2}d_{3}d_{4}\times10^{n}$},
+then add a comma after \texttt{4} and enter on a new line (recommended
+but not strictly necessary; the comma is the crucial thing), \texttt{output-sci-notation=1,}
+(note the comma) and on another new line, \texttt{output-exponent-char=x}.
+
+Perhaps you also want a non-zero setting for the final query terms
+for infinite sums and products. This makes sense if you are largely
+dealing with non-monotonic series – like Fourier series. Even the
+Euler transformation of the exponential series for $e^{-1}$ discussed
+above required a non-zero \texttt{S?}. If you wish to make this change
+then add a comma and on a new line add (for instance) \texttt{sum-query-terms
+= 1,} and again on a new line, \texttt{prod-query-terms = }1. If this
+is all you wish to change, then no comma is necessary after this final
+entry. Your newly created file should look something like
+\begin{lyxcode}
+rounding~~~~~~~~~~~~~=~4,
+
+output-sci-notation~~=~1,
+
+output-exponent-char~=~x,
+
+sum-query-terms~~~~~~=~1,
+
+prod-query-terms~~~~~=~1
+\end{lyxcode}
+The white spacing may be different; white space is ignored by \texttt{numerica}
+when reading the file. Using it to align the equals signs helps \emph{us}
+read the file. Note that the last entry, because it is the last entry,
+lacks a comma. Now save the file with the name \texttt{numerica.cfg}.
+This file will be read by \texttt{numerica} near the end of its loading
+process. These settings will be \texttt{numerica}'s defaults for the
+relevant keys.
+
+\subsection{Location of \texttt{numerica.cfg}}
+
+Save, yes, but where to? If the new settings are likely to apply only
+to your current document, then the document's directory is a sensible
+place to put it and \texttt{numerica} will certainly find it there
+since it is part of \LaTeX 3 file handling that file searches are
+not limited to the \TeX{} distribution (including your personal texmf
+tree) but also include the current document directory. But what happens
+when you start working on another document? Will you remember to copy
+\texttt{numerica.cfg} to its new location? That is why your \emph{personal
+texmf tree} is a better place.
+
+\subsubsection{Personal texmf tree? }
+
+This is a directory for `waifs and strays' of the \TeX{} system that
+are not included in the standard distributions like MiK\TeX{} or \TeX Live.
+Here you place personal packages designed for your own particular
+circumstances. These may include your own \TeX{} or \LaTeX{} package,
+say \texttt{mypackage.sty}, achieving some small or singular effect
+that doesn't warrant wider distribution on CTAN. Here you might place
+configuration files for other packages with your preferences (unless
+the package requires some specific location). Here you can put your
+personal bibliography files.
+
+Your personal texmf tree is structured like the standard MiK\TeX{}
+or \TeX Live hierarchy but placed in another location so that there
+is no chance of its being overwritten when packages in MiK\TeX{} or
+\TeX Live are updated. But these distributions need to be alerted
+to its existence.
+
+For example, in the MiK\TeX{} console, click on \textsf{Settings},
+and then on the \textsf{Directories} tab of the resulting dialog.
+Here you get to add your personal texmf hierarchy to the list of paths
+that MiK\TeX{} searches, by clicking on the \textsf{+} button, browsing
+to your texmf folder and selecting it. By using the up and down arrow
+keys that the MiK\TeX{} console provides, ensure that it lies \emph{above
+}the the entry for the main MiK\TeX{} tree. That way, files in your
+personal texmf tree will be found first and loaded. Now go to the
+\textsf{Tasks} menu and click on \textsf{Refresh the filename database}.
+This will let MiK\TeX{} know what is held in your personal texmf tree.
+Files there can then be used like standard \LaTeX{} packages.
+
+\subsection{Rounding in \textquoteleft int-ifying\textquoteright{} calculations}
+
+\label{subsec:defaultsIntifyingRounding}Factorials, binomial coefficients,
+summation and product variables, and (in \texttt{numerica}) $n$-th
+roots from the \texttt{\textbackslash sqrt} command, all require
+integer arguments. These integers may indeed be entered explicitly
+as integers, but they can also be determined as the result of a calculation.
+Rounding errors may mean the result is not an exact integer. How much
+leeway should be allowed before it is clear that the calculation did
+not give an integer result? In the default setup, \texttt{numerica}
+is generous. A number is considered an integer if it rounds to an
+integer when the rounding value is $14$. Since \texttt{l3fp} works
+to $16$ significant figures this provides more than enough `elbowroom'
+for innocuous rounding errors to be accommodated. If a calculation
+does not round to an integer at a rounding value of $14$ then it
+seems reasonable to conclude that it has \emph{really }not given an
+integer answer, not just that rounding errors have accumulated. If
+you want to change this `int-ifying' value for a particular calculation,
+then add a line to \texttt{numerica.cfg} like
+\begin{lyxcode}
+intify-rounding~=~<integer>
+\end{lyxcode}
+Since \texttt{l3fp} works to $16$ significant figures, values of
+\texttt{integer} greater than $16$ are pointless. Generally int-ifying
+rounding values will be less than but close to $16$ (although when
+testing the code I used some ridiculous values like $3$ or $4$).
+If other entries follow this one in the file, then conclude the line
+with a comma.
+
+\section{Parsing mathematical arguments}
+
+\label{sec:Argument-parsing} A main aim of the \texttt{numerica}
+package is to require minimal, preferably no, adjustment to the \LaTeX{}
+form in which an expression is typeset in order to evaluate it. But
+mathematicians do not follow codified rules of the kind programming
+languages insist on when writing formulas – like parenthesizing the
+arguments of functions, or inserting explicit multiplication signs
+({*}) between juxtaposed terms. Hence the question of where the arguments
+of mathematical functions end is acute. For a few functions \LaTeX{}
+delimits the argument: think of \verb`\sqrt`, \verb`\frac`, \verb`\binom`;
+also \verb`^`. But for functions like \verb`\sin` or \verb`\tanh`
+or \verb`\ln`, unary functions, this is not so. Nor is it for sums
+and products, and comparisons.
+
+Before discussing the parsing rules for different groups of functions,
+I discuss the means \texttt{numerica} provides to handle exceptions
+to those rules, when one \emph{does} need to make some adjustment
+to a formula.
+
+\subsection{The cleave commands \texttt{\textbackslash q} and \texttt{\textbackslash Q}}
+
+\label{subsec:parseCleave}The word \emph{cleave} has two opposed
+meanings: to adhere or cling to, and to split or sever. \texttt{numerica}
+defines two commands, \verb`\q` and \verb`\Q` to achieve these opposite
+effects. When a mathematical argument is being parsed, the \verb`\q`
+command joins the next token to the argument (\emph{cleaves to});
+the \verb`\Q` command severs the next token from the argument (\emph{cleaves
+apart}). Neither command is added to the argument nor leaves a visible
+trace in the output.
+
+Thus, without \verb`\q`,
+\begin{centred}
+\verb`\eval{$ \sin(n+\tfrac12)(x-t) $}[n=3,x=t+\pi,t=1.234]` $\Longrightarrow$
+\eval{$ \sin(n+\tfrac12)(x-t) $}[n=3,x=t+\pi,t=1.234],
+\end{centred}
+which is $(\sin\tfrac{7}{2})\times\pi$. With \verb`\q` between the
+bracketed factors,
+\begin{centred}
+\verb`\eval{$ \sin(n+\tfrac12)\q(x-t) $}[n=3,x=t+\pi,t=1.234]` $\Longrightarrow$
+\eval{$ \sin(n+\tfrac12)\q(x-t) $}[n=3,x=t+\pi,t=1.234],
+\end{centred}
+which is $\sin(\tfrac{7}{2}\pi)$. Similarly, without \verb`\q`,
+\begin{centred}
+\verb`\eval[p]{\[ \cos\frac{2\pi}{T}n(t+\tfrac12T) \]}[T=2,t=1,n=3]`
+$\Longrightarrow$ \eval[p]{\[ \cos\frac{2\pi}{T}n(t+\tfrac12T) \]}[T=2,t=1,n=3]
+\end{centred}
+which is $(\cos\pi)\times3\times(1+\tfrac{1}{2}\times2)$. With \verb`\q`
+used twice, once after the fraction and once before the left parenthesis,
+\begin{centred}
+\verb`\eval[p]{\[ \cos\frac{2\pi}{T}\q n\q(t+\tfrac12T) \]}[T=2,t=1,n=3]`
+$\Longrightarrow$ \eval[p]{\[ \cos\frac{2\pi}{T}\q n\q(t+\tfrac12T) \]}[T=2,t=1,n=3]
+\end{centred}
+which is $\cos(\pi\times3\times2)$.
+
+It should be noted that for \emph{trigonometric }functions, because
+of their use in Fourier series especially, there is another way of
+handling arguments with parentheses (and fractions). This is discussed
+in §\ref{subsec:parseTrigFns} below.
+
+For the \verb`\Q` command which splits an argument we have, without
+it,
+\begin{centred}
+\verb`\eval{$ 1/2e $}` $\Longrightarrow$ \eval{$ 1/2e $},
+\end{centred}
+which is the reciprocal of $2e$, whereas with the \verb`\Q` command
+inserted before \verb`e`,
+\begin{centred}
+\verb`\eval{$ 1/2\Q e $}` $\Longrightarrow$ \eval{$ 1/2\Q e $},
+\end{centred}
+which is a half of $e$. Of course, the meaning in this example would
+be clearer if $1/2$ were parenthesized or presented as a \verb`\tfrac`.
+
+\subsubsection{Mnemonic}
+
+As mnemonic, best seen in sans serif for the Latin Modern fonts used
+in this document, think of the letter \textsf{q} as a circle \emph{clinging
+}to a vertical descender; think of the letter \textsf{Q} as a circle
+\emph{cut} by the diagonal stroke.
+
+\subsection{Parsing groups}
+
+The arguments of different groups of functions are handled in different
+ways. The criterion used for deciding when an argument ends for one
+group will not be that used for the others. Table §\ref{subsec:defaultsIntifyingRounding}
+lists the different groups that \texttt{numerica} takes account of.
+At the top are functions or operations that have the smallest reach
+when determining where their arguments end; at the bottom are operations
+that have the greatest reach. The denominator of a slash fraction
+is treated as a unary function and is assigned to group II. By default
+trigonometric functions are treated the same as other unary functions
+but there is a setting which enables the direct (rather than inverse)
+trigonometric functions to accept a wider range of arguments, as occurs
+in Fourier series. Hence they are separated into their own group.
+
+\begin{wraptable}{o}{0.5\columnwidth}%
+\noindent \centering{}\caption{Parsing groups}\label{tab:settingsParsing-groups}
+{\ttfamily\noindent \begin{center}
+{\ttfamily{}%
+\begin{tabular}{ll}
+\toprule
+{\small\textrm{group}} & {\small\textrm{function/operation}}\tabularnewline
+\midrule
+{\small\textrm{I}} & {\small\textrm{surd, logical Not}}\tabularnewline
+{\small\textrm{II}} & {\small\textrm{unary functions, /}}\tabularnewline
+{\small\textrm{III}} & {\small\textrm{direct trig. functions}}\tabularnewline
+{\small\textrm{IV}} & {\small\textrm{sums, products}}\tabularnewline
+{\small\textrm{V}} & {\small\textrm{comparisons}}\tabularnewline
+{\small\textrm{VI}} & {\small\textrm{logical And, logical Or}}\tabularnewline
+\bottomrule
+\end{tabular}}
+\par\end{center}}\end{wraptable}%
+ A formula is a sequence of tokens and brace groups. All parsing occurs
+from the left, \LaTeX{} argument by \LaTeX{} argument, where \emph{argument}
+means either a token (an N-type argument in \verb`expl3`-speak) or
+a brace group (an n-type argument). To distinguish \LaTeX{} arguments
+from mathematical arguments I shall when necessary refer to L-args
+and M-args. A mathematical argument may end \emph{at} an L-arg, meaning
+immediately before the L-arg, or end \emph{with} the L-arg, meaning
+immediately after the L-arg. Ending or not will in general depend
+on whether the argument is in \emph{first position} – the position
+immediately following a function token like \verb`\sin` or \verb`\log`
+– or in \emph{general position} – any later position (although for
+trigonometric functions we will also need to consider \emph{second}
+and even \emph{third }position).
+
+For counting position, we need to allow for formatting elements and
+multi-token numbers – in both decimal and scientific formats. Formatting
+elements do not change the position count. This applies to things
+like thin spaces or phantoms (and their arguments) or modifiers like
+\verb`\left` or \verb`\biggl`. Multi-token numbers (in decimal or
+scientific formats) are treated as single items; they advance the
+position count by exactly one. \LaTeX{} functions – like \verb`\frac`
+– which take \LaTeX{} arguments again advance the position count only
+by one. Mathematically, the fraction is viewed as a single unit.
+
+I shall refer to a token or a token and its \LaTeX{} arguments – like
+\verb`\frac` and its arguments – as an \emph{item}. Similarly, a
+(possibly multi-token) number is an item. Also it will help to distinguish
+tokens within brackets where both brackets lie to the right of a function
+from those that do not. The former I call \emph{clothed}; the latter
+are \emph{naked}. Thus the plus sign in $(\sin x+y)$ is naked relative
+to the sine (one bracket to the left of the function), but is clothed
+in $\sin(x+y)$ (both brackets to the right of the function).
+
+\subsubsection{Parsing group I}
+
+The only functions in this category are the surd and logical Not.
+
+Why distinguish the surd from other unary functions? Surely we all
+agree that \verb`\sin2\pi`, displaying as $\sin2\pi$, vanishes?
+The argument of the sine extends beyond the $2$ to include the $\pi$.
+But \verb`\surd2\pi`, displaying as $\surd2\pi$, is understood to
+be the product $\surd2\times\pi$. The argument of the surd ends with
+the $2$. The surd binds more tightly to its argument than is true
+of unary functions generally.
+
+For parsing group I
+\begin{enumerate}
+\item if a left bracket is in first position, the mathematical argument
+ends with the matching right bracket; otherwise
+\item the argument ends with the item in first position and any L- or M-args
+required by that item.
+\end{enumerate}
+If the factorial sign \verb`!` \emph{preceded} its argument, it too
+would belong to this parsing state, for it also binds tightly like
+the surd. This means that an expression like $\surd4!$ is intrinsically
+ambiguous. Is it the square root of $24$ or the factorial of $2$?
+In \texttt{numerica} it produces the (perhaps rather odd) error
+\begin{centred}
+\verb`\eval{$ \surd 4! $}` $\Longrightarrow$ \eval{$ \surd4! $}
+\end{centred}
+The surd has seized the argument; there is nothing for the factorial
+to operate on. The same error arises if the $4$ is parenthesized,
+but parenthesizing like either \verb`(\surd 4)!` or \verb`\surd(4!)`
+repairs the situation. Because other unary functions (like the sine
+or logarithm) do not bind as tightly, this ambiguity does not arise
+for them.
+
+Exponents cause no problem because taking square roots and raising
+to a power are commutative operations – the result is the same whichever
+is performed first.
+\begin{centred}
+\verb`\eval{$ \surd 3^4 $}` $\Longrightarrow$ \eval{$ \surd 3^4 $}.
+\end{centred}
+
+\subsubsection{Parsing group II: unary functions, slash fractions}
+
+In the default setup this category includes the trigonometric and
+hyperbolic functions, their inverses, the various logarithms and the
+exponential functions, the signum function \verb`\sgn`, and the slash
+fraction \verb`/` where the argument to be determined is its denominator.
+Note however that there is a setting switch which enables trigonometric
+ functions to handle parentheses in arguments more generally; see
+§\ref{subsec:parseTrigFns}.
+\begin{itemize}
+\item In parsing group II we wish to accommodate usages like $\ln z^{n}=n\ln z$
+(\emph{HMF} 4.1.11), or $\gd z=2\arctan e^{z}-\frac{1}{2}\pi$ (\emph{HMF}
+4.3.117), defining the Gudermannian. The exponent is included in the
+argument. Considering $\ln(1+1/n)^{n}$ exponents must also be part
+of parenthesized arguments.
+\item An approximation to Stirling's formula for the factorial is often
+written $\ln N!\approx N\ln N-N$ (widely used in texts on statistical
+mechanics). Hence the factorial sign should also be considered part
+of the argument.
+\item $\ln xy=\ln x+\ln y$ means the argument must reach over a product
+of variables. Identities like $\sin2z=2\sin z\cos z$ mean the argument
+also reaches over numbers, and expressions like $\sin\tfrac{1}{2}\pi x$
+(\emph{HMF} 4.3.104) mean that it further reaches over \verb`\tfrac`-s
+and constants.
+\item Essentially \emph{anything }can be in first position, and without
+parentheses; e.g.
+\begin{itemize}
+\item unary functions: $\ln\ln z$ (\emph{HMF} 4.1.52), $\ln\tan\dfrac{z}{2}$
+(\emph{HMF} 4.3.116),
+\item fractions: $\ln\dfrac{z_{1}}{z_{2}}$ (\emph{HMF} 4.1.9), $\arcsin\dfrac{(2ax+b)}{(b^{2}-4ac)^{1/2}}$
+(\emph{HMF} 3.3.36), $\ln\dfrac{\tan z}{z}$ (\emph{HMF} 4.3.73),
+\item absolute values: $\ln\abs*{\dfrac{a+x}{a-x}}$ (\emph{HMF} 3.3.25),
+\item square roots: $\arctan\sqrt{\dfrac{\nu_{1}}{\nu_{2}}F}$ (\emph{HMF
+}26.6.8)
+\end{itemize}
+\end{itemize}
+With these examples in mind, for parsing group II
+\begin{enumerate}
+\item if a left bracket is in first position, the mathematical argument
+ends with the matching right bracket and any attached exponent, or
+factorial or double factorial sign; otherwise
+\item the mathematical argument includes the item in first position and
+any L- or M-args required by that item;
+\begin{enumerate}
+\item if the item in first position is a number, variable, constant or \verb`\tfrac`
+\begin{enumerate}
+\item the argument appends the next item if it is a number, variable, constant
+or \verb`\tfrac`, and so on recursively; or
+\item the argument appends the next item if it is an exponent, or facorial
+or double factorial sign, and ends there; otherwise
+\item the argument ends.
+\end{enumerate}
+\item if the item in first position is not a number, variable, constant
+or \verb`\tfrac`
+\begin{enumerate}
+\item the argument appends the next item if it is an exponent, or factorial
+or double factorial sign, and ends there; otherwise
+\item the argument ends.
+\end{enumerate}
+\end{enumerate}
+\end{enumerate}
+An argument may extend over (see 2(a)i) numbers, constants, variables
+and \verb`\tfrac`-s: $\sin2\tfrac{p}{q}\pi x$ exhibits all elements.
+
+Illustrating 1, the exponent is included in the argument but not the
+following variable:
+\begin{centred}
+\verb`\eval{$ \log_{10}(1+2+3+4)^3n $}[n=5]` $\Longrightarrow$ \eval{$ \log_{10}(1+2+3+4)^3n $}[n=5].
+\end{centred}
+For the sake of the reader, and as one naturally does in any case
+to avoid ambiguity, the formula should be written with the variable
+$n$ preceding the logarithm: $n\log_{10}(1+2+3+4)^{3}$. The way
+the example is written suggests that the writer wished the $n$ to
+be considered part of the argument. If that is the case, an outer
+set of parentheses would make intentions clear, but it is possible
+to leave the argument as written but insert a \verb`\q` command before
+$n$:
+\begin{centred}
+\verb`\eval{$ \log_{10}(1+2+3+4)^3\q n $}[n=5]` $\Longrightarrow$
+\eval{$ \log_{10}(1+2+3+4)^{3}\q n $}[n=5],
+\end{centred}
+which is $\log_{10}5000.$
+
+Illustrating 2(a)ii, again the exponent is included in the argument
+but not the following variable:
+\begin{centred}
+\verb`\eval{$ \log_{10}m^3n $}[m=10,n=5]` $\Longrightarrow$ \eval{$ \log_{10}m^3n $}[m=10,n=5].
+\end{centred}
+Again, for the sake of the reader and as one naturally does to avoid
+ambiguity, the variable $n$ should precede the logarithm. If in fact
+one wants the $n$ included in the argument of the logarithm, the
+\verb`\q` command is again available or, better in this case, the
+$n$ can be shifted to precede the $m$, which illustrates 2(a)i:
+\begin{centred}
+\verb`\eval{$ \log_{10}nm^3 $}[m=10,n=5]` $\Longrightarrow$ \eval{$ \log_{10}nm^3 $}[m=10,n=5].
+\end{centred}
+Is \texttt{numerica} being too strict when $nm^{3}$ is included in
+the argument of the logarithm, but $m^{3}n$ is not? Any criterion
+is going to miss some instances where a different outcome might be
+desirable. Where an argument ends is affected by visual appearance.
+It is simple and easy to remember if it is understood that anything
+that breaks the visual appearance of juxtaposed numbers, variables,
+constants and \verb`\tfrac`-s ends the argument. An exponent does
+just that.
+
+Illustrating 2(b)ii, the argument stops with the \verb`\dfrac` and
+its arguments and does not extend to the following constant:
+\begin{centred}
+\verb`\eval{$ \sin\dfrac12\pi $}` $\Longrightarrow$ \eval{$ \sin\dfrac12\pi $}.
+\end{centred}
+Obviously, someone writing an expression like this intends the $\pi$
+to be part of the argument. In that case, a \verb`\tfrac` should
+be used. The \verb`\dfrac` breaks the `visual flow' of an argument.
+\begin{description}
+\item [{Fractions}]~
+
+But why not a plain \verb`\frac`? After all, for an inline expression
+it displays in the same way as a \verb`\tfrac`. I considered making
+the argument-behaviour of \verb`\frac` the same as \verb`\tfrac`
+for text-style contexts, and the same as \verb`\dfrac` for display-style
+contexts, but that would have meant the same expression evaluating
+to different results depending on the context, text-style or display-style,
+which ruled it out. Because \verb`\frac` sometimes displays as \verb`\dfrac`,
+it necessarily is treated like \verb`\dfrac` (but see §\ref{subsec:parseTrigFns},
+specifically \texttt{()=2}).
+\item [{Slash~fractions}]~
+
+It is easy to write ambiguous expressions using the slash $/$ to
+indicate fractions or division. How should $\pi/2n$ be interpreted?
+With from-the-left evaluation and calculator precedence rules which
+give equal precedence to {*} (multiplication) and / (division), this
+would be interpreted as $(\pi/2)\times n$, but most people will instinctively
+interpret it as $\pi/(2n)$. By placing \verb`/` in parsing group
+II, this is what \texttt{numerica} does.
+
+It treats the right-hand argument of the slash \emph{as if it were
+the argument of a named function}. This means that $1/2\sin(\pi/6)$
+is parsed as $(1/2)\sin(\pi/6)$ rather than as $1/(2\sin(\pi/6))$.
+It also means that $1/2\exp(1)$ and $1/2e$ give different results,
+which is acceptable since (in the author's view) they display differently
+and are not instinctively read in the same way.
+\end{description}
+
+\subsubsection{Parsing group III}
+
+\label{subsec:parseTrigFns}By default trigonometric functions are
+set to parsing group II. This accommodates many instances of how arguments
+are used with these functions, but Fourier series in particular require
+more. For them we need to take account of how \emph{parentheses} are
+used in arguments. I find $\tan\tfrac{1}{2}(A+B)$ (\emph{HMF }4.3.148),
+$\sec\pi(\tfrac{1}{4}+\tfrac{1}{2}az)$ (\emph{HMF }19.3.3), $\cos(2m+p)z$
+(\emph{HMF }20.2.3), $\sin(2n+1)v$ (\emph{HMF }16.38.1). Looking
+through various texts discussing Fourier series it is easy to find
+examples like
+\[
+\cos\frac{2\pi}{T}nt,\quad\cos\frac{2\pi}{T}n(t+\tfrac{1}{2}T),
+\]
+and
+\[
+\cos(N+\tfrac{1}{2})\frac{2\pi\tau}{T},\quad\sin2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right).
+\]
+In the last of these \verb`\left` and \verb`\right` have been used
+to enlarge the parentheses.
+
+All these usages can be accommodated by adjusting a setting in the
+settings option (§\ref{sec:settingsOption}) of the \verb`\eval`
+command:
+\begin{lyxcode}
+()~=~integer
+\end{lyxcode}
+where \texttt{integer} is one of \texttt{0, 1, 2}. For convenience
+of statement in what follows call parentheses, square brackets or
+braces \emph{brackets}. If preceded by a \verb`\left` or \verb`\right`
+or \verb`\biggl` or \verb`\biggr` etc. modifier, call them \emph{Brackets},
+with an uppercase `B'. Modifiers do not contribute to the position
+count, so that a left Bracket in first position means the modifier
+and left bracket are both considered to be in first position. When
+it is immaterial whether it is a bracket or a Bracket I write b/Bracket.
+The rules that follow do not prescribe what mathematicians \emph{ought}
+to do but are intended to be descriptive of certain patterns of mathematical
+practice as discerned in \emph{HMF} and a number of texts (about half
+a dozen) on Fourier series.
+\begin{description}
+\item [{\texttt{()=0}}] is the \emph{default} setting; b/Brackets are included
+in the argument only if
+\begin{itemize}
+\item the left b/Bracket is in first position;
+\begin{itemize}
+\item if the first item beyond the matching right b/Bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, otherwise
+\item the argument ends with the right b/Bracket.
+\end{itemize}
+\end{itemize}
+\item [{\texttt{()=1}}] includes a b/Bracketed expression in the argument,
+provided
+\begin{itemize}
+\item the left Bracket is in first position;
+\begin{itemize}
+\item if the first item beyond the matching right Bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, otherwise
+\item the argument ends with the right Bracket.
+\end{itemize}
+\item or the item in first position is a number, variable, constant or \verb`\tfrac`
+and the left bracket is in second position;
+\begin{itemize}
+\item if the first item beyond the matching right bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, or
+\item if the first item beyond the matching right bracket is a number, variable,
+constant, or \verb`\tfrac` it is appended to the argument, and so
+on recursively, until
+\begin{itemize}
+\item an exponent, or factorial or double factorial sign is met, which is
+appended to the argument which ends there, or
+\item an item is met which is \emph{not} an exponent, or factorial or double
+factorial sign, or a number, variable, constant or \verb`\tfrac`,
+at which point the argument ends, or
+\item the end of the formula is reached.
+\end{itemize}
+\end{itemize}
+\end{itemize}
+\item [{\texttt{()=2}}] includes a b/Bracketed expression in the argument
+provided
+\begin{itemize}
+\item the left b/Bracket is in first position, or the item in first position
+is a number, variable, constant, \verb`\dfrac`, \verb`\frac` or
+\verb`\tfrac` and the left b/Bracket is in second position, or the
+items in first and second positions are numbers, variables, constants,
+\verb`\dfrac`-s, \verb`\frac`-s or \verb`\tfrac`-s and the left
+b/Bracket is in third position;
+\begin{itemize}
+\item if the first item beyond the matching right b/Bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, or
+\item if the first item beyond the matching right b/Bracket is a number,
+variable, constant, \verb`\dfrac`, \verb`\frac` or \verb`\tfrac`
+it is appended to the argument, and so on recursively, until
+\begin{itemize}
+\item an exponent, or factorial or double factorial sign is met, which is
+appended to the argument which ends there, or
+\item an item is met which is \emph{not} an exponent, or factorial or double
+factorial sign, or a number, variable, constant, \verb`\dfrac`, \verb`\frac`
+or \verb`\tfrac`, at which point the argument ends, or
+\item the end of the formula is reached.
+\end{itemize}
+\end{itemize}
+\end{itemize}
+\end{description}
+The default setting is \texttt{()=0} which is parsing group II behaviour.
+
+\texttt{()=1} allows $\tan\tfrac{1}{2}(A+B)$ and $\sec\pi(\tfrac{1}{4}+\tfrac{1}{2}az)$,
+and $\cos(2m+p)z$ and $\sin(2n+1)v$, and also items on \emph{both}
+sides of the bracketed part like $\sin\tfrac{1}{2}(m+n)\pi$ provided
+there is only one item between the function and the left bracket:
+\begin{centred}
+\verb`\eval[()=1]{$ \sin\tfrac16(m+n)\pi $}[m=1,n=2]`. $\Longrightarrow$
+\eval[()=1]{$ \sin\tfrac16(m+n)\pi $}[m=1,n=2].
+\end{centred}
+Note that \texttt{numerica} does not check what is included between
+the brackets – it could be anything. However inserting \verb`\left`,
+\verb`\right` modifiers before the parentheses restricts the argument
+of the sine in this example to the \verb`\tfrac`:
+\begin{centred}
+\verb`\eval[()=1]{$ \sin\tfrac16\left(m+n\right)\pi $}[m=1,n=2]`.
+$\Longrightarrow$ \eval[()=1]{$ \sin\tfrac16\left(m+n\right)\pi $}[m=1,n=2].
+\end{centred}
+\texttt{()=2} draws no distinction between brackets and Brackets.
+It allows all \texttt{()=1} possibilities but also \emph{two }items
+(of a suitable kind) before the left b/Bracket; it also allows \verb`\dfrac`-s
+and \verb`\frac`-s in addition to \verb`\tfrac`-s.
+
+The following examples are taken from different texts on Fourier series.
+The first shows a \verb`\frac` being included in the argument, the
+second shows \emph{two} items – including a \verb`\frac` – preceding
+the left parenthesis, the third shows a \verb`\frac` to the right
+of the parentheses, and the fourth shows parentheses using \verb`\left`-\verb`\right`
+modifiers with two items preceding them:
+\[
+\cos\frac{2\pi}{T}nt,\quad\cos\frac{2\pi}{T}n(t+\tfrac{1}{2}T),\quad\text{\ensuremath{\sin(N+\tfrac{1}{2})\frac{2\pi\tau}{T}}\ensuremath{\quad}and}\quad\sin2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right).
+\]
+All these usages are accommodated by the \verb`()=2` setting. For
+instance
+\begin{verbatim}
+ \eval[p,()=2]
+ {
+ \[ \sin(N+\tfrac12)\frac{2\pi\tau}T \]
+ }[N=1,\tau=2,T=3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p,()=2]
+ {
+ \[ \sin(N+\tfrac12)\frac{2\pi\tau}T \]
+ }[N=1,\tau=2,T=3]which is the sine of $(\tfrac{3}{2})\times(\tfrac{4}{3}\pi)=2\pi$
+(and \emph{not }$(\sin\tfrac{3}{2})(\tfrac{4}{3}\pi)$ ), where a
+\verb`\frac` trailing the parentheses has been included in the argument.
+Or consider
+\begin{verbatim}
+ \eval[p,()=2]
+ {\[
+ \sin2\pi\left(\frac{x}{\lambda}
+ -\frac{t}{T}\right)
+ \]}[x=1,\lambda=2,t=3,T=4]
+\end{verbatim}
+$\Longrightarrow$ \eval[p,()=2]
+ {\[
+ \sin2\pi\left(\frac{x}{\lambda}
+ -\frac{t}{T}\right)
+ \]}[x=1,\lambda=2,t=3,T=4] which is the sine of $2\pi\times(-\tfrac{1}{4})=-\tfrac{1}{2}\pi$
+(and \emph{not} $\sin2\pi$ times the parenthesised expression) where
+there are two items before the parentheses which surround two \verb`\frac`-s
+and \verb`\left` and \verb`\right` modifiers have been used with
+the parentheses.
+
+However a usage like $\sin(n+\tfrac{1}{2})(x-t)$, noted in two different
+texts, is not available without explicit use of the \verb`\q` command
+between the parenthesized groups.
+
+\subsubsection{Parsing group IV}
+
+The only members of this group are \verb`\sum` and \verb`\prod`.
+
+For parsing group IV
+\begin{enumerate}
+\item the argument ends
+\begin{enumerate}
+\item at the first naked plus or minus sign encountered, or
+\item at the first comparison sign or comparison command encountered, or
+\item at the first logical And or logical Or sign encountered, or
+\item at the end of the formula.
+\end{enumerate}
+\end{enumerate}
+In practice this means mainly (a) and (d), and seems to be the instinctive
+practice. \emph{HMF} has multiple examples in multiple chapters of
+the argument to a sum ending at a naked plus sign: 7.3.12 \& 7.3.14,
+9.1.11 \& 9.1.77, 9.6.35 \& 9.6.43, 11.1.9, \ldots{} (at that point
+I stopped looking). They were all of the form
+\[
+\sum\text{argument}+\ldots
+\]
+ A minus sign serving the same purpose was harder to find but \emph{HMF}
+10.4.65 \& 10.4.67 are two instances. I considered whether a \verb`\times`
+or slash fraction sign \verb`/` might end the argument of a sum,
+but surely we need to allow things like $\sum1/n^{2}$ which rules
+out the slash and \emph{HMF} 9.9.11 provides two of a number of instances
+in \emph{HMF} of sum arguments continuing past explicit \verb`\times`
+signs (at line breaks when a summand spills onto a second line).
+
+Because they are evaluated using the same code as sums I (unthinkingly)
+placed products with sums but doubts later intruded. In \emph{HMF}
+products occur only occasionally and are almost all of the form
+\[
+\prod\left(\text{argument}\right)
+\]
+where the argument is bracketed (often with \verb`\left \right` modifiers)
+and the multiplicand ends with the right bracket. At least twice (\emph{HMF
+}6.1.25 and 24.2.2.1) an exponent ($-1$) is attached to the right
+bracket and the argument ends there. Looking further afield, a text
+on number theory has examples where the argument of the product extends
+to \emph{three} parenthesised factors, $\prod\left(\text{arg}1\right)\left(\text{arg2}\right)\left(\text{arg3}\right)$
+and a number of others where it extends to two. A text on theory of
+functions has
+\[
+\prod_{n=1}^{\infty}\left(1+\frac{z}{n}\right)e^{z/n}
+\]
+although \emph{HMF}, for the same expression, encloses the two factors
+within (large) square brackets, as if some ambiguity existed as to
+how far the reach of the \verb`\prod` extended.
+
+\emph{Tentatively} I retain products here in the same group as sums.
+
+\subsubsection{Parsing group V}
+
+Comparison symbols compose this group: \texttt{=}, \texttt{<}, \texttt{>},
+\verb`\ne`, \verb`\le`, \verb`\ge`, \verb`\leq`, \verb`\geq`,
+and the various comparison commands from the \texttt{amssymb} package
+listed in §\ref{subsec:evalAmssymb-comparisons}. Because of the way
+\texttt{numerica} handles comparisons, it is the argument on the right-hand
+side of the relation that needs determining.
+
+For parsing group V
+\begin{enumerate}
+\item the argument ends at
+\begin{enumerate}
+\item the first logical And or logical Or encountered, or
+\item the first comparison sign or command encountered, or
+\item the end of the formula.
+\end{enumerate}
+\end{enumerate}
+
+\subsubsection{Parsing group VI}
+
+Logical And and logical Or are the sole members of this group. It
+is the right-hand side of the And or Or command that needs determining.
+
+For parsing group VI
+\begin{enumerate}
+\item the argument ends at
+\begin{enumerate}
+\item the first logical And or logical Or encountered, or
+\item the end of the formula.
+\end{enumerate}
+\end{enumerate}
+
+\subsubsection{Disclaimer}
+
+The parsing rules of the different groups are not normative; they
+are not statements of how mathematical formulas should be written.
+Rather they are attempts to discern regularities in how mathematicians
+often do write formulas. It is how things look in the pdf, not \LaTeX ,
+that is the guide. You are always free to parenthesize as you see
+fit and to insert cleave commands (\verb`\q` or \verb`\Q`) to force
+outcomes.
+
+(But note that parenthesizing has its limits. For sums, writing
+\[
+\sum\left(\mathtt{<stuff>}\right)\mathtt{<more-stuff>}
+\]
+ does not necessarily end the summand at the right parenthesis: it
+ends at the first naked $+$ or $-$ sign, or \verb`\Q` command,
+encountered.)
+
+The rule should always be to write expressions that are clear to the
+reader of the pdf. An expression that is ambiguous to the reader,
+even if it fits within the parsing rules, is to be deplored. The \emph{intent}
+is that \verb`\eval` can parse unambiguous expressions correctly.
+
+\chapter{Nesting \texttt{\textbackslash eval} commands}
+
+\label{chap:Nesting}\verb`\eval` commands can be used within \verb`\eval`
+commands, both as part of the formula being evaluated or as part of
+the vv-list or both. Only in special circumstances is this likely
+to be useful – perhaps when a calculation can be divided into two
+or more parts where different settings are appropriate for the different
+parts. One can imagine cases in which trigonometric functions are
+involved and different \texttt{()} settings would be helpful in different
+parts of the formula. Nesting of command within command becomes especially
+significant with the additional commands available when \texttt{numerica}
+is loaded with the \texttt{plus} or \texttt{tables} options; see the
+associated documents \texttt{numerica-plus.pdf} and \texttt{numerica-tables.pdf}.
+Since those additional commands are not available in this document,
+I restrict myself here to some `toy' examples of \verb`\eval` commands
+within \verb`\eval` commands to show how things work.
+
+\section{Star option for inner \texttt{\textbackslash eval}}
+
+The \verb`\eval` command `digests' a \LaTeX{} formula to produce
+an \verb`l3fp`-readable formula. This is then fed to \verb`l3fp`
+to be evaluated. The evaluated output is then formatted in various
+ways to be displayed. If the inner \verb`\eval` command produces
+formatted output, it is \emph{that} that the outer \verb`\eval` command
+will attempt to digest – and fail. Hence \emph{always} use the star
+option for the inner \verb`\eval` command. That means the outer \verb`\eval`
+is feeding on a number only:
+\begin{centred}
+\verb`\eval{$ \sin(\eval*{\sin x}[x=\pi/6]\pi) + 1 $}` $\Longrightarrow$
+\eval{$ \sin(\eval*{\sin x}[x=\pi/6]\pi) + 1$}.
+\end{centred}
+Also no math delimiters are used in the inner command. These are irrelevant
+with the star option in any case, but in the present context would
+cause error if included because they would be treated as part of the
+formula and thereby produce an `unknown token' error message. In
+the presentation of the overall result that the inner \verb`\eval`
+command is evaluated, showing as $0.5$.
+
+\section{Nesting in the vv-list}
+\begin{centred}
+\verb`\eval{$ \sin k\pi + 1 $}[k=\eval*{\sin x},x=\pi/6]` $\Longrightarrow$
+\eval{$ \sin k\pi + 1 $} [k=\eval*{\sin x},x=\pi/6].
+\end{centred}
+When the inner \verb`\eval` command is in the vv-list of the outer
+command and has a vv-list of its own, then the entire inner command
+needs to be placed in braces:
+\begin{centred}
+\verb`\eval{$ \sin k\pi + z $}[k={\eval*{y\sin x}[x=\pi/4,y=1/\surd2]},z=1]`
+$\Longrightarrow$ \eval{$ \sin k\pi + z $} [k={\eval*{y\sin x}[x=\pi/4,y=1/\surd2]},z=1].
+\end{centred}
+The vv-list of the inner \verb`\eval` command contains both a comma
+and square brackets. Both elements need to be hidden from the outer
+\verb`\eval` in order that \emph{its }vv-list be parsed correctly.
+Hence braces surround the inner \verb`\eval` and its arguments. The
+same need arises if the inner \verb`\eval` has a non-empty settings
+option – another comma-separated square-bracketed option.
+
+The values of variables used in an inner \verb`\eval` command are
+restricted to that command; they do not `leak' into the outer calculation.
+But variables and their values in the outer vv-list are available
+for use in the inner \verb`\eval` command (unless a value is explicitly
+changed in the inner vv-list).
+
+\subsection{Debugging}
+
+\label{subsec:nestDebugging}It is worth looking at the debug display
+when \verb`\eval` commands are nested. For the outer \verb`\eval`
+command:
+\begin{centred}
+\verb`\eval[dbg=210]{$ \sin \eval*{\sin x}[x=\pi/6]\pi + 1 $}` $\Longrightarrow$
+\eval[dbg=210]{$ \sin \eval*{\sin x}[x=\pi/6]\pi + 1 $}
+\end{centred}
+and when the inner \verb`\eval` is in the vv-list,
+\begin{centred}
+\verb`\eval[dbg=210]{$ \sin k\pi + 1 $}[k=\eval*{\sin x},x=\pi/6]`
+$\Longrightarrow$ \eval[dbg=210]{$ \sin k\pi + 1 $} [k=\eval*{\sin x},x=\pi/6]
+\end{centred}
+For the inner \verb`\eval` command debugging still works but in an
+idiosyncratic way. To clarify exactly what is going on I have added
+a \verb`\left( \right)` pair around the entire inner \verb`\eval`
+command. Note that I have also used a \emph{negative} \texttt{dbg}
+value. With a positive value, the right parenthesis is pressed toward
+the right margin of the page. The negative value limits the display
+to the text width and gives the much neater result shown.
+\begin{verbatim}
+ \eval[()=2]{$
+ \sin\left(
+ \eval*[dbg=-210]{ \sin x }[x=\pi/6]
+ \right)\pi + 1 $}
+\end{verbatim}
+$\Longrightarrow$ \eval[()=2]{$
+ \sin\left(
+ \eval*[dbg=-210]{ \sin x }[x=\pi/6]
+ \right)\pi + 7 $}
+
+\medskip{}
+The debug display from the inner \verb`\eval` command has been inserted
+into the formula of the outer \verb`\eval` in the position occupied
+by the inner \verb`\eval`. I did not deliberately code for this,
+but have decided to leave it as is, since there can be no confusion
+about which \verb`\eval` command is being `debugged', despite the
+potential for some rather odd displays. In this last example, in order
+to both use \verb`\left(...\right)` and have the calculation give
+the previous result I have employed the setting \verb`()=2` in the
+outer \verb`\eval`; see §\ref{subsec:parseTrigFns}.
+
+\chapter{\texttt{\textbackslash nmcInfo (\textbackslash info)}}
+
+\label{sec:info}Used after the evaluation of an `infinite' process,
+the \verb`\nmcInfo` command, or its equivalent short-name form \verb`\info`
+will tell you how many terms or factors or iterations or steps were
+needed to arrive at the result. The syntax of the \verb`\nmcInfo`
+command is
+\begin{lyxcode}
+\textbackslash nmcInfo\{<arg>\}~
+\end{lyxcode}
+where \texttt{<arg>} is restricted to one of two choices at present,
+either \texttt{sum} or \texttt{prod}. If the package \texttt{numerica-plus.def}
+is loaded (see §\ref{subsec:introPackagesOptions} and the associated
+document \texttt{numerica-plus.pdf}) two further arguments are possible:
+\texttt{iter} and \texttt{solve}.
+
+There is a starred form of the command:
+\begin{lyxcode}
+\textbackslash nmcInfo{*}\{<arg>\}
+\end{lyxcode}
+(or \verb`\info*{<arg>}`). As with the \verb`\eval` command the
+star has the effect of suppressing anything other than the numerical
+result from the display.
+
+As an example, let's test a standard identity, $\cosh^{2}x-\sinh^{2}x=1$,
+`the hard way'. We know that $\cosh x=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$
+and $\sinh x=x\prod_{k=1}^{\infty}\left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)$.
+The difference of their squares should be $1$:
+\begin{verbatim}
+ \eval{\[
+ \left[\sum_{n=0}^{\infty}
+ \frac{x^{2n}}{(2n)!}
+ \right]^2-
+ \left[x\prod_{k=1}^{\infty}
+ \left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)
+ \right]^2
+ \]}[x=1][3] \info{sum}\quad \info{prod}
+\end{verbatim}
+$\Longrightarrow$ \eval{\[
+ \left[\sum_{n=0}^{\infty}
+ \frac{x^{2n}}{(2n)!}\right]^2-
+ \left[x\prod_{k=1}^{\infty}
+ \left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)\right]^2
+ \]}[x=1][3] \info{sum}\quad \info{prod}.
+
+Nearly right. Obviously the product converges only slowly which is
+where the error comes from (see the discussion in §\ref{sec:settingsInfiniteSumsProds},
+where we needed the extra rounding setting \texttt{P+=3} and $350$
+factors to get a correct 3-figure value). The point of the example
+is to show the information command being used for both sum and product
+in the one evaluation. One does not exclude the other.
+
+The information command can also be placed in the settings option
+as the value of the punctuation setting. An example of this has already
+been provided earlier which I'll repeat here:
+\begin{verbatim}
+ \eval[p=\mbox{,\quad\nmcInfo{sum}.}]
+ {\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=\mbox{,\quad\nmcInfo{sum}.}]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+
+Because of the \texttt{\textbackslash{[} \textbackslash{]}} delimiters,
+if the information command had been placed \emph{after} the \texttt{\textbackslash eval}
+command, it would have slid down to the next line. As it is, it resides
+\emph{inside} the \texttt{\textbackslash{[} \textbackslash{]}}
+delimiters, on the same line as the expression. This may be significant
+for adjusting vertical spacing of subsequent elements of the document.
+
+\section{Errors}
+
+Should the wrong argument be used in the \verb`\nmcInfo` command,
+no harm is done:
+\begin{verbatim}
+ \eval{$
+ \sum_{k=0}^{\infty}\binom \alpha k x^k
+ $}[x=1/2,\alpha=3], \ \info{prod}
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \sum_{k=0}^{\infty}\binom \alpha k x^k $}[x=1/2,\alpha=3],\ \info{prod}.
+
+$119$ \emph{factors}? The information command is remembering a previous
+result, the last time \verb`prod` was used as its argument. Changing
+the argument from \verb`prod` to \verb`sum` reveals the correct
+number of \emph{terms}.
+
+Should a non-existent argument be used, an error message is generated:
+\begin{verbatim}
+ \eval{$
+ \sum_{k=0}^{\infty}\binom \alpha k x^k
+ $}[x=1/2,\alpha=3], \\ \info{Fred}
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \sum_{k=0}^{\infty}\binom \alpha k x^k $}[x=1/2,\alpha=3],\\ \info{Fred}
+
+\chapter{Saving and reusing results }
+
+\label{chap:Saving-and-retrieving}You may want to use at some place
+in a document a result calculated earlier. It would be good to be
+able to do so without having to do the calculation again at the new
+location. \texttt{numerica} offers a command \verb`\nmcReuse` which
+saves a result to a control sequence which can be used elsewhere in
+the document, expanding to the saved result. The control sequence
+and its content are also saved to file for use on other occasions.
+
+The syntax of \verb`\nmcReuse` is simple. The command takes two optional
+arguments, a star (asterisk) and a conventional square-bracket delimited
+argument. If both are used it looks like
+\begin{lyxcode}
+{\small\textbackslash nmcReuse{*}{[}csname{]}}{\small\par}
+\end{lyxcode}
+where \texttt{csname} is the proposed name of what will become the
+control sequence \verb`\csname` containing the latest result from
+the \verb`\eval` command. The name should be composed of letters
+only.
+
+As with \verb`\nmcEvaluate` and \verb`\nmcInfo` there is a short-name
+form, \verb`\reuse`, for \verb`\nmcReuse`. If a conflict with another
+package arises, it should be possible to fall back on \verb`\nmcReuse`.
+
+\section{Use without optional argument: \texttt{\textbackslash nmcReuse}}
+
+Suppose your document is \verb`mydoc.tex` (so that the \LaTeX{} command
+{\small\verb`\jobname`} expands to \verb`mydoc`). If \verb`\nmcReuse`
+is used without optional arguments, then \texttt{numerica} checks
+for the existence of a file \verb`mydoc.nmc` in the current document
+directory\texttt{ }and if found loads and records the contents of
+\verb`mydoc.nmc`. The contents should be a comma separated list of
+control sequences and braced values like \texttt{\textbackslash csname1
+\{value1\},\textbackslash csname2 \{value2\},...} The control sequences
+can then be used elsewhere in the document. In particular, control
+sequences containing numerical values can be used in expressions within
+\verb`\eval` commands and in vv-lists.\footnote{The associated document \texttt{numerica-plus.pdf} describes how other
+quantities like tables and sequences of numerical values (iterates,
+recurrences) can also be saved and reused.}
+
+Note that the control sequences are given \LaTeX{} definitions using
+\verb`xparse`'s \verb`\NewDocumentCommand` (a little more general
+than \LaTeXe 's \verb`\newcommand`) and can be used at any later
+point in the document simply by entering the control sequence (e.g.,
+\verb`\csname1`) there (but see §\ref{subsec:reuseGroup-level}).
+Should there already be a control sequence with the same name, \LaTeX{}
+will generate an error and halt compilation.
+
+\section{Use with optional name: \texttt{\textbackslash nmcReuse{[}csname{]}}}
+
+If a name \emph{is} supplied, say \verb`\nmcReuse[csname]`, then
+not only does \texttt{numerica} first look for \verb`mydoc.nmc` (assuming
+your document is called \verb`mydoc.tex`) and load the values stored
+in that file if they have not already been loaded, but it also defines
+\verb`\csname` to contain the latest result from the \verb`\eval`
+command. Should \verb`\csname` already be present in \verb`mydoc.nmc`
+and so have been loaded with the other values from \verb`mydoc.nmc`,
+the old value is overwritten with the new value using \verb`xparse`'s
+\verb`\RenewDocumentCommand` and the new value is saved to the file
+\verb`mydoc.nmc`.
+
+\subsection{Group level}
+
+\label{subsec:reuseGroup-level}Control sequences like \verb`\csname`
+defined by \verb`\nmcReuse` are defined within the current group
+level. A usage like
+\begin{centred}
+\verb`$ \eval{1+1}\reuse[two] $`
+\end{centred}
+confines the definition of \verb`\two` to the environment delimited
+by the dollar signs. A usage like
+\begin{centred}
+\verb`$ \eval{1+1} $ \reuse[two]`,
+\end{centred}
+where the command has been moved outside the math delimiters, still
+confines the definition of \verb`\two` to whatever larger environment
+the \verb`\reuse` command might lie within. If in fact the definition
+occurs at document level then \verb`\two` (in the present example)
+is available for use throughout the document, otherwise it is available
+only within the confines of the current environment.
+
+This is likely to be \emph{not }what is wanted. The remedy is simple:
+precede the saved control sequence – \verb`\two` in the present instance
+– with a `naked' \verb`\reuse` command. This loads the value stored
+in \verb`mydoc.nmc`. In the following example, \verb`\reuse` is
+used within a math environment which is followed by some text (\verb`blah blah result:`)
+then a `naked' \verb`\reuse` command and the control sequence \verb`\two`
+(between math delimiters, which are necessary for reasons explained
+in the next section).
+\begin{centred}
+\verb`$ \eval{1+1} \reuse[two] $, blah blah result: \reuse $\two$.`
+$\Longrightarrow$ $ \eval{1+1} \reuse[two]$, blah blah result: \reuse $\two$.
+\end{centred}
+
+\section{What is saved?}
+
+\label{sec:reuseWhat-is-saved}In the default configuration \verb`\nmcReuse`
+saves the entirety of the display resulting from the latest \verb`\eval`-uation.
+This may include invisible formatting elements meaning that what one
+expects to be only a number cannot be inserted into text without causing
+a \LaTeX{} error; it requires a math environment to print – see the
+last example.
+
+The simplest way to avoid this awkwardness is to use \verb`\eval`
+with the star option. This produces a numerical result with \emph{no
+}formatting. In the following example, I calculate \verb`11+11` with
+\verb`\eval*` and store the value in the control sequence \verb`\twos`.
+I then enter some text (\verb`Blah blah blah:`) and insert the control
+sequence \verb`\twos` into the text without math delimiters. As you
+can see, \verb`\twos` has expanded to $22$, the result of the calculation.
+\begin{centred}
+\verb`\eval*{$ 11+11 $}. \nmcReuse[twos] Blah blah blah: \twos` $\Longrightarrow$
+ \eval*{$ 11+11 $}. \nmcReuse[twos] Blah blah blah: \twos
+\end{centred}
+
+\subsection{Viewing what is saved: \textbackslash reuse\texttt{{*}}}
+
+To view what is saved in the \verb`.nmc` file append a star (asterisk)
+to the \verb`\reuse` command. (This makes particular sense when using
+\texttt{numerica} in a program like \LyX{} with a preview facility;
+see Chapter~\ref{chap:LyX}.)
+\begin{centred}
+\verb`\reuse*` $\Longrightarrow$ \reuse*
+\end{centred}
+Particularly notable here is the invisible formatting \verb`\mathchoice{}{}{}{}`
+accompanying \verb`2` in the value of \verb`\two`.
+
+The two options of the\verb`\reuse` command can be used together,
+in which case the control sequence resulting from the name supplied
+in the square-bracketed option will appear in the list resulting from
+the star option, i.e. the list contains not just what has been saved
+earlier but also the current control sequence saved.
+
+\subsection{\texttt{\textbackslash eval}'s \texttt{reuse} setting}
+
+\label{subsec:reuse=00005Ceval's-reuse-setting}The star option of
+the \verb`\eval` command allows a purely numerical result to be saved,
+but also only a number is displayed. By using the \verb`reuse` setting
+of the \verb`\eval` command it is possible to have both a full display
+of an evaluation, vv-list and all, and to save only a numerical result.
+
+For the \emph{starred} form of the \verb`\eval` command it is always
+\emph{only the numerical result} that is saved, whatever the value
+of the \texttt{reuse} key in the settings option of the \verb`\eval`
+command.
+
+For the \emph{unstarred} form of the \verb`\eval` command exactly
+what is saved with \verb`\nmcReuse` depends on the \texttt{reuse}
+setting:
+\begin{lyxcode}
+reuse~=~<integer>
+\end{lyxcode}
+where \verb`<integer>` can take one of two values,
+\begin{itemize}
+\item \texttt{reuse=0} (the default) saves\emph{ the form that is displayed}
+including a formatting component. If the result is displayed in the
+form \emph{formula=result (vv-list)} then that is what is saved; if
+the display is of the form \emph{result (vv-list)} then that is what
+is saved; if the vv-list is empty, an empty formatting component is
+still present in the saved result;
+\item \texttt{reuse=1} (or, indeed, any non-zero integer) saves only the
+numerical result with no other elements of the display (meaning no
+formatting component).
+\end{itemize}
+Thus, with the default setting (\texttt{reuse=0}) the full content
+of the display is saved:
+\begin{centred}
+\verb`\eval{$ x + y $}[x=1,y=1] \reuse[iandi]` $\Longrightarrow$
+\eval{$ x + y $}[x=1,y=1] \reuse[iandi].
+\end{centred}
+To check that this is the case, \verb`\reuse \iandi` $\Longrightarrow$
+\reuse \iandi .
+
+On the other hand, with \texttt{reuse=1} only the numerical value
+is saved:
+\begin{centred}
+\verb`\eval[reuse=1]{$ x + y $}[x=1,y=1] \reuse[iandi]` $\Longrightarrow$
+\eval[reuse=1]{$ x + y $}[x=1,y=1] \reuse[iandi],
+\end{centred}
+which we can check here: \verb`\reuse\iandi` $\Longrightarrow$ \reuse\iandi .
+
+\subsubsection{\texttt{\textbackslash reuse} in the preamble}
+
+To gain access from the outset to the control sequences stored in
+the file \texttt{mydoc.nmc}, place \verb`\nmcReuse` without an optional
+argument in the preamble (but after \verb`\usepackage{numerica}`).
+
+\chapter{Using \texttt{numerica} with \protect\LyX}
+
+\label{chap:LyX}The document processor \LyX{} has a facility that
+enables snippets from a larger document to be compiled separately
+and the results presented to the user without having to compile the
+entire document. The present document was written in \LyX . The demonstration
+calculations were evaluated using this \emph{instant preview} facility.
+
+To use \texttt{numerica} in \LyX{} go to \textsf{Document \lyxarrow{}
+Settings \lyxarrow{} LaTeX Preamble} and enter
+\begin{lyxcode}
+\textbackslash usepackage\{numerica\}
+\end{lyxcode}
+then click \textsf{OK}. However preview poses problems for the straightforward
+use of the \verb`\nmcReuse` command. If you wish to use this command
+in \LyX{} then \texttt{numerica} should be loaded with the \texttt{lyx}
+package option. Thus in \textsf{Document \lyxarrow{} Settings \lyxarrow{}
+LaTeX Preamble} enter
+\begin{lyxcode}
+\textbackslash usepackage{[}lyx{]}\{numerica\}
+\end{lyxcode}
+then click \textsf{OK}, or you may wish to follow the above line in
+the preamble with \verb`\nmcReuse`,
+\begin{lyxcode}
+\textbackslash usepackage{[}lyx{]}\{numerica\}
+
+\textbackslash nmcReuse
+\end{lyxcode}
+and \emph{then} click \textsf{OK}. The additional line ensures all
+saved values are available in your document from the outset.
+
+\section{Instant~preview}
+
+Preview performs localised mini-\LaTeX{} runs on selected parts of
+a document (for instance, the mathematical parts) and displays the
+results in \LyX{} while the user continues to work on the surrounding
+document.\texttt{ numerica} uses these local \LaTeX{} runs to do its
+evaluations and display their results. That means you get feedback
+on your calculations almost immediately.
+
+To use this facility first ensure that instant preview is turned on.
+This means selecting \textsf{Tools \lyxarrow Preferences \lyxarrow Look
+\& Feel \lyxarrow{} Display} and against \textsf{Instant preview} selecting
+\textsf{On}, then clicking \textsf{OK}.
+
+\subsection{Conflict with hyperref support?}
+
+There may be a conflict in \LyX{} between hyperref support and preview,
+not that the previews do not form but that their formation takes a
+circuitous path that noticeably slows their display.\footnote{At least there is on the author's Windows 10 system, but I'm not sure
+that this is general.} If this occurs on your system, go to \textsf{Document \lyxarrow{}
+Settings \lyxarrow{} PDF Properties} and ensure the check box \textsf{Use
+Hyperref Support} is cleared. By all means reset the check box when
+you come finally to compile your document, but until then it should
+result in a noticeably brisker display of previews if the checkbox
+is cleared.
+
+\section{Mathed}
+
+(Mathed = the \LyX{} mathematics editor.) If you have instant preview
+\emph{on} then one way to use \texttt{numerica} in \LyX{} is to enter
+an \verb`\eval` command in mathed. Clicking the cursor outside the
+editor with the mouse or moving it outside with the arrow keys will
+then trigger formation of a preview of the editor's contents – a snippet
+of what will be shown in the pdf. This will be displayed in mathed's
+place after a generally short `pause for thought' as the mini-\LaTeX{}
+run progresses behind the scenes.
+
+The original expression can be recovered by clicking on the preview.
+The content of mathed is immediately displayed and can be edited.
+
+\subsection{\protect\LaTeX{} braces~\{~~\}}
+
+\LyX{} does not support \texttt{numerica}'s \verb`\eval` command `out
+of the box' as it does, say, \verb`\frac` or \verb`\sqrt`. To use
+the \verb`\eval` command in mathed you will need to supply the braces
+used to delimit its mandatory argument. (For \verb`\frac` and \verb`\sqrt`
+by contrast, \LyX{} supplies these automatically.) Unfortunately the
+\verb`{` key\footnote{\textsf{Shift+{[}} on my keyboard.} does not
+insert a left brace into the document but rather an escaped left brace
+\verb`\{` as you can see by looking at \textsf{View \lyxarrow{} Code
+Preview Pane}. Escaped braces like this are used for grouping terms
+in \emph{mathematics}; they are not the delimiters of a \LaTeX{} argument.
+
+The brace delimiters for \LaTeX{} arguments are entered in mathed by
+typing a backslash \textsf{\textbackslash{} }then\textsf{ \{} (two
+separate key presses rather than a single combined press). This enters
+a balanced pair of (unescaped) braces with the cursor sitting between
+them waiting for input. Alternatively, if you have already written
+an expression that you want to place between braces, select it, then
+type \textsf{\textbackslash{} }then\textsf{ \{}.
+
+\section{Preview insets vs mathed}
+
+There are problems with using mathed for calculations.
+\begin{itemize}
+\item Expressions entered in mathed are necessarily of the form \verb`$ \eval... $`
+or more generally \verb`delimiter` \verb`\eval...` \verb`delimiter`.
+But you may wish to wrap the \verb`\eval` command \emph{around} the
+math delimiters to produce a \emph{formula=result} form of display.
+In mathed the only way to do that is to write the \emph{formula= }part
+yourself – which may involve no more than copy and paste but is still
+additional mouse work/key pressing.
+\item Mathed does not accept carriage returns. If you want to format a complicated
+expression for readability by breaking it into separate lines, you
+can't. The expression is jammed into the one line, along with the
+settings option content and the vv-list.
+\end{itemize}
+For these reasons I have come to prefer \emph{not} using mathed for
+calculations but instead to use preview insets wrapped around \TeX -code
+(ERT) insets. \LyX{} uses the shortcut \textsf{Ctrl+L} to insert an
+ERT inset. Since \LyX{} now does no printing itself, the shortcut \textsf{Ctrl+P}
+that was formerly used for printing is available for other purposes.
+On my keyboard, the \textsf{P} key lies diagonally up and to the right
+but adjacent to the \textsf{L} key. I suggest assigning \textsf{Ctrl+P}
+to inserting a preview inset. Then typing \textsf{Ctrl+P Ctrl+L} –
+which means holding the \textsf{Ctrl} key down and tapping two adjacent
+keys, \textsf{P} followed immediately by \textsf{L} – will insert
+an ERT inset inside a preview inset with the cursor sitting inside
+the ERT inset waiting for input. In the ERT inset you can enter carriage
+returns, and so format complicated expressions. You can place the
+vv-list on a separate line or onto consecutive lines. And when you
+have finished, clicking outside the preview inset will trigger preview
+into doing its thing and present the result `before your eyes'.
+
+To assign the suggested shortcut, go to \textsf{Tools \lyxarrow{} Preferences
+\lyxarrow{} Editing \lyxarrow{} Shortcuts}. Under \textsf{Cursor, Mouse
+and Editing Functions} in the main window on the right, scroll down
+until you come to \textsf{preview-insert}, select it, then click \textsf{Modify}.
+Now press \textsf{Ctrl+P}. The shortcut will magically appear in the
+greyed, depressed key.\textsf{ }Click \textsf{OK} and then \textsf{OK}
+in the \textsf{Preferences} window to close it. (Most of the examples
+in this document have been evaluated in this way, using \textsf{Ctrl+P
+Ctrl+L.)}
+
+\section{Errors }
+
+Instant preview will display \texttt{numerica} error messages in \LyX{}
+just as it does the results of calculations. Clicking on the message
+will show the underlying expression which can then be edited. However
+\LaTeX{} errors will \emph{not} produce a preview; formation of the
+preview will stall. To find precisely what has gone wrong, you will
+need to look at the \LaTeX{} log, but not the log of the overall document;
+rather the \emph{preview} log. Unfortunately this is tucked away in
+a temporary directory and is not immediately accessible in \LyX{} (unlike
+the main \LaTeX{} log from \textsf{Document \lyxarrow{} \LaTeX{} Log}).
+When \LyX{} is started, it sets up a temporary directory in which to
+perform various tasks. On Windows systems this will be located in
+\texttt{C:\textbackslash Users\textbackslash <your name>\textbackslash AppData\textbackslash Local\textbackslash Temp}
+and will have a name like \texttt{lyx\_tmpdir.XOsSGhBc1344}.
+
+One of the tasks \LyX{} uses this temporary directory for is to create
+preview images when a document is opened. If you look inside \LyX 's
+temporary directory when a document is first loaded, you will see
+a subdirectory created, with a name like \texttt{lyx\_tmpbuf0}. There
+may already be such directories there, in which case the number on
+the end will be greater than \texttt{0} – it depends on whether other
+documents are or have been open in the current instance of \LyX .
+Inside the appropriate \texttt{lyx\_tmpbuf}\texttt{\emph{n}} folder
+will be the preview log with a name like \texttt{lyxpreviewZL1344.log}.
+It will usually be accompanied by other files with extensions like
+\texttt{.dvi}, \texttt{.tex}, and perhaps quite a number with the
+extension \texttt{.png}, each one of which is a preview, or part of
+a preview. For a document just loaded there will be only the one preview
+log, but if you have added preview insets or math insets to your document\textsf{
+}in the current editing session there will be a number of such logs
+and you will need to determine the relevant one by the time stamp.
+
+The log files are text files and can be opened in a text editor. The
+relevant part of the log is towards the end (just before the final
+statistical summary) where you will find a list of entries like \texttt{Preview: Snippet
+1 641947 163840 7864588}. If there is an error, it will be noted here
+among these snippets and will generally make clear what needs remedying.
+
+\subsection{CPU usage, \protect\LaTeX{} processes}
+
+It is possible when a preview fails to resolve that the \LaTeX{} process
+associated with the preview will continue to run, using CPU cycles,
+slowing overall computer performance, and perhaps resulting in extra
+fan use giving a different sound to the computer. In Windows 10, the
+\textsf{Task Manager} (\textsf{Ctrl+Shift+esc}) under the \textsf{Details}
+tab shows the current executables running. The \textsf{CPU} column
+will show which processes are preoccupying the CPU. Check whether
+one or more of these processes looks \LaTeX -related (e.g. \texttt{latex.exe}
+or \texttt{pdflatex.exe}, or \texttt{miktex-pdftex.exe} if using MiK\TeX ).
+Click the \textsf{Name} column to sort the processes by name and look
+for the relevant name in the list, select it, and end the process
+(click the \textsf{End Task} button).
+
+I am not familiar with the corresponding situation on Linux or Mac.
+
+\section{Using \texttt{\textbackslash nmcReuse}}
+
+As noted, \LyX{} creates its previews in a temporary directory, not
+the document directory. If you want to save values from your current
+document – say, \texttt{mydoc.lyx} – to \texttt{mydoc.nmc} then you
+can do so without drama, but \texttt{mydoc.nmc} will be located in
+the temporary directory, and when \LyX{} is closed will be deleted
+along with the temporary directory.
+
+Suppose first that at the end of a session you manually copy \texttt{mydoc.nmc}
+back to the document directory. How can you ensure that the values
+saved in this file are available the next time you open \texttt{mydoc.lyx}?
+As noted at the start of this chapter entering
+\begin{lyxcode}
+\textbackslash usepackage{[}lyx{]}\{numerica\}
+
+\textbackslash nmcReuse
+\end{lyxcode}
+in the preamble ensures that these saved values are available for
+use from the outset – available to the mini-\LaTeX{} runs creating
+previews in the temporary directory.
+
+That leaves the problem of saving new values from the current session,
+which are stored in \texttt{mydoc.nmc} in the \emph{temporary} directory,\texttt{
+}back to \texttt{mydoc.nmc} in the \emph{document} directory. When
+\LyX{} is closed the temporary directory with all its contents is deleted.
+As suggested already we could manually copy \texttt{mydoc.nmc} from
+the temporary directory to the document directory but that means remembering
+to do so before closing \LyX . Inevitably we will sometimes forget.
+
+Fortunately \LyX{} has a copying mechanism for getting files out of
+the temporary directory. When a document is exported – say to pdf
+– it is possible to specify a \emph{copier} to automatically copy
+back to the document directory or subdirectory various files in the
+temporary directory. We want the \texttt{.nmc} file containing the
+saved values to be copied back. Go to \textsf{Tools \lyxarrow{} Preferences
+\lyxarrow{} File Handling \lyxarrow{} File Formats} and find \textsf{PDF
+(pdflatex)} (assuming export to \texttt{pdf} by this route) in the
+list of formats. In\textsf{ }the \textsf{Copier} slot of the dialogue
+insert the following line of code:
+\begin{lyxcode}
+{\small python~-tt~\$\$s/scripts/ext\_copy.py~-e~nmc,pdf~-d~\$\$i~\$\$o}{\small\par}
+\end{lyxcode}
+\verb`ext_copy.py` is a python script that is supplied with \LyX .
+The \texttt{-e nmc,pdf -d} part of the line tells \texttt{ext\_copy.py}
+that on export to \texttt{pdf} by the \texttt{pdflatex} route\texttt{
+}to copy any files with the extensions \texttt{.nmc} or \texttt{.pdf}
+from the temporary directory where \LyX{} does its work back to the
+document directory – the \verb`-d` option (which became available
+from \LyX{} 2.3.0).
+
+But if you have a complex document, it may take too much time to want
+to export to pdf before closing \LyX , particularly if there are a
+lot of evaluations in the document. Much faster is to export to \emph{plain
+text}, not because you want a plain text version of your document
+but because it too can be used to trigger the copier mechanism. Go
+to \textsf{Tools \lyxarrow{} Preferences \lyxarrow{} File Handling \lyxarrow{}
+File Formats} and find \textsf{Plain text} in the list of formats.
+In the \textsf{Copier} slot enter
+\begin{lyxcode}
+{\small python~-tt~\$\$s/scripts/ext\_copy.py~-e~nmc~-d~\$\$i~\$\$o}{\small\par}
+\end{lyxcode}
+The only difference from the previous copier command is the absence
+of \texttt{pdf}.\footnote{I'm assuming that you don't actually want the plain text version of
+the file copied back. If you do, then change \texttt{-e nmc} to \texttt{-e
+nmc,txt}.} This will copy \texttt{mydoc.nmc} with its saved values from the
+temporary directory back to the document directory. To effect the
+export, go to \textsf{File \lyxarrow{} Export }and find \textsf{Plain
+text} in the list of formats and click on it.
+
+A shortcut would be nice. For that go to \textsf{Tools \lyxarrow{}
+Preferences \lyxarrow{} Editing \lyxarrow{} Shortcuts}, click on \textsf{New},
+enter \texttt{buffer-export text} in the \textsf{Function:} slot,
+click on the blank key against \textsf{Shortcut:} and type your shortcut.
+You may have to try a number before you find one that hasn't already
+been assigned. (I'm using \textsf{Ctrl+}; for no particular reason
+beyond the fact that it fits under the fingers easily and saving values
+to the document directory has a punctuation-like feel to it, a pause
+in the process of writing.) It is now an easy matter to press the
+shortcut at the end of a \LyX{} session to copy all the values saved
+in \texttt{mydoc.nmc} back to a file of the same name in the document
+directory. And it is brisk, not least because plain text export ignores
+ERT insets (and hence preview insets wrapped around ERT insets), nor
+does it evaluate \verb`\eval` commands in math insets.
+
+\subsection{A final tweak?}
+
+But one still needs to \emph{remember} to press the shortcut. The
+thought arises: can \emph{closing} the current document trigger the
+copying process? \LyX{} provides a means of linking two commands and
+assigning a keyboard shortcut to them with its \texttt{command-sequence}
+\LyX{} function. I suggest assigning a shortcut\textsf{ }to
+\begin{lyxcode}
+command-sequence~buffer-export~text;~view-close
+\end{lyxcode}
+Indeed, why not reassign the current shortcut for \texttt{view-close},\texttt{
+}which is \textsf{Ctrl+W} on my system, to this command sequence?
+(I use the \texttt{cua} key bindings – check the \textsf{Bind file:}
+slot in \textsf{Tools \lyxarrow{} Preferences \lyxarrow{} Editing \lyxarrow{}
+Shortcuts}.)
+
+Please note, however, that \emph{this will work as intended only from
+\LyX{} 2.4.0}.\footnote{Due for release in the first half of 2021.}
+For \LyX{} 2.3 and earlier, the command sequence will generally fail
+because of `asynchronous' processing – \texttt{buffer-export }and
+\texttt{view-close} use different threads and the latter may well
+start before the former is complete. From \LyX{} 2.4.0 this defect
+has been fixed. You press your shortcut, the export to plain text
+occurs and the \texttt{.nmc} file is copied back to the document directory,
+then the current view is closed.
+
+\section{Using \protect\LyX{} notes}
+
+The central fact about a \LyX{} note is that it does not contribute
+to the pdf. But instant preview still works there. This suggests a
+possibility: that a calculation be performed within a \LyX{} note and
+the result saved using \verb`\nmcReuse` within the same preview inset.
+The saved value is now available \emph{from file} for use elsewhere
+in the document. In this way, some selected content from a LyX note
+\emph{can} find its way into the pdf when the document is compiled.
+
+\chapter{Reference summary}
+
+\section{Commands defined in \texttt{numerica}}
+\begin{enumerate}
+\item \texttt{\textbackslash nmcEvaluate, \textbackslash eval }
+\item \texttt{\textbackslash nmcInfo, \textbackslash info, }
+\item \texttt{\textbackslash nmcReuse, \textbackslash reuse}
+\item \texttt{\textbackslash q, \textbackslash Q }(`cleave' commands)
+\end{enumerate}
+Provided they have not already been defined when \texttt{numerica}
+is loaded, the following commands are defined in \texttt{numerica}
+using \verb`\DeclareMathOperator` from \texttt{amsmath} :
+\begin{enumerate}
+\item \texttt{\textbackslash arccsc, \textbackslash arcsec, \textbackslash arccot}
+\item \texttt{\textbackslash csch, \textbackslash sech}
+\item \texttt{\textbackslash asinh, \textbackslash acosh, \textbackslash atanh,
+\textbackslash acsch, \textbackslash asech, \textbackslash acoth}
+\item \texttt{\textbackslash sgn, \textbackslash lb}
+\end{enumerate}
+Provided they have not already been defined, the following commands
+are defined in \texttt{numerica} using \verb`\DeclarePairedDelimiter`
+from \texttt{mathtools}:
+\begin{lyxcode}
+\textbackslash abs,~\textbackslash ceil,~\textbackslash floor
+\end{lyxcode}
+The following commands have been redefined in \texttt{numerica} to
+give more spacing around the underlying \verb`\wedge` and \verb`\vee`
+symbols:
+\begin{lyxcode}
+\textbackslash land,~\textbackslash lor
+\end{lyxcode}
+
+\section{\textquoteleft Digestible\textquoteright{} content}
+
+\texttt{numerica} knows how to deal with the following content, meaning
+that any of these elements occurring within an \verb`\eval` command
+should not of itself cause a \texttt{numerica} error. Not all formatting
+commands affect display of the output.
+\begin{enumerate}
+\item variable names (sequences of tokens given values in the variable~=~value
+list)
+\item digits, decimal point
+\begin{enumerate}
+\item \texttt{1, 2, 3, 4, 5, 6, 7, 8, 9, 0, .}
+\end{enumerate}
+\item constants
+\begin{enumerate}
+\item \texttt{e, \textbackslash pi, \textbackslash gamma, \textbackslash phi,
+\textbackslash deg, \textbackslash infty }(sometimes)
+\end{enumerate}
+\item arithmetic operators
+\begin{enumerate}
+\item \texttt{+, -, {*}, /, \textasciicircum , \textbackslash times, \textbackslash cdot,
+\textbackslash div}
+\end{enumerate}
+\item logical operators
+\begin{enumerate}
+\item \texttt{\textbackslash wedge, \textbackslash land, \textbackslash vee,
+\textbackslash lor, \textbackslash neg, \textbackslash lnot}
+\end{enumerate}
+\item comparisons
+\begin{enumerate}
+\item \texttt{=, <, >, \textbackslash ne, \textbackslash neq, \textbackslash le,
+\textbackslash leq, \textbackslash ge, \textbackslash geq}
+\item (if \texttt{amssymb} loaded) \texttt{\textbackslash nless, \textbackslash ngtr,
+\textbackslash geqq, \textbackslash geqslant, \textbackslash leqq,
+\textbackslash leqslant, \textbackslash ngeq, \textbackslash ngeqq,
+\textbackslash ngeqslant, \textbackslash nleq, \textbackslash nleqq,
+\textbackslash nleqslant}
+\end{enumerate}
+\item brackets, bracket-like elements, modifiers
+\begin{enumerate}
+\item \texttt{( ), {[} {]}, \textbackslash\{ \textbackslash\}}
+\item \texttt{\textbackslash lparen \textbackslash rparen} (from \texttt{mathtools})\texttt{,
+\textbackslash lbrack \textbackslash rbrack, \textbackslash lbrace
+\textbackslash rbrace}
+\item \texttt{\textbackslash lvert \textbackslash rvert, \textbackslash lfloor
+\textbackslash rfloor, \textbackslash lceil \textbackslash rceil}
+\item \texttt{| |} (no nesting, deprecated)
+\item \texttt{\textbackslash left \textbackslash right, \textbackslash bigl
+\textbackslash bigr, \textbackslash Bigl \textbackslash Bigr, \textbackslash biggl
+\textbackslash biggr, \textbackslash Biggl \textbackslash Biggr}
+\item \texttt{.} \texttt{/ |} (used with a modifier)
+\item \texttt{\textbackslash abs{[}{]}\{\}, \textbackslash abs{*}\{\},
+\textbackslash floor{[}{]}\{\}, \textbackslash floor{*}\{\}, \textbackslash ceil{[}{]}\{\},
+\textbackslash ceil{*}\{\}}
+\end{enumerate}
+\item unary functions (in the mathematical sense)
+\begin{enumerate}
+\item \texttt{\textbackslash sin, \textbackslash cos, \textbackslash tan,
+\textbackslash csc, \textbackslash sec, \textbackslash cot}
+\item \texttt{\textbackslash arcsin, \textbackslash arccos, \textbackslash arctan,
+arccsc, \textbackslash arcsec, \textbackslash arccot }
+\item \texttt{\textbackslash sin\textasciicircum\{-1\}, \textbackslash cos\textasciicircum\{-1\},
+\textbackslash tan\textasciicircum\{-1\}, \textbackslash csc\textasciicircum\{-1\},
+\textbackslash sec\textasciicircum\{-1\}, \textbackslash cot\textasciicircum\{-1\}}
+\item \texttt{\textbackslash sinh, \textbackslash cosh, \textbackslash tanh,
+\textbackslash csch, \textbackslash sech, \textbackslash coth }
+\item \texttt{\textbackslash asinh, \textbackslash acosh, \textbackslash atanh,
+\textbackslash csch, \textbackslash sech, \textbackslash acoth}
+\item \texttt{\textbackslash sinh\textasciicircum\{-1\}, \textbackslash cosh\textasciicircum\{-1\},
+\textbackslash tanh\textasciicircum\{-1\}, \textbackslash csch\textasciicircum\{-1\},
+\textbackslash sech\textasciicircum\{-1\}, \textbackslash acoth\textasciicircum\{-1\}}
+\item \texttt{\textbackslash exp, \textbackslash lb, \textbackslash lg,
+\textbackslash ln, \textbackslash log, \textbackslash log\_\{\},
+\textbackslash sgn, \textbackslash surd}
+\item \texttt{\textbackslash sqrt\{\}, \textbackslash abs{[}{]}\{\}, \textbackslash abs{*}\{\},
+\textbackslash floor{[}{]}\{\}, \textbackslash floor{*}\{\}, \textbackslash ceil{[}{]}\{\},
+\textbackslash ceil{*}\{\}}
+\item \texttt{!, !! }(prepended argument)
+\end{enumerate}
+\item binary functions
+\begin{enumerate}
+\item \texttt{\textbackslash tfrac\{\}\{\}, \textbackslash frac\{\}\{\},
+\textbackslash dfrac\{\}\{\}}
+\item \texttt{\textbackslash tbinom\{\}\{\}, \textbackslash binom\{\}\{\},
+\textbackslash dbinom\{\}\{\}}
+\item \texttt{\textbackslash sqrt{[}{]}\{\}}
+\end{enumerate}
+\item $n$-ary functions
+\begin{enumerate}
+\item \texttt{\textbackslash min, \textbackslash max, \textbackslash gcd}
+\end{enumerate}
+\item sum, prod
+\begin{enumerate}
+\item \texttt{\textbackslash sum\_\{\}\textasciicircum , \textbackslash prod\_\{\}\textasciicircum{} }
+\end{enumerate}
+\item formatting commands
+\begin{enumerate}
+\item \texttt{,} (comma, in $n$-ary functions)
+\item \texttt{\{\}, \textbackslash\textbackslash , \&, \textbackslash to}
+\item \texttt{\textbackslash dots, \textbackslash ldots, \textbackslash cdots,}
+\item \texttt{\textbackslash{} , \textbackslash ,{}, \textbackslash ;,
+\textbackslash :, \textbackslash !, \textbackslash >}
+\item \texttt{\textbackslash thinspace, \textbackslash quad, \textbackslash qquad
+, \textbackslash hfill, \textbackslash hfil}
+\item \texttt{\textbackslash phantom\{\}, \textbackslash vphantom\{\},
+\textbackslash hphantom\{\}}
+\item \texttt{\textbackslash xmathstrut{[}{]}\{\}} \texttt{, \textbackslash splitfrac\{\}\{\},
+\textbackslash splitdfrac\{\}\{\} }(from \texttt{mathtools}), \texttt{\textbackslash mathstrut}
+\item \texttt{\textbackslash displaystyle, \textbackslash textstyle, \textbackslash scriptstyle,
+\textbackslash scriptscriptstyle}
+\item \texttt{\textbackslash label\{\}, \textbackslash ensuremath\{\},
+\textbackslash text\{\}, \textbackslash mbox\{\}}
+\item \texttt{\textbackslash begin\{\}, \textbackslash end\{\}}
+\end{enumerate}
+\item font commands
+\begin{enumerate}
+\item \texttt{\textbackslash mathrm\{\}, \textbackslash mathit\{\}, \textbackslash mathcal\{\},
+\textbackslash mathtt\{\}, \textbackslash mathbf\{\}, \textbackslash mathbb\{\},
+\textbackslash mathsf\{\}, \textbackslash mathfrak\{\}, \textbackslash mathscr\{\},
+\textbackslash mathnormal\{\}, \textbackslash boldsymbol\{\}}
+\end{enumerate}
+\end{enumerate}
+
+\section{Settings}
+
+\subsection{Available \texttt{\textbackslash nmcEvaluate} settings}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{xx}} & {\small int (0/1)} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{()}} & {\small int (0/1/2)} & {\small trig. function arg. parsing} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{o}} & & {\small degree switch for trig. funcions} & \tabularnewline
+{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{vvmode}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvd}} & {\small token(s)} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{vvi}} & {\small token(s)} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{*} & & {\small switch to suppress equation numbering (if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
+in }{\small\texttt{vvd}}{\small )} & \tabularnewline
+{\small\texttt{p}} & char(s) & {\small punctuation (esp. in display-style)} & {\small\texttt{,}}\tabularnewline
+{\small\texttt{S+}} & {\small int} & {\small extra rounding for stopping criterion, sums} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{S?}} & {\small$\text{int}\ge0$} & {\small query stopping with these final terms, sums} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{P+}} & {\small int} & {\small extra rounding for stopping criterion, products} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{P?}} & {\small$\text{int}\ge0$} & {\small query stopping with these final terms, products} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{reuse}} & {\small int} & {\small form of result saved with }{\small{\small\verb`\nmcReuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+\subsection{Available configuration file settings}
+\begin{center}
+\bigskip{}
+\begin{tabular}{ll}
+\toprule
+key & default\tabularnewline
+\midrule
+{\small\texttt{rounding}} & {\small\texttt{6}}\tabularnewline
+{\small\texttt{pad}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{output-sci-notation}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{output-exponent-char}} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{input-sci-notation}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{input-exponent-char}} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{multitoken-variables}} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{logarithm-base}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{intify-rounding}} & {\small\texttt{14}}\tabularnewline
+{\small\texttt{vv-display}} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{\small\texttt{vv-inline}} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{sum-extra-rounding}} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{sum-query-terms}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{prod-extra-rounding}} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{prod-query-terms}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{eval-reuse}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{document}
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@@ -0,0 +1,1973 @@
+%% LyX 2.4.0-alpha1 created this file. For more info, see https://www.lyx.org/.
+%% Do not edit unless you really know what you are doing.
+\documentclass[english,tableposition=top]{report}
+\usepackage{lmodern}
+\renewcommand{\sfdefault}{lmss}
+\renewcommand{\ttdefault}{lmtt}
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\usepackage[latin9]{inputenc}
+\setcounter{secnumdepth}{3}
+\usepackage{color}
+\definecolor{shadecolor}{rgb}{0.667969, 1, 1}
+\usepackage{babel}
+\usepackage{array}
+\usepackage{wrapfig}
+\usepackage{booktabs}
+\usepackage{framed}
+\usepackage{url}
+\usepackage{amsmath}
+\usepackage[unicode=true,pdfusetitle,
+ bookmarks=true,bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=2,
+ breaklinks=true,pdfborder={0 0 1},backref=section,colorlinks=true]
+ {hyperref}
+
+\makeatletter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+%% Because html converters don't know tabularnewline
+\providecommand{\tabularnewline}{\\}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
+\newenvironment{centred}%
+ {\begin{center}\baselineskip=13pt\parskip=1pt}{\end{center}}
+\newenvironment{lyxcode}
+ {\par\begin{list}{}{
+ \setlength{\rightmargin}{\leftmargin}
+ \setlength{\listparindent}{0pt}% needed for AMS classes
+ \raggedright
+ \setlength{\itemsep}{0pt}
+ \setlength{\parsep}{0pt}
+ \normalfont\ttfamily}%
+ \item[]}
+ {\end{list}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
+\usepackage[plus]{numerica}
+
+\newcommand\rel{\,\varrho\;}
+\DeclareMathOperator{\erf}{erf}
+\DeclareMathOperator{\gd}{gd}
+
+\reuse
+
+\usepackage{upquote}
+
+\makeatother
+
+\begin{document}
+\title{\texttt{numerica-plus}~\\
+}
+\author{Andrew Parsloe\\
+(\url{ajparsloe at gmail.com})}
+\maketitle
+\begin{abstract}
+In this module of the \verb`numerica` package, commands are defined
+which iterate and find fixed points of functions of a single variable,
+which find the zeros or extrema of such functions, and which calculate
+the terms of recurrence relations.\\
+\\
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\paragraph*{Note:}
+\begin{itemize}
+\item {\normalsize This document applies to version 1.0.0 of }{\normalsize\texttt{numerica-plus}}{\normalsize .def.}{\small\par}
+\item {\normalsize Reasonably recent versions of the \LaTeX 3 bundles }{\normalsize\texttt{l3kernel}}{\normalsize{}
+and }{\normalsize\texttt{l3packages}}{\normalsize{} are required.}{\small\par}
+\item {\normalsize I refer a number of times in this document to }{\normalsize\emph{Handbook
+of Mathematical Functions}}{\normalsize , edited by Milton Abramowitz
+and Irene A. Segun, Dover, 1965. This is abbreviated to }{\normalsize\emph{HMF}}{\normalsize .}{\small\par}
+\end{itemize}
+\end{shaded}%
+\end{minipage}
+\end{abstract}
+\tableofcontents{}
+
+\chapter{Introduction}
+
+By calling \texttt{numerica} with the \texttt{plus} package option
+in the preamble,
+\begin{lyxcode}
+\textbackslash usepackage{[}plus{]}\{numerica\}
+\end{lyxcode}
+a number of additional commands are loaded:
+\begin{itemize}
+\item \verb`\nmcIterate`, a command to iterate a function (apply it repeatedly
+to itself), including finding fixed points (values $x$ where $f(x)=x$);
+\item \verb`\nmcSolve`, a command to find the zeros of functions of a single
+variable (values $x$ for which $f(x)=0$) or, failing that, local
+maxima or minima of such functions;
+\item \verb`\nmcRecur`, a command to calculate the values of terms in recurrence
+relations in a single (recurrence) variable (like the terms of the
+Fibonacci sequence or Legendre polynomials).
+\end{itemize}
+These commands are defined in the package \texttt{numerica-plus.def}
+which is loaded with \texttt{numerica.sty} when the \texttt{plus}
+option is used. The new commands all share the syntax of the \verb`\nmcEvaluate`
+command. I will discuss them individually in later chapters but turn
+first to an example that illustrates their use and gives a sense of
+`what they are about'.
+
+\section{Example of use: the rotating disk}
+
+\label{sec:introExampleOfUse}Consider a disk rotating uniformly with
+angular velocity $\omega$ in an anticlockwise sense in an inertial
+system in which the disk's centre \textbf{0} is at rest. Three distinct
+points \textbf{1}, \textbf{2}, \textbf{3} are fixed in the disk and,
+in a co-rotating polar coordinate system centred at \textbf{0}, have
+polar coordinates $(r_{i},\theta_{i})$ ($i,j=1,2,3$). Choose \textbf{01}
+as initial line so that $\theta_{1}=0$.
+
+The cosine rule for solving triangles tells us that the time $t_{ij}$
+in the underlying inertial system for a signal to pass from \textbf{i}
+to \textbf{j} satisfies the equation
+\[
+t_{ij}=c^{-1}\sqrt{r_{i}^{2}+r_{j}^{2}-2r_{i}r_{j}\cos(\theta_{j}-\theta_{i}+\omega t_{ij})}\equiv f(t_{ij}),
+\]
+where $c$ is the speed of light. (Equally, we could be describing
+an acoustic signal between points on a disk rotating uniformly in
+a still, uniform atmosphere \textendash{} in which case $c$ would
+be the speed of sound.) Although the equation doesn't solve algebraically
+for the time $t_{ij},$ it does tell us that $t=t_{ij}$ is a \emph{fixed
+point} of the function $f(t)$. To calculate fixed points we use the
+command \verb`\nmcIterate`, or its short-name form \verb`\iter`,
+with the star option, \verb`\iter*`. For \verb`\iter` the star option
+means: continue iterating until a fixed point has been reached and,
+as with the \verb`\eval` command, suppress all elements from the
+display save for the numerical result.
+
+First, though, values need to be assigned to the various parameters.
+Suppose we use units in which $c=30,$ $r_{1}=10$ and $r_{3}=20$;
+also $\omega=0.2$ radians per second and $\theta_{13}\equiv\theta_{3}-\theta_{1}=0.2$
+radians. We shall give $t$ an initial value $1$ (plucked from the
+air). Its position as the rightmost item in the vv-list tells \verb`\iter`
+that $t$ is the iteration variable:
+\begin{verbatim}
+ \iter*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)}
+ }[ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=1 ], \quad\info{iter}.
+\end{verbatim}
+$\Longrightarrow$ \iter*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)}
+ }[ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=1 ],
+\quad\info{iter}. The \verb`\nmcInfo` command or, more briefly \verb`\info`, can
+be used to display the number of iterations required to attain the
+fixed-point value.
+
+To six figures, only five iterations are needed, which seems rapid
+but we can check this by substituting $t=0.356899$ back into the
+formula and \verb`\eval`-uating it:
+\begin{verbatim}
+ \eval*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)}
+ }[ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=0.356899 ]
+\end{verbatim}
+$\Longrightarrow$ \eval*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)}
+ }[ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=0.356899 ], confirming that we have indeed calculated a fixed point. That it
+took only $5$ iterations can be checked by omitting the asterisk
+from the \verb`\iter` command and specifying the total number of
+iterations to perform. I choose \texttt{do=}7 to show not just the
+$5$th iteration but also the next two just to confirm that the result
+is stable. We shall view all $7$: \texttt{see=7}. I have also suppressed
+display of the vv-list by giving the key \texttt{vvd}\emph{ }an empty
+value,\footnote{See the associated document \texttt{numerica-basics.pdf}, the chapter
+on settings.} since there are too many variables to display on a line:
+\begin{verbatim}
+ \iter[do=7,see=7,vvd=]
+ {\[ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)} \]}
+ [ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=1 ]
+\end{verbatim}
+$\Longrightarrow$ \iter[do=7,see=7,vvd=]
+ {\[ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)} \]}
+ [ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=1 ]
+\noindent \begin{flushleft}
+This display makes clear that on the $5$th iteration, the $6$-figure
+value has been attained.
+\par\end{flushleft}
+
+Alternatively, we could use the \verb`\nmcRecur` command, or its
+short-name form \verb`\recur`, to view the successive iterations
+since an iteration is a first-order recurrence: $f_{n+1}=f(f_{n})$:
+\begin{verbatim}
+ \recur[do=8,see1=0,see2=5,vvd={,\\(vv)\\},*]
+ {\[ f_{n+1}=c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega f_{n})} \]}
+ [ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2, f_{0}=1 ]
+\end{verbatim}
+$\Longrightarrow$ \recur[do=8,see1=0,see2=5,vvd={\\(vv)\\},*]
+ {\[ f_{n+1}=c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega f_{n})} \]}
+ [ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2, f_{0}=1 ]
+
+\noindent I have specified \texttt{do=8} terms rather than $7$ since
+the zero-th term ($f_{0}=1$) is included in the count. I've chosen
+to view the last $5$ of them but none prior to those by writing \texttt{see1=0,see2=5}.
+The vv-list has been suppressed from the display (\texttt{vvd=} )
+and the numerical results forced onto a new line (without an equation
+number \textendash{} the \texttt{{*}} in the vv-list).
+
+Another and perhaps more obvious way to find the value of $t_{13}$,
+is to look for a zero of the function $f(t)-t$. That means using
+the command \verb`\nmcSolve` or its short-name form \verb`\solve`.
+I shall do so with the star option \verb`\solve*` which suppresses
+display of all but the numerical result. A trial value for $t$ is
+required. I have chosen \texttt{t=0}:
+\begin{verbatim}
+ \solve*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)} - t }
+ [ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=0 ], \quad\nmcInfo{solve}.
+\end{verbatim}
+$\Longrightarrow$ \solve*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)}-t
+ }[ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=0],
+\quad\nmcInfo{solve}.
+
+Nearly the same answer as before is attained but this time many more
+steps have been required. This is to be expected. The \verb`\solve`
+command uses the bisection method. Since $1/2^{10}\approx1/10^{3}$
+or $10$ bisections for $3$ decimal places, we can expect about $20$
+bisections being needed for a $6$-decimal-place answer. The particular
+form of the \verb`\nmcInfo` command display, `$1+20$ steps', indicates
+that it took $1$ search step to find an interval in which the function
+vanished and, having found that interval, $20$ bisections to narrow
+the position of the zero to $6$-figures.
+
+\subsection{Circuits}
+
+Okay, so we can calculate the time taken in the underlying inertial
+system for a signal to pass from one point of the rotating disk to
+another. How long does it take to traverse the circuit \textbf{1}
+to \textbf{2} to \textbf{3} back to \textbf{1}? That means forming
+the sum $t_{12}+t_{23}+t_{31}$, hence calculating the separate $t_{ij}$
+and then using \verb`\eval` to calculate their sum.
+
+To make things neater and easier to read, I assume a little symmetry.
+Let the coordinates of \textbf{1} be $(a,0),$ of \textbf{2} be $(r,-\theta)$,
+and of \textbf{3} be $(r,\theta)$: \textbf{2} and \textbf{3} are
+at the same radial distance from the centre \textbf{0} and at the
+same angular distance but on opposite sides of the line \textbf{01},
+\textbf{3} an angle $\theta$ ahead of \textbf{1}, \textbf{2} an angle
+$\theta$ behind \textbf{1}. The rotation is in the direction of positive
+$\theta$. Rather than just calculate $t_{12}+t_{23}+t_{31}$ for
+the circuit \textbf{1231}, I also calculate the time $t_{13}+t_{32}+t_{21}$
+for a signal to traverse the opposite circuit \textbf{1321} and compare
+them (form the difference).
+
+Note that with \textbf{2} and \textbf{3} positioned as they are relative
+to \textbf{1},
+\[
+t_{31}=c^{-1}\sqrt{r^{2}+a^{2}-2ra\cos(\theta-\omega t_{31})}
+\]
+and
+\[
+t_{12}=c^{-1}\sqrt{a^{2}+r^{2}-2ar\cos(\theta-\omega t_{12})}.
+\]
+Both $t_{12}$ and $t_{31}$ are fixed points of the same function
+\[
+F_{-1}(t)=c^{-1}\sqrt{r^{2}+a^{2}-2ra\cos(\theta-\omega t).}
+\]
+Similarly, both $t_{13}$ and $t_{21}$ are fixed points of
+\[
+F_{+1}(t)=c^{-1}\sqrt{r^{2}+a^{2}-2ra\cos(\theta+\omega t)}.
+\]
+ Geometrically one can show that $t_{31}=t_{12}$ and $t_{13}=t_{21}$.
+Analytically, with the values $c=30,a=10,r=20$,
+\[
+0<F_{\pm1}(t)\le c^{-1}\sqrt{r^{2}+a^{2}+2ar}=c^{-1}(r+a)=1,
+\]
+so that we need look for fixed points only in the interval $[0,1]$.
+Differentiating $F_{-1}(t)$ shows the derivative is negative on $[0,1]$,
+hence that $F_{-1}$ is strictly decreasing there and can intersect
+the identity mapping at most once on the interval. Similarly, differentiating
+$F_{+1}(t)$ shows the derivative
+\[
+\frac{ar\omega\sin(\theta+\omega t)}{c\sqrt{r^{2}+a^{2}-2ar\cos(\theta+\omega t)}}
+\]
+is positive on $[0,1]$, but with the given values for $c,a$ and
+$r$, and with $\theta=0.2,\omega=0.2$, $F'_{+1}(t)<0.1$ on $[0,1]$:
+hence $F_{+1}$ increases sufficiently slowly on the interval that
+the identity mapping can intersect it at most once there. In both
+cases, there can be at most one fixed point.
+
+It follows that $t_{12}=t_{31}$ and $t_{13}=t_{21}$ and the round
+trip times are $2t_{12}+t_{23}$ and $2t_{13}+t_{32}$.
+
+\subsubsection{Nesting commands}
+
+The calculation to compare the round trip times can be done `in one
+go' by nesting the \verb`\iter*` commands to calculate the $t_{ij}$
+inside an \verb`\eval*` command to form the sum of their results
+and then take the difference.
+\begin{verbatim}
+ $
+ \eval*{ % circuit 1231
+ 2\iter*{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta-\omega t)} }
+ + \iter*{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta+\omega t)} }
+ % circuit 1321
+ - 2\iter*{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta+\omega t)} }
+ - \iter*{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta-\omega t)} }
+ }[ c=30,a=10,r=20,\theta=0.2,\omega=0.2, t=1 ]
+ $
+\end{verbatim}
+$\Longrightarrow$ $
+ \eval*{ % circuit 1231
+ 2\iter*{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta-\omega t)} }
+ + \iter*{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta+\omega t)} }
+ % circuit 1321
+ - 2\iter*{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta+\omega t)} }
+ - \iter*{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta-\omega t)} }
+ }[ c=30,a=10,r=20,\theta=0.2,\omega=0.2, t=1 ]
+ $.
+
+By itself this result is of little interest beyond seeing that \texttt{numerica}
+can handle it. What \emph{is} interesting is to find values of $\theta$,
+say, given the other parameters and especially the value of $r$,
+such that the time difference vanishes. Is there a circuit, despite
+the rotation of the disk, such that it takes a signal the same time
+to travel in opposite directions around the circuit? That is a job
+for \verb`\nmcSolve` (or \verb`\solve`):
+\begin{verbatim}
+ \solve[p=.,var=\theta,vvi=]{$ % circuit 1231
+ 2\times\iter*[var=t]{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta-\omega t)} }
+ + \iter*[var=t]{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta+\omega t)} }
+ % circuit 1321
+ - 2\times\iter*[var=t]{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta+\omega t)} }
+ - \iter*[var=t]{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta-\omega t)} }
+ $}[ c=30,a=10,r=20,\theta=0.1,\omega=0.2, {t}=1 ]
+\end{verbatim}
+$\Longrightarrow$ \solve[p=.,var=\theta,vvi=]{$ % circuit 1231
+ 2\times\iter*[var=t]{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta-\omega t)} }
+ + \iter*[var=t]{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta+\omega t)} }
+ % circuit 1321
+ - 2\times\iter*[var=t]{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta+\omega t)} }
+ - \iter*[var=t]{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta-\omega t)} }
+ $}[ c=30,a=10,r=20,\theta=0.1,\omega=0.2, {t}=1 ]
+
+One point to note here is the use of \verb`\times` (in \verb`2\times\iter*`)
+as against the previous example with \verb`\eval*` where \verb`\times`
+was not used. In the present case the formula is displayed (\verb`\solve`
+wraps around math delimiters); the star on the \verb`\eval` command
+in the previous example suppressed display of the formula. Without
+the \verb`\times` the `$2$'s would be juxtaposed against the following
+decimals, making it look as if the signal travel times were $20.537778$
+and $20.614419$ and no doubt causing perplexity. Also note the \texttt{vvi=}
+to suppress display of the vv-list.
+
+But this is not a research paper on the rotating disk. I wished to
+show how the different commands of \texttt{numerica-plus} can be used
+to explore a meaningful problem. And although it looks as if a lot
+of typing is involved, once $c^{-1}\sqrt{r^{2}+a^{2}-2ra\cos(\theta-\omega t)}$
+has been formed in \LaTeX{} and values specified in the vv-list, much
+of the rest is copy-and-paste followed by minor editing.
+
+\section{Shared syntax of the new commands}
+
+\texttt{numerica-plus} offers three new commands for three processes:
+\verb`\nmcIterate` (short-name form \verb`\iter`) for iterating
+functions, \verb`\nmcSolve` (short-name form \verb`\solve`) for
+finding the zeros or (local) extrema of functions, and \verb`\nmcRecur`
+(short-name form \verb`\recur`) for calculating terms of recurrence
+relations.
+
+All three commands share the syntax of the \verb`\nmcEvaluate` (or
+\verb`\eval`) command detailed in the associated document \texttt{numerica-basics.pdf}.
+When all options are used then the command looks like
+\begin{centred}
+\noindent \verb`\nmcIterate*[settings]{expr.}[vv-list][num. format]`
+\end{centred}
+You can substitute \verb`\nmcSolve`, or \verb`\nmcRecur` for \verb`\nmcIterate`
+here. The arguments are similar to those for \verb`\nmcEvaluate`.
+\begin{enumerate}
+\item \verb`*` optional switch; if present ensures a single number output
+with no formatting, or an appropriate error message if the single
+number cannot be produced;
+\item \verb`[settings]` optional comma-separated list of \emph{key=value
+}settings for this particular command and calculation;
+\item \verb`{expr.}` the only mandatory argument; the mathematical expression
+in \LaTeX{} form that is the object of interest;
+\item \verb`[vv-list]` optional comma-separated list of \emph{variable=value
+}items; for \verb`\iter` and \verb`\solve` the \emph{rightmost}
+(or innermost) variable in the vv-list may have special significance;
+\item \verb`[num. format]` optional format specification for presentation
+of the numerical result (rounding, padding with zeros, scientific
+notation); boolean output is suppressed for these commands.
+\end{enumerate}
+Boolean output makes little sense for these three commands and is
+turned \emph{off}. Entering a question mark in the trailing number-format
+option of each command has no effect \textendash{} see \xA7.
+
+All commands also share with \verb`\nmcEvaluate` changes in the way
+the result is displayed depending on whether they wrap around math
+delimiters, or are wrapped within math delimiters, or are employed
+in the absence of math delimiters. These distinctions are relevant\emph{
+only if the optional star {*} is absent.}
+\begin{itemize}
+\item When the star option is used, the \emph{result} is a number only without
+any formatting or vv-list display, or an error message is displayed.
+\item When the star option is not used and one of the following is the case
+\begin{itemize}
+\item the command wraps around math delimiters, e.g. \verb`\iter{$ expr. $}`,
+then the result is displayed in the form \emph{expression~=~result}
+or the form \emph{expression~$\rightarrow$~result} as appropriate,
+where \emph{result} may be multi-line and may include the vv-list,
+\item the command is used within math delimiters, e.g. \verb`\[\iter...\]`,
+then only the \emph{result} is presented, where \emph{result} may
+be multi-line and may include the vv-list,
+\item the command is used in the absence of delimiters, then the result
+is presented as if it had been used between \verb`\[` and \verb`\]`.
+\end{itemize}
+\end{itemize}
+Looking at the various examples in the preceding section on the rotating
+disk you will see illustrations of all these situations.
+
+\subsection{Settings}
+
+\begin{table}
+\noindent \centering{}\caption{\protect\label{tab:introSettingsInherited}Inherited settings options}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{xx}} & {\small int (0/1)} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{()}} & {\small int (0/1/2)} & {\small trig. function arg. parsing} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{o}} & & {\small degree switch for trig. funcions} & \tabularnewline
+{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{vvmode}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvd}} & {\small tokens} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{vvi}} & {\small token(s)} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{*} & & {\small suppress equation numbering if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
+in }{\small\texttt{vvd}} & \tabularnewline
+{\small\texttt{p}} & char(s) & {\small punctuation (esp. in display-style)} & {\small\texttt{, (comma)}}\tabularnewline
+{\small\texttt{S+}} & {\small int} & {\small extra rounding for stopping criterion for sums} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{S?}} & {\small$\text{int}\ge0$} & {\small stopping criterion query terms for sums} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{P+}} & {\small int} & {\small extra rounding for stopping criterion for products} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{P?}} & {\small$\text{int}\ge0$} & {\small stopping criterion query terms for products} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+Nearly all the settings available to the \verb`\eval` command are
+available to these other commands. To save switching between documents
+I reproduce in Table~\ref{tab:introSettingsInherited} the options
+found in \texttt{numerica-basics.pdf}, although for discussion of
+the options you will need to refer to that document. In addition,
+each of the present commands also has settings of its own, discussed
+at the relevant parts of the following chapters.
+
+\subsection{Nesting}
+
+Provided the starred form of a command actually does produce a numerical
+result and not an error message then it can be nested within the main
+argument of any one of the commands, including itself. The example
+of use, \xA7\ref{sec:introExampleOfUse} above, shows several examples
+of this. The starred form can also be used in the vv-list of any one
+of the commands, including itself. The associated document \texttt{numerica-basics.pdf}
+shows examples of an \verb`\eval*` command being used in the vv-list
+of an \verb`\eval` command.
+
+\chapter{Iterating functions: \texttt{\textbackslash nmcIterate}}
+
+\label{chap:Iterating-functions}Only in desperation would one try
+to evaluate a continued fraction by stacking fraction upon fraction
+upon fraction like so:
+\[
+1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1}}}}}}}}}}}}
+\]
+\texttt{numerica-plus} provides a command for tackling problems like
+this sensibly. In such problems a function is repeatedly applied to
+itself (\emph{iterated}). This can be achieved in \texttt{numerica}
+through the command \verb`\nmcIterate` or (short-name form) \verb`\iter`.
+Thus to evaluate this continued fraction we write
+\begin{centred}
+\verb`\iter[do=15,see=5]{\[ 1+1/x \]}[x=1]` $\Longrightarrow$ \iter[do=15,see=5]{\[ 1+1/x \]}[x=1]
+\end{centred}
+The \verb`\iter` command evaluates $1+1/x$ when $x=1$ and then
+uses this value as a new $x$-value to substitue into $1+1/x$, to
+once again evaluate and use as a new $x$-value, and so on. It looks
+as if the repeated iterations are approaching \verb`\eval{$ \tfrac{\sqrt{5}+1}2 $}`
+$\Longrightarrow$ \eval{$ \tfrac{\sqrt{5}+1}2 $}.
+
+Increasing the number of iterations in the example from \texttt{do=15}
+to, say, \texttt{do=18}, shows that this is indeed the case. I've
+taken the opportunity to also put a question mark in the number-format
+option to get boolean output in order to show that it has \emph{no}
+effect:
+\begin{centred}
+\verb`\iter[do=18,see=5]{\[ 1+1/x \]}[x=1][?]` $\Longrightarrow$
+\iter[do=18,see=5]{\[ 1+1/x \]}[x=1][?]
+\end{centred}
+But iteration of functions is not limited to continued fractions.
+Particularly since the emergence of chaos theory, iteration has become
+an important study in its own right. Any function whose range lies
+within its domain can be iterated \textendash{} repeatedly applied
+to itself. The cosine, for instance,
+\begin{centred}
+\verb`\iter[do=20,see=4]{\[ \cos x \]}[x=\pi/2]` $\Longrightarrow$
+\iter[do=20,see=4]{\[ \cos x \]}[x=\pi/2]
+\end{centred}
+which displays the first one and last four of 20 iterations of $\cos x$
+when $x=\tfrac{\pi}{2}$. It looks as if the cosine is `cautiously'
+approaching a limit, perhaps around $0.738$ or $0.739$. You need
+to nearly double the number of iterations (\texttt{do=40}) to confirm
+that this is so.
+
+The logistic function $kx(1-x)$ exhibits a variety of behaviours
+depending on the value of $k$. For instance, with $k=3.5$ we get
+a period-4 cycle:
+\begin{centred}
+\verb`\iter[do=12,see=8]{\[ kx(1-x) \]}[k=3.5,x=0.5]` $\Longrightarrow$
+ \iter[do=12,see=8]{\[ kx(1-x) \]}[k=3.5,x=0.5]
+\end{centred}
+and with $k=3.1$ we get a period-2 cycle, although it takes many
+more iterations to stabilize there:
+\begin{centred}
+\verb`\iter[do=44,see=8]{\[ kx(1-x) \]}[k=3.1,x=0.5]` $\Longrightarrow$
+\iter[do=44,see=8]{\[ kx(1-x) \]}[k=3.1,x=0.5]
+\end{centred}
+
+\section{Star (\texttt{{*}}) option: fixed points}
+
+In the first two of these examples, iteration eventually ended at
+a \emph{fixed point}. This is a point $x$ where $f(x)=x$ and, by
+induction, $f^{n}(x)=x$. Appending a star (asterisk) to the \verb`\iter`
+command is the signal for iteration to continue until a fixed point
+has been reached at the specified rounding value:
+\begin{centred}
+\verb`\iter*{ 1+a/x }[a=n(n+1),n=1,x=1]` $\Longrightarrow$ \iter*{ 1+a/x }[a=k(k+1),k=1,x=1]
+\end{centred}
+(with the default rounding value $6$).\footnote{For your own interest try also putting $n=2,3,4,\ldots$ .}
+The star overrides any value for the number of iterations to perform
+(the \texttt{do} key) that may have been entered in the settings option.
+It also overrides any elements of the display other than the numerical
+result. With the star option math delimiters are irrelevant \textendash{}
+other than displaying minus signs correctly when \verb`\iter*` is
+between them.
+
+A function may not approach a fixed point when iterated \textendash{}
+see the examples with the logistic function above. To prevent an infinite
+loop \verb`\iter*` counts the number of iterations performed and
+when that number reaches a certain cut-off value \textendash{} the
+default is $100$ \textendash{} the loop terminates and a message
+is displayed:
+\begin{centred}
+\verb`\iter*{kx(1-x)}[k=3.5,x=0.5]` $\Longrightarrow$ \iter*{kx(1-x)}[k=3.5,x=0.5]
+\end{centred}
+In this case we \emph{know} that a fixed point does not exist, but
+that may not always be the case. One response is to change parameter
+values or starting value of the iteration variable. For instance,
+changing the parameter value to $k=1.5$,
+\begin{centred}
+\verb`\iter*{kx(1-x)}[k=1.5,x=0.5]` $\Longrightarrow$ \iter*{kx(1-x)}[k=1.5,x=0.5],
+\end{centred}
+and a fixed point is now attained.
+
+But should a fixed point still not eventuate after `fiddling' with
+parameter and start values, there are two general adjustments one
+might try: either
+\begin{enumerate}
+\item reduce the rounding value, from the default $6$ (or the one specified),
+to a smaller value, or
+\item increase the cut-off figure from the default 100 to some higher value.
+\end{enumerate}
+The former is done via the trailing \verb`[num format]` optional
+argument; the latter is done via the settings option, see \xA7\ref{sec:iterSettings-option}.
+
+\subsection{Use with \texttt{\textbackslash nmcInfo}}
+
+It is of interest to know how many iterations are required to reach
+a fixed point at a particular rounding value. That knowledge allows
+a good guess as to whether a fixed point will be attained at a greater
+rounding value. Thus when iterating the function
+\[
+f(t_{ij})=c^{-1}\sqrt{r_{i}^{2}+r_{j}^{2}-2r_{i}r_{j}\cos(\theta_{j}-\theta_{i}+\omega t_{ij})}
+\]
+in \xA7\ref{sec:introExampleOfUse} only $5$ iterations were required
+to attain $6$-figure accuracy for the fixed point. That information
+came by following the \verb`\iter*` command with \verb`\nmcInfo`
+(or \verb`\info`) with the argument \verb`iter`. And generally,
+for any `infinite' process, follow the command with an \verb`\info`
+command if you want to know how many `steps' \textendash{} in the
+present case iterations \textendash{} are required to achieve the
+result. So, if $5$ iterations achieve $6$-figure accuracy, presumably
+something like $10$ iterations will achieve $12$-figure accuracy:
+\begin{verbatim}
+ \iter*{ c^{-1}\sqrt{r_i^2+r_j^2-2r_i r_j
+ \cos(\theta_{ij}+\omega t)}
+ }[ c=30,r_i=10,r_j=20,\theta_{ij}=0.2,
+ \omega=0.2,t=1 ][12],
+ \quad\info{iter}.
+\end{verbatim}
+$\Longrightarrow$ \iter*{ c^{-1}\sqrt{r_i^2+r_j^2-2r_i r_j
+ \cos(\theta_{ij}+\omega t)}
+ }[ c=30,r_i=10,r_j=20,\theta_{ij}=0.2,
+ \omega=0.2,t=1 ][12],
+\quad\info{iter}. Only $9$ iterations suffice to achieve $12$-figure accuracy. Or
+again, with another earlier example,
+\begin{centred}
+\verb`$ \iter*{\cos x}[x=\pi/2] $,\ \info{iter}.` $\Longrightarrow$
+$\iter*{\cos x}[x=\pi/2] $,\ \info{iter}.
+\end{centred}
+That suggests that around $74$ iterations will give $12$-figure
+accuracy, well within the cut-off figure of $100$:
+\begin{centred}
+\verb`$ \iter*{\cos x}[x=\pi/2][12] $,\ \info{iter}.` $\Longrightarrow$
+$\iter*{\cos x}[x=\pi/2][12] $,\ \info{iter}.
+\end{centred}
+
+\section{Settings option}
+
+\label{sec:iterSettings-option}The settings option is a comma-separated
+list of items of the form \emph{key~=~value}.
+
+\subsection{Inherited settings}
+
+Nearly all of the keys discussed in the settings\emph{ }option for
+\verb`\nmcEvaluate` are available for \verb`\nmcIterate`. Table~\ref{tab:introSettingsInherited}
+above lists these, repeating a table from \texttt{numerica-basics.pdf}.
+Thus should a quantity in the vv-list depend on the iteration variable,
+forcing an implicit mode calculation, simply enter (as with \verb`\eval`)
+\texttt{vvmode=1} in the settings option:
+\begin{centred}
+\verb`\iter*[vvmode=1]{$ 1+f(x) $}[f(x)=a/x,a=12,x=1]` $\Longrightarrow$
+\iter*[vvmode=1]{$ 1+f(x) $}[f(x)=a/x,a=12,x=1].
+\end{centred}
+Implicit in this example is the default multi-token setting \texttt{xx=1}
+inherited from \verb`\eval` and ensuring that the multi-token variable
+$f(x)$ is treated correctly.
+
+Let's add \verb`dbg=1` to the example:
+\begin{centred}
+\verb`\iter*[dbg=1,vvmode=1]{$ 1+f(x) $}[f(x)=a/x,a=12,x=1]` $\Longrightarrow$
+\iter*[dbg=1,vvmode=1]{$ 1+f(x) $}[f(x)=a/x,a=12,x=1]
+\end{centred}
+The multi-token variable \verb`f(x)` has been changed to the single-token
+\verb`\_nmcu`. The values of \verb`x` and \verb`\_nmcu` shown are
+those of the \emph{final} iteration.
+
+\subsection{\texttt{\textbackslash iter}-specific settings}
+
+In addition to the inherited settings there are some specific to \verb`\nmcIterate`.
+These are listed in Table~\ref{tab:iterSettings}.
+
+\subsubsection{Iteration variable}
+
+In nearly all of the examples so far, the iteration variable has been
+the rightmost variable in the vv-list and has not needed to be otherwise
+specified. However it is sometimes not feasible to indicate the variable
+in this way. In that case, entering
+\begin{lyxcode}
+var~=~<variable~name>
+\end{lyxcode}
+in the settings option enables the variable to be specified, irrespective
+of what the rightmost variable in the vv-list is. Here, \texttt{<variable
+name>} will generally be a character like \texttt{x} or \texttt{t}
+or a token like \verb`\alpha`, but it could also be a multi-token
+name like {\ttfamily\verb`x'`}\texttt{ }or \texttt{\textbackslash beta\_\{ij\}}
+(or even \texttt{Fred} if you so chose). Although the iteration variable
+can be independently specified like this, it must still be given an
+initial \emph{value} in the vv-list \textendash{} only it need not
+be the rightmost variable.
+
+In the following example the rightmost variable is $a$ which is clearly
+\emph{not} the iteration variable:
+\begin{centred}
+\verb`\iter[var=x'',do=40,see=5]{$ 1+a/x'' $}[x''=a/6,a=6][*]` $\Longrightarrow$
+\iter[var=x'',do=40,see=5]{$ 1+a/x'' $}[x''=a/6,a=6][*]
+\end{centred}
+
+\subsubsection{Extra rounding for fixed-point calculations}
+
+\begin{table}
+\centering{}\caption{\protect\label{tab:iterSettings}Settings for \texttt{\textbackslash nmcIterate}}
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{var}} & {\small token(s)} & {\small iteration variable} & \tabularnewline
+{\small\texttt{+}} & {\small int} & {\small fixed point extra rounding} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{max}} & {\small int > 0} & {\small max. iteration count (fixed points)} & {\small\texttt{100}}\tabularnewline
+{\small\texttt{do}} & {\small int > 0} & {\small number of iterations to perform} & {\small\texttt{5}}\tabularnewline
+{\small\texttt{see}} & {\small int > 0} & {\small number of final iterations to view} & {\small\texttt{4}}\tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}/\mathtt{2}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+\texttt{numerica} determines that a fixed point has been reached when
+the difference between successive iterations vanishes when rounded
+to the current rounding value. One might want reassurance that this
+really is the correct value by seeking a fixed point at a higher rounding
+value than that displayed. This extra rounding is achieved by entering
+\begin{lyxcode}
++~=~<integer>
+\end{lyxcode}
+in the settings option. By default this extra rounding is set to zero.
+
+We have seen before that $\cos x$ starting at $x=\tfrac{1}{2}\pi$
+takes $37$ iterations to reach a $6$-figure fixed point $0.739085$,
+about $6$ iterations per decimal place. By entering \texttt{+=1}
+in the settings option the number of iterations is increased to $43$,
+$6$ more than $37$ but, reassuringly, the $6$-figure result that
+is displayed remains unchanged:
+\begin{centred}
+\verb`$ \iter*[+=1]{\cos x}[x=\pi/2] $,\ \info{iter}.` $\Longrightarrow$
+$ \iter*[+=1]{\cos x}[x=\pi/2] $,\ \info{iter}.
+\end{centred}
+
+\subsubsection{Maximum {\small iteration count for fixed point searches}}
+
+To prevent a fixed-point search from continuing indefinitely when
+no fixed point exists, there needs to be a maximum number of iterations
+specified after which point the search is called off. By default this
+number is $100$. To change it enter
+\begin{lyxcode}
+max~=~<positive~integer>
+\end{lyxcode}
+in the settings option.
+
+\subsubsection{Number of iterations to perform}
+
+To specify the number of iterations to perform enter
+\begin{lyxcode}
+do~=~<positive~integer>
+\end{lyxcode}
+in the settings option. Note that if the \verb`*` option is present
+this value will be ignored and iteration will continue until either
+a fixed point or the maximum iteration count is reached. By default
+\texttt{do} is set to $5$. (Note that \texttt{do} can be set to a
+greater number than \texttt{max}; \texttt{max} applies only to \verb`\iter*`.)
+
+\subsubsection{Number of iterations to view}
+
+To specify the number of final iteations to view enter
+\begin{lyxcode}
+see~=~<positive~integer>
+\end{lyxcode}
+in the settings option. By default \texttt{see} is set to $4$. Always
+it is the \emph{last} \texttt{see} iterations that are displayed.
+If \texttt{see} is set to a greater value than \texttt{do}, all iterations
+are shown. If the star option is used the \texttt{see} value is ignored.
+
+\subsubsection{Form of result saved by \texttt{\textbackslash reuse}}
+
+By entering
+\begin{lyxcode}
+reuse~=~<integer>
+\end{lyxcode}
+in the settings option of the \verb`iter` command it is possible
+to specify the form of result that is saved when using \verb`\nmcReuse`.
+(This setting has no effect when the star option is used with \verb`\nmcIterate`.
+In that case only the numerical result of the fixed point calculation
+\textendash{} if successful \textendash{} is saved.) The possibilities
+are:
+\begin{itemize}
+\item \texttt{int=0} (or any integer $\mathtt{\ne1,2}$) saves the display
+resulting from the \verb`\iter` command (the default);
+\item \texttt{int=1} saves a comma-separated list of pairs of the form:\texttt{ }~\\
+\texttt{\{$k$, value-of-$k$-th-iterate\}};
+\item \texttt{int=2} saves a comma-separated list of iterate values.
+\end{itemize}
+Note that the number and content of the items in the lists are those
+resulting from the \texttt{see} setting (the number of iterations
+to view).
+\begin{verbatim}
+ \iter[reuse=1,do=12,see=4]
+ {\[ kx(1-x) \]}[k=3.5,x=0.5]
+ \reuse[logistic]
+\end{verbatim}
+$\Longrightarrow$ \iter[reuse=1,do=12,see=4]{\[ kx(1-x) \]}[k=3.5,x=0.5] \reuse[logistic]
+
+\noindent whence \verb`\logistic` $\Longrightarrow$ \logistic.
+As you can see the control sequence \verb`\logistic` displays as
+a comma-separated list of numbers, alternating between the ordinal
+of the iterate, and the iterate value. In fact these values are grouped
+as braced pairs separated by commas in \verb`\logistic`, as can be
+seen by using \TeX 's \verb`\meaning` command:
+\begin{centred}
+\verb`\meaning\logistic` $\Longrightarrow$ \meaning\logistic
+\end{centred}
+
+\subsection{Changing default values}
+
+\begin{wraptable}[12]{o}{0.5\columnwidth}%
+\centering{}\caption{\protect\label{tab:iterSettingsDefaults}Defaults for \texttt{\textbackslash nmcIterate}}
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small iter-extra-rounding} & {\small\texttt{0}}\tabularnewline
+{\small iter-max-iterations} & {\small\texttt{100}}\tabularnewline
+{\small iter-do} & {\small\texttt{5}}\tabularnewline
+{\small iter-see-last} & {\small\texttt{4}}\tabularnewline
+{\small iter-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}\end{wraptable}%
+If you wish to change the default values of the various settings for
+\verb`\nmcIterate` this can be done by entering new values in a configuration
+file \texttt{numerica.cfg} as described in the chapter on settings
+in the associated document \texttt{numerica-basics.pdf}. The relevant
+keys are listed in Table~\ref{tab:iterSettingsDefaults}, corresponding
+to the \verb`+`, \verb`max`, \verb`do`, \verb`see` and \verb`reuse`
+settings of the \verb`\iter` command. (Obviously it makes no sense
+to have a default setting for the iteration variable. That will change
+from case to case.)
+
+\section{Errors}
+
+By errors I refer to \texttt{numerica} errors rather than \LaTeX{}
+errors. We have already met one in the discussion of fixed points:
+\begin{centred}
+\verb`\iter*{kx(1-x)}[k=3.5,x=0.5]` $\Longrightarrow$ \iter*{kx(1-x)}[k=3.5,x=0.5]
+\end{centred}
+For a function to be iterated indefinitely, its range must lie within
+or be equal to its domain. If even part of the range of a function
+lies outside its domain, then on repeated iteration there is a chance
+that a value will eventually be calculated which lies in this `outside'
+region. Iteration cannot continue beyond this point and an error message
+is generated. As an example consider the inverse cosine, \verb`\arccos`.
+This can be iterated only so far as the iterated values lie between
+$\pm1$ inclusive. If we try to iterate \verb`\arccos` at 0 for example,
+since $\cos\frac{1}{2}\pi=0$, $\arccos0=\eval{0.5\pi}[4]$ (which
+is $\tfrac{1}{2}\pi$) so only a first iterate is possible. But we
+could choose an initial value more carefully; $37$ iterations of
+the cosine at $\tfrac{1}{2}\pi$ led to a fixed point $0.739085$,
+so let's choose $0.739085$ as initial point and perform $37$ iterations:
+\begin{centred}
+\verb`\iter[do=37,see=4]{\[ \arccos x \]}[x=0.739085]` $\Longrightarrow$
+\iter[do=37,see=4]{\[ \arccos x \]}[x=0.739085]
+\end{centred}
+The result of the $37$th iteration is greater than $1$. Thus increasing
+the number of iterations to 38 should generate an error message:
+\begin{centred}
+\verb`\iter[do=38,see=4]{\[ \arccos x \]}[x=0.739085]` $\Longrightarrow$\iter[do=38,see=4]{\[ \arccos x \]} [x=0.739085]
+\end{centred}
+\verb`l3fp` objects when asked to find the inverse cosine of a number
+greater than $1$.
+
+\chapter{Finding zeros and extrema: \texttt{\textbackslash nmcSolve}}
+
+\texttt{numerica} provides a command\textbf{ }\verb`\nmcSolve` (short-name
+form \verb`\solve`) for finding a zero of a function, should it have
+one. In the following example,
+\begin{centred}
+\verb`\solve[p]{\[ e^{ax}-bx^2 \]}[a=2,b=3,{x}=0]` $\Longrightarrow$
+\solve[p]{\[ e^{ax}-bx^2 \]}[a=2,b=3,{x}=0]
+\end{centred}
+I have sought and found a solution $x$ to the equation $e^{ax/2}-bx^{2}=0$
+when $a=2$ and $b=3$, starting with a trial value $x=0$, entered
+as the \emph{rightmost} variable in the vv-list (and em-braced since
+I don't want this trial value displaying in the presentation of the
+result). Although $x$ has been found to the default six-figure accuracy,
+it is evident that the function vanishes only to five figures. Let's
+check:
+\begin{centred}
+\verb`\eval{$ bx^2 $}[b=3,x=x=-0.390647]` $\Longrightarrow$ \eval{$ bx^2 $}[b=3,x=-0.390647],
+
+\verb`\eval{$ e^{ax} $}[a=2,x=-0.390646]` $\Longrightarrow$ \eval{$ e^{ax} $}[a=2,x=-0.390647];
+\end{centred}
+the values agree save in the final digit.
+
+This discrepancy in the final decimal place or places is a general
+feature of solutions found by \verb`\solve`. It is the value of $x$,
+not the value of $f(x)$, that is being found (in this case) to six
+figures. If the graph of a function crosses the $x$-axis steeply
+then the $x$ value (the zero) may be located to a higher precision
+than the function value. Conversely, if the graph of a function crosses
+the $x$-axis gently (at a shallow angle) then the function value
+will vanish to a greater number of decimal places than the zero (the
+$x$ value) is located.
+
+A second example, which we can check with values tabulated in \emph{HMF},
+is
+\begin{centred}
+\verb`\solve{$ \tan x - \lambda x $}[\lambda=-1/0.8,{x}=1][5]` $\Longrightarrow$
+\solve{$ \tan x - \lambda x $}[\lambda=-1/0.8,{x}=1][5].
+\end{centred}
+Table 4.19 of \emph{HMF }lists values of $x$ against $\lambda$ and
+this is the value tabulated there.
+
+\section{Extrema}
+
+A function may not have a zero or for the given initial trial value
+and initial step in the search for a zero there may be a local extremum
+in the way. In that case \texttt{numerica} may well locate the local
+extremum (maximum or minimum but not a saddle point). For example
+for the quadratic $(2x-1)^{2}+3x+1$ the \verb`\solve` command gives
+the result
+\begin{centred}
+\verb`\solve[vvi=]{$ (2x-1)^2+3x+1 $}[x=2]` $\Longrightarrow$\solve[vvi=]{$ (2x-1)^2+3x+1 $}[{x}=2].
+\end{centred}
+Since $(2x-1)^{2}+3x+1\ne0$ for any (real number) $x$, we deduce
+that the quadratic takes a minimum value $1.9375$ at $x=0.125$ \textendash{}
+easily confirmed analytically. This particular minimum is a global
+minimum but in general any extremum found is only \emph{local}. The
+function may well take larger or smaller values (or vanish for that
+matter) further afield.
+
+It is also worth noting in this example the \verb`vvi=` in the settings
+option which suppresses display of the vv-list. (The only member of
+the vv-list is the trial value \verb`x=2` which we do not want to
+display.)
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+Note that the function for which a zero is being sought is \emph{not}
+equated to zero when entered in the \verb`\solve` command. It is
+\verb`\solve{ f(x) }`, not \verb`\solve{ f(x)=0 }`. This is precisely
+because it may be an extremum that is found rather than a zero (if
+extremum or zero is found at all \textendash{} think $e^{x}$). The
+display of the result makes clear which is which, equating $f(x)$
+to its value, zero or extremum depending on what has been found, as
+you can see in the preceding examples.\end{shaded}%
+\end{minipage}
+
+\subsection{The search strategy}
+
+\label{subsec:solveSearch-strategy}If you have some sense of where
+a function has a zero, then choose a trial value in that vicinity.
+\verb`\solve` uses a bisection method to home in on the zero. It
+therefore needs \emph{two} initial values. For the first it uses the
+trial value you specify, call it $a$ and for the second, by default,
+it uses $a+1$. (The default value $1$ for the initial step from
+the trial value can be changed in the settings option; see \xA7\ref{sec:solveSettings-option}.)
+If $f(a)$ and $f(a+1)$ have opposite signs then that is good. Bisection
+of the interval $[a,a+1]$ can begin immediately in order to home
+in on the precise point where $f$ vanishes. Write $b=a+1$.
+\begin{itemize}
+\item Let $c=\tfrac{1}{2}(a+b)$; if $f(c)=0$ the zero is found; otherwise
+either $f(a),f(c)$ are of opposite signs or $f(c),f(b)$ are of opposite
+signs. In the former case write $a_{1}=a,$ $b_{1}=c$; in the latter
+case write $a_{1}=c$, $b_{1}=b$ and then redefine $c=\tfrac{1}{2}(a_{1}+b_{1})$.
+Continue the bisection process, either until an exact zero $c$ of
+$f$ is reached ($f(c)=0$) or a value $c$ is reached where the difference
+between $a_{n+1}$ and $b_{n+1}$ is zero at the specified rounding
+value. (But note, $f(c)$ may not vanish at that rounding value \textendash{}
+the zero might be elsewhere in the interval and $f$ might cross the
+axis at a steep slope.)
+\end{itemize}
+However $f(a)$ and $f(b)=f(a+1)$ may not have opposite signs. If
+we graph the function $y=f(x)$ and suppose $f(a),f(b)$ are distinct
+but of the same sign, then the line through the points $(a,f(a))$,
+$(b,f(b))$ will intersect the $x$-axis to the left of $a$ or the
+right of $b$ depending on its slope. We search always \emph{towards
+the $x$-axis} in steps of $b-a$ ($=1$ with default values).
+\begin{itemize}
+\item If the line intersects the axis to the left of $a$ then $c=a-(b-a)$
+and we set $a_{1}=c,b_{1}=a$; if the line intersects the axis to
+the right of $b$ then $c=b+(b-a)$ and we set $b_{1}=c,a_{1}=b$.
+The hope is that by always taking steps in the direction towards the
+$x$-axis that eventually $f(c)$ will be found to lie on the \emph{opposite}
+side of the axis from $f(a_{n})$ or $f(b_{n})$, at which point the
+bisection process begins.
+\item Of course this may not happen. At some point $c$ may lie to the left
+of $a_{n}$ but $\left|f(c)\right|>\left|f(a_{n})\right|$, or $c$
+may lie to the right of $b_{n}$ but $\left|f(c)\right|>\left|f(b_{n})\right|$.
+The slope has reversed. In that case we halve the step value to $\tfrac{1}{2}(b-a)$
+and try again in the same direction as before from the same point
+as before ($a_{n}$ or $b_{n}$ as the case may be).
+\item Should we find at some point that $f(a_{n})=f(b_{n})$ then the previous
+strategy does not apply. In this case we choose $a_{n+1}$ and \textbf{$b_{n+1}$}
+at the quarter and three-quarter marks between $a_{n}$ and $b_{n}$.
+Either $f(a_{n+1})$ and $f(b_{n+1})$ will differ and the previous
+search strategy can start again or we are on the way to finding an
+extremum of $f$.
+\end{itemize}
+As already noted it is also possible that our function has neither
+zeros nor extrema. To prevent the search continuing indefinitely,
+\texttt{numerica} uses a cut-off value for the maximum number of steps
+pursued \textendash{} by default set at 100.
+
+\subsubsection{Elusive extrema}
+
+The strategy `search always towards the $x$-axis' has a consequence:
+it means that a local maximum above the $x$-axis will almost certainly
+not be found, since `towards the $x$-axis' pulls the search away
+from the maximum. Similarly a local minimum below the $x$-axis will
+also not be found since `towards the $x$-axis' pulls the search
+away from the minimum.
+
+One way of countering this elusiveness is to add a constant value
+(possibly negative) to the function whose zeros and extrema are being
+sought. The zeros of the function will change but the abscissae ($x$
+values) of the extrema remain unchanged. If the constant is big enough
+it will push a local minimum above the axis where it can be found
+or, for a negative constant, push a local maximum below the axis where
+it can be found.
+
+For example $f(x)=x^{3}-x$ has roots at $-1,0,1$, a local maximum
+at $-\tfrac{1}{\surd3}$ and a local minimum at $\tfrac{1}{\surd3}$.
+To locate the minimum, I have added an unnecessarily large constant
+$k$ to $f(x)$. ($k=1$ would have sufficed, but note, $k=0$ fails.)
+\begin{centred}
+\verb`\solve{$ x^3-x+k $}[k=5,{x}=0.5]` $\Longrightarrow$ \solve{$ x^3-x+k $}[k=5,{x}=0.5].
+\end{centred}
+Checking, \verb`\eval{$\tfrac1{\surd 3}$}` $\Longrightarrow$ \eval{$\tfrac1{\surd 3}$}.
+There is a discrepancy in the $6$th decimal place which can be eliminated
+by using the extra rounding setting; see \xA7\ref{subsec:solveExtraRounding}.
+
+\section{Star (\texttt{{*}}) option}
+
+A starred form of the\textbf{ }\verb`\nmcSolve` command suppresses
+all elements of display of the result apart from the numerical value.
+When nesting a \verb`\solve` command within another command this
+is the form to use. Thus with the `elusive' extremum example above,
+we can find the actual value of the minimum by nesting the starred
+\verb`\solve` command within the vv-list of an \verb`\eval` command:
+\begin{centred}
+\verb`\eval{$ x^3-x $}[x={\solve*{y^3-y+k}[k=5,y=0.5]}]` $\Longrightarrow$
+\eval{$ x^3-x $}[x={\solve*{y^3-y+k}[k=5,y=0.5]}].
+\end{centred}
+(Note the braces around the \verb`\solve*` and arguments to hide
+\emph{its} square-bracketed vv-list from the parsing of the vv-list
+of the \verb`\eval` command.) The result is to be compared with \verb`$\eval*{x^3-x}[x=\tfrac1{\surd3}]$`
+$\Longrightarrow$ $\eval*{x^3-x}[x=\tfrac1{\surd3}]$.
+
+\section{Settings option}
+
+\label{sec:solveSettings-option}The settings option is a comma-separated
+list of items of the form \emph{key~=~value}.
+
+\subsection{Inherited settings}
+
+The keys discussed in the settings\emph{ }option for \verb`\nmcEvaluate`
+are also available for \verb`\nmcSolve`. The very first example in
+this chapter used the punctuation option \texttt{p} (\verb`\solve[p]{\[... `)
+inherited from the \verb`\eval` command to ensure a comma after the
+display-style presentation of the result. We also saw in the quadratic
+example illustrating extrema the use of \texttt{vvi} with no value
+to suppress display of the vv-list: \verb`\solve[vvi=]{$ ...`.
+
+Putting \verb`dbg=1` produces a familiar kind of display. Using the
+function
+\[
+ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+\]
+from the rotating disk problem,
+\begin{verbatim}
+ \solve[dbg=1,var=t,p=.,vvd={,}\\(vv)\\,*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4]
+\end{verbatim}
+$\Longrightarrow$ \solve[dbg=1,var=t,p=.,vvd={,}\\(vv)\\,*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=1][4]
+
+\subsubsection{Multi-line display of the result}
+
+\label{subsec:solveMulti-line-display}By default the result is presented
+on a single line. Unless the star option is being used, this can be
+of the form \emph{function = function value, (vv-list) $\rightarrow$
+result}. It takes only a slightly complicated formula and only a few
+variables in the vv-list before this becomes a crowded line, likely
+to exceed the line width and extend into the margin. To split the
+display over two lines choose a \texttt{vvd} specification in the
+vv-list like, for instance, \texttt{vvd=\{,\}\textbackslash\textbackslash (vv)}.
+The \texttt{\textbackslash\textbackslash} is a trigger for \texttt{numerica}
+to replace whatever environment the \texttt{\textbackslash eval}
+command is wrapped around with a \texttt{multline} environment. An
+asterisk in the vv-list replaces \texttt{multline} with \texttt{multline{*}}
+so that no equation number is used:
+\begin{verbatim}
+ \solve[p=.,vvd={,}\\(vv),*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4]
+\end{verbatim}
+$\Longrightarrow$ \solve[p=.,vvd={,}\\(vv),*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4]
+\begin{verbatim}
+
+\end{verbatim}
+You could introduce a third line if you wished to display the result
+on a line of its own by using the spec. \texttt{vvd=\{,\}\textbackslash\textbackslash (vv)\textbackslash\textbackslash}:
+\begin{verbatim}
+ \solve[p=.,vvd={,}\\(vv)\\,*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4]
+\end{verbatim}
+$\Longrightarrow$ \solve[p=.,vvd={,}\\(vv)\\,*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4]
+
+The function evaluates to $-0.0015$. Is this a zero that has been
+found or an extremum? To find out, the calculation needs to be carried
+out to a higher rounding value which is the reason why \verb`\nmcSolve`
+has an extra rounding setting; see \xA7\ref{subsec:solveExtraRounding}
+below.
+
+\subsection{\texttt{\textbackslash solve}-specific settings}
+
+In addition there are some settings peculiar to \verb`\nmcSolve`.
+These are listed in Table~\ref{tab:solveSettings}.
+
+\begin{table}
+\centering{}\caption{\protect\label{tab:solveSettings}Settings for \texttt{\textbackslash nmcSolve}}
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{var}} & {\small token(s)} & {\small equation variable} & \tabularnewline
+{\small\texttt{dvar}} & {\small real $\ne0$ } & {\small initial step size} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{+}} & {\small int} & {\small extra rounding} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{max}} & {\small int > 0} & {\small max. number of steps before cut off} & {\small\texttt{100}}\tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+
+
+\subsubsection{Equation variable}
+
+By default the equation variable is the \emph{rightmost} variable
+in the vv-list. This may not always be convenient. A different equation
+variable can be specified by entereing
+\begin{lyxcode}
+var~=~<variable~name>
+\end{lyxcode}
+in the vv-list. \texttt{<variable name>} will generally be a single
+character or token \textendash{} $x$, $t$, $\alpha$, $\omega$
+\textendash{} but is not necessarily of this kind. Multi-token names
+are perfectly acceptable (with the default \texttt{xx=1} multi-token
+setting).
+
+\subsubsection{Initial step size}
+
+The vv-list must contain the equation variable set to some trial value.
+But \verb`\solve` needs \emph{two} initial values to begin its search
+for a zero or extremum; see \xA7\ref{subsec:solveSearch-strategy}. If
+the equation variable is set to a trial value $a$ then by default
+the second value is $a+1$. The `$+1$' here can be changed by entering
+in the settings option
+\begin{lyxcode}
+dvar~=~<non-zero~real~number>
+\end{lyxcode}
+For instance, \texttt{dvar=-1}, or \texttt{dvar=\textbackslash pi}
+are two valid specifications of initial step size.
+
+The notation is prompted by the use of expressions like $x+dx$ in
+calculus.
+
+An example where the default step value is too big and a smaller one
+needs to be specified is provided by Planck's radiation function (\emph{HMF
+}Table 27.2),
+\[
+f(x)=\frac{1}{x^{5}(e^{1/x}-1)}.
+\]
+From the (somewhat coarse-grained) table in \emph{HMF }it is clear
+that there is a maximum of about 21.2 when $x$ is a little more than
+$0.2$. This is a maximum above the $x$-axis and hence `elusive'.
+To find it, substract $100$ (say) from the formula and again use
+the ability to nest commands to display the result. Note the \verb`dvar=0.1`
+in the settings option of the \verb`\solve*` command:
+\begin{verbatim}
+ \eval[p=.]{\[ \frac1{x^5(e^{1/x}-1)} \]}
+ [ x={ \solve*[dvar=0.1]
+ { \frac1{y^5(e^{1/y}-1)}-100 }[y=0.1]
+ } ]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]{\[ \frac1{x^5(e^{1/x}-1)} \]}
+ [ x={ \solve*[dvar=0.1]
+ { \frac1{y^5(e^{1/y}-1)}-100 }[y=0.1]
+ } ]
+
+\noindent The maximum is indeed a little over $21.2$ and the $x$
+value a little more than $0.2$.
+
+The default \verb`dvar=1` is too big for this problem. From the table
+in \emph{HMF},\emph{ }$f(0.1)=4.540$ and $f(1.1)=0.419$. By subtracting
+$100$ from $f(x)$ the `towards the $x$-axis' search strategy
+leads to negative values of $x$, an artifact of the subtraction.
+
+\subsubsection{Extra rounding}
+
+\label{subsec:solveExtraRounding}\verb`\solve` determines that a
+zero or an extremum has been reached when the difference between two
+successive bisection values or step values vanishes at the specified
+rounding value (the value in the final trailing optional argument
+of the \verb`\solve` command; $6$ by default). If our function is
+$f(x)$ then $\abs{x_{n+1}-x_{n}}=0$ to the specified rounding value
+and (assuming $x_{n+1}>x_{n}$) there is a critical value $x_{c}\in[x_{n},x_{n+1}]$
+such that $f(x_{c})=0$ (to all 16 places of decimals that \texttt{l3fp}
+works to). But in general the critical value $x_{c}$ will not coincide
+with $x_{n}$ or $x_{n+1}$. If $f(x)$ crosses the $x$-axis at a
+steep angle it may well be that although $f(x_{c})$ vanishes to all
+$16$ places of decimals, $f(x_{n}),f(x_{n+1})$ may well not vanish
+at the (generally smaller) specified rounding value. For instance,
+suppose $f(x)=1000x-3000$.
+\begin{centred}
+\verb`\solve[vvi=]{$ 1000x-3000 $}[x=e][4*]` $\Longrightarrow$ \solve[vvi=]{$ 1000x-3000 $}[x=e][4*].
+\end{centred}
+In this example, although the difference between successive $x$ values
+vanishes to $4$ places of decimals, $f(x)$ does not, not even to
+$2$ places.
+
+This suggests it would be helpful to be able to use two rounding values,
+one to determine when a zero has been found and one for the visible
+display. This is done through the extra rounding key in the settings
+option. Enter
+\begin{lyxcode}
++~=~<integer>
+\end{lyxcode}
+in the settings option of the \verb`\solve` command to add \texttt{<integer>}
+to the rounding value determining the conclusion of the calculation.
+By default the setting is \texttt{+=}0. The extra rounding setting
+leaves the display rounding unaffected.
+
+With this option available it is easy to check that \verb`+=3` suffices
+to ensure that both $x$ and $f(x)$ vanish to $4$ places of decimals,
+\begin{centred}
+\verb`\solve[+=3]{$ 1000x-3000 $}[x=e][4*]` $\Longrightarrow$ \solve[+=3]{$ 1000x-3000 $}[x=e][4*],
+\end{centred}
+and that \texttt{+=2} does not, i.e., we need to locate the zero to
+$4+3=7$ figures to ensure the function vanishes to $4$ figures.
+
+There is no need for the \texttt{<integer>} to be positive. In fact
+negative values can illuminate what is going on. In the first of the
+following, the display is to $10$ places but the calculation is only
+to $10-4=6$ places. In the second, the display is again to $10$
+places, but the calculation is to $10-3=7$ places.
+\begin{centred}
+\verb`\solve[+=-4]{$ 1000x-3000 $}[x=e][10*]` $\Longrightarrow$
+\solve[+=-4]{$ 1000x-3000 $}[x=e][10*],
+
+\verb`\solve[+=-3]{$ 1000x-3000 $}[x=e][10*]` $\Longrightarrow$
+\solve[+=-3]{$ 1000x-3000 $}[x=e][10*].
+\end{centred}
+Only in the second does $f(x)=1000x-3000$ vanish when rounded to
+$4$ figures.
+
+Returning to an earlier example (\xA7\ref{subsec:solveMulti-line-display})
+in which it was not entirely clear whether a zero or an extremum had
+been found, we can now resolve the confusion. Use the extra rounding
+setting (and pad with zeros to emphasize the $4$-figure display by
+adding an asterisk in the trailing optional argument):
+\begin{verbatim}
+ \solve[+=2,vvd={,}\\(vv),*]
+ {$
+ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4*]
+\end{verbatim}
+$\Longrightarrow$ \solve[+=2,vvd={,}\\(vv),*]
+ {$
+ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4*]
+
+\subsubsection{Maximum number of steps before cut-off}
+
+Once two function values have been found of opposite sign, bisection
+is guaranteed to arrive at a result. The problem is the \emph{search}
+for two such values. This may not terminate \textendash{} think of
+functions like $e^{x}$or $1/x$ which lack both zeros and extrema.
+To prevent an infinite loop, \verb`\solve` cuts off the search after
+$100$ steps. This cut-off value can be changed for a calculation
+by entering
+\begin{lyxcode}
+max~=~<positive~integer>
+\end{lyxcode}
+in the settings option.
+
+To illustrate, we know that $1/x$ has neither zero nor extremum,
+but we do not get an infinite loop; we get an error message if we
+attempt to `solve' $1/x$:
+\begin{centred}
+\verb`\solve{ 1/x }[x=1]` $\Longrightarrow$ \solve{ 1/x }[x=1]
+\end{centred}
+
+\subsubsection{Form of result saved by \texttt{\textbackslash reuse}}
+
+As wth \verb`\eval` and \verb`\iter` it is possible to specify to
+some extent what is saved to file when using \verb`\reuse` after
+a \verb`\solve` command. The form of entry in the settings option
+is
+\begin{lyxcode}
+reuse~=~<integer>
+\end{lyxcode}
+If the star option is used with the \verb`\solve` command the numerical
+result is the only thing saved, but in the absence of the star option,
+\begin{itemize}
+\item \texttt{reuse=0} saves\emph{ the form that is displayed}. For example,
+if the display is of the form \emph{function = function value, (vv-list)
+$\rightarrow$ result} then that is what is saved; this is the default
+behaviour;
+\item \texttt{reuse=1} (or any non-zero integer) saves only the numerical
+result.
+\end{itemize}
+
+\subsection{Changing default values}
+
+\begin{wraptable}[12]{o}{0.5\columnwidth}%
+\centering{}\caption{\protect\label{tab:solveSettingsDefaults}Defaults for \texttt{\textbackslash nmcSolve}}
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small solve-first-step} & 1\tabularnewline
+{\small solve-extra-rounding} & {\small\texttt{0}}\tabularnewline
+{\small solve-max-steps} & {\small\texttt{100}}\tabularnewline
+{\small solve-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}\end{wraptable}%
+If you wish to change the default values of the various settings for
+\verb`\nmcSolve` this can be done by entering new values in a configuration
+file \texttt{numerica.cfg} as described in the chapter on settings
+in the associated document \texttt{numeric-basics.pdf}. The relevant
+keys are listed in Table~\ref{tab:solveSettingsDefaults}, corresponding
+to the \verb`dvar`, \verb`+`, \verb`max` and \verb`reuse` settings
+of the \verb`\solve` command. (Obviously it makes no sense to have
+a default setting for the solution variable. That will change from
+case to case.)
+
+\chapter{Recurrence relations: \texttt{\textbackslash nmcRecur}}
+
+One of the simplest recurrence relations is that determining the Fibonacci
+numbers, $f_{n+2}=f_{n+1}+f_{n}$, with initial values $f_{0}=f_{1}=1$.
+The command \verb`\nmcRecur`, short-name form \verb`\recur`, allows
+calculation of the terms of this sequence:
+\begin{verbatim}
+ $ \nmcRecur[do=8,see1=8,...]
+ { f_{n+2}=f_{n+1}+f_{n} }
+ [f_{1}=1,f_{0}=1] $
+\end{verbatim}
+$\Longrightarrow$ $\nmcRecur[do=8,see1=8,...]
+ { f_{n+2}=f_{n+1}+f_{n} }
+ [f_{1}=1,f_{0}=1]$
+
+The recurrence relation is entered in the main argument (between braces),
+the initial values in the vv-list trailing the main argument, and
+the display specification is placed in the settings option: \texttt{do=8}
+terms to be calculated, all $8$ to be viewed (\texttt{see1=8}), and
+the display to be concluded by an ellipsis to indicate that the sequence
+continues (but those are three dots/periods/full stops in the settings
+option).
+
+A more complicated recurrence relation determines the Legendre polynomials:
+\[
+(n+2)P_{n+2}(x)-(2n+3)xP_{n+1}(x)+(n+1)P_{n}(x)=0.
+\]
+For the purposes of \verb`\recur` we need $P_{n+2}$ expressed in
+terms of the lower order terms:
+\[
+P_{n+2}(x)=\frac{1}{n+2}\left((2n+3)xP_{n+1}(x)-(n+1)P_{n}(x)\right).
+\]
+ It is this standard form \textendash{} the term to be calculated
+on the left, equated to an expression involving a fixed number of
+lower-order terms on the right \textendash{} that \texttt{numerica}
+works with. For $P_{0}(x)=1,~P_{1}(x)=x$ and $x=0.5$, the terms
+are calculated thus:
+\begin{verbatim}
+ \recur[p,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl((2n+3)xP_{n+1}(x)-(n+1)P_{n}(x)\Bigr)
+ \]}[P_{1}(x)=x,P_{0}(x)=1,x=0.5]
+\end{verbatim}
+$\Longrightarrow$ \recur[p,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl((2n+3)xP_{n+1}(x)-(n+1)P_{n}(x)\Bigr)
+ \]}[P_{1}(x)=x,P_{0}(x)=1,x=0.5]
+
+\noindent where $P_{9}(0.5)$ and $P_{10}(0.5)$ are the last two
+displayed values (and to $6$-figures are the values listed in \emph{HMF
+}Table 8.1).
+
+These examples also illustrate a common behaviour of the \texttt{numerica}
+commands: when wrapped around math delimiters: the display is of the
+\emph{expression=result} form, and when placed between math delimiters
+the display is simply of the \emph{result}. When used without math
+delimiters, \texttt{numerica} treats the command as if it had been
+placed between \texttt{\textbackslash{[} \textbackslash}{]}.
+
+\section{Notational niceties}
+
+More than the other commands in \texttt{numerica}, \verb`\nmcRecur`
+depends on getting the notation into a standard form.
+\begin{itemize}
+\item The terms of the recurrence must be \emph{subscripted}: $f_{n}$,
+$P_{n}(x)$ are examples.
+\item The recurrence relation is placed in the main (mandatory) argument
+of \verb`\nmcRecur` in the form: \emph{high-order term=function of
+lower-order terms}.
+\item The initial-value terms in the vv-list must occur left-to-right in
+the order \emph{high }to \emph{low} order.
+\item The recurrence variable changes by $1$ between successive terms.
+\end{itemize}
+The example for Legendre polynomials in particular shows what is required.
+The Fibonacci example is simpler, since the recurrence variable does
+not occur independently in the recurrence relation as it does with
+the Legendre polynomials. In both cases though the recurrence variable
+is absent from the vv-list.
+
+\subsection{Vv-list and recurrence variable}
+
+The recurrence variable is required in the vv-list only when an implicit
+mode calculation is undertaken. Suppose we write $A$ and $B$ for
+the coefficients $2n+3$ and $n+1$ respectively in the Legendre recurrence.
+$A$ and $B$ will now need entries in the vv-list which means the
+recurrence variable will need a value assigned to it there too, and
+we will need to add \texttt{vvmode=1} to the settings option.
+\begin{verbatim}
+ \recur[p,vvmode=1,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(AxP_{n+1}(x)-BP_{n}(x)\Bigr)
+ \]}[P_{1}(x)=x,P_{0}(x)=1,x=0.5,A=2n+3,B=n+1,n=0]
+\end{verbatim}
+$\Longrightarrow$ \recur[p,vvmode=1,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(AxP_{n+1}(x)-BP_{n}(x)\Bigr)
+ \]}[P_{1}(x)=x,P_{0}(x)=1,x=0.5,A=2n+3,B=n+1,n=0]
+
+Since the vv-list is evaluated from the right, the left-to-right high-to-low
+ordering of the initial-value terms means the value of the lowest
+order term is read first. Although \texttt{numerica} depends on this
+order of occurrence of the terms, they do not need to be \emph{consecutive}
+as in the examples so far (although it is natural to enter them in
+this way). \texttt{numerica} reads the value of the subscript of only
+the right-most term (the lowest order term), increments it by $1$
+when reading the next recurrence term to the left, and so on. The
+reading of the subscript of the lowest order term in the vv-list provides
+the initial value of the recurrence variable.
+
+In the following example I have placed other items between $P_{1}(x)$
+and $P_{0}(x)$ in the vv-list (but maintained their left-to-right
+order) and given the recurrence variable $n$ a ridiculous initial
+value $\pi^{2}/12$. (Because of the order in which things get done
+`behind the scenes', \emph{some} value is necessary so that the
+$n$ in `$B=n+1$' does not generate an `unknown token' message.)
+The result is unchanged.
+\begin{verbatim}
+ \recur[p,vvmode=1,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(AxP_{n+1}(x)-BP_{n}(x)\Bigr)
+ \]}[A=2n+3,P_{1}(x)=x,B=n+1,n=\pi^2/12,P_{0}(x)=1,x=0.5]
+\end{verbatim}
+$\Longrightarrow$ \recur[p,vvmode=1,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(AxP_{n+1}(x)-BP_{n}(x)\Bigr)
+ \]}[A=2n+3,P_{1}(x)=x,B=n+1,n=\pi^2/12,P_{0}(x)=1,x=0.5]
+
+\subsection{Form of the recurrence relation}
+
+As noted earler, the form of the recurrence must be entered in the
+main argument in the form: \emph{highest order term = function of
+consecutive lower order terms}. The number of lower\emph{ }order terms
+is the order of the recurrence. The Fibonacci and Legendre polynomial
+recurrences are both second order and presented in the form: \emph{$n+2$-th
+term = function of $n+1$-th term and $n$-th term}. We could equally
+have done
+\begin{verbatim}
+ \nmcRecur[p,do=8,see1=8,...]
+ {$ f_{n}=f_{n-1}+f_{n-2} $}
+ [f_{1}=1,f_{0}=1]
+\end{verbatim}
+$\Longrightarrow$ \nmcRecur[p,do=8,see1=8,...]
+ {$ f_{n}=f_{n-1}+f_{n-2} $}
+ [f_{1}=1,f_{0}=1] where now the recurrence is of the form $n$\emph{-th term = function
+of $n-1$-th term and $n-2$-th term}, or (adjusting the coefficients
+as well as the recurrence terms),
+\begin{verbatim}
+ \recur[p=.,do=10,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ P_{n+1}(x)=\frac{1}{n+1}
+ \Bigl((2n+1)xP_{n}(x)-nP_{n-1}(x)\Bigr)
+ \]}[P_{2}(x)=-0.125,P_{1}(x)=x,x=0.5]
+\end{verbatim}
+$\Longrightarrow$ \recur[p=.,do=10,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ P_{n+1}(x)=\frac{1}{n+1}
+ \Bigl((2n+1)xP_{n}(x)-nP_{n-1}(x)\Bigr)
+ \]}[P_{2}(x)=-0.125,P_{1}(x)=x,x=0.5]
+
+\noindent The recurrence here is of the form $n+1$\emph{-th term
+= function of $n$-th term and $n-1$-th term}. This last example
+has one further `wrinkle'. I've made $P_{1}(x)$ the lowest order
+term and decreased the number of terms to calculate by $1$ accordingly.
+
+\subsection{First order recurrences (iteration)}
+
+The recurrence relations for both the Fibonacci sequence and Legendre
+polynomials are second order. There is no reason why the recurrence
+should not be of third or higher order or, indeed, lower. A first
+order recurrence provides an alternative means of iterating functions.
+\verb`\recur` therefore provides a means to display the results of
+an iteration in a different form from \verb`\iter`.
+
+Iterating $1+a/x$ in this way, $16$ terms gives the sequence
+\begin{verbatim}
+ \recur[do=16,see1=0,see2=3,...]{$
+ x_{n+1}=1+a/x_{n}
+ $}[x_{0}=1,a=1]
+\end{verbatim}
+$\Longrightarrow$ \recur[do=16,see1=0,see2=3,...]{$
+ x_{n+1}=1+a/x_{n}
+ $}[x_{0}=1,a=1]
+
+\noindent to be compared with the example near the start of Chapter~\ref{chap:Iterating-functions}.
+(\emph{That} effected $15$ iterations; \emph{this} uses $16$ terms
+because of the extra $x_{0}=1$ term.)
+
+\section{Star (\texttt{{*}}) option}
+
+When the star option is used with the \verb`\nmcRecur` command, only
+a single term, the \emph{last}, is presented as the result. Repeating
+the last calculation, but with the star option produces
+\begin{verbatim}
+ \recur*[p=.,do=10]{\[
+ P_{n+1}(x)=\frac{1}{n+1}
+ \Bigl((2n+1)xP_{n}(x)-nP_{n-1}(x)\Bigr)
+ \]}[P_{2}(x)=-0.125,P_{1}(x)=x,x=0.5]
+\end{verbatim}
+$\Longrightarrow$ \recur*[p=.,do=10]{\[
+ P_{n+1}(x)=\frac{1}{n+1}
+ \Bigl((2n+1)xP_{n}(x)-nP_{n-1}(x)\Bigr)
+ \]}[P_{2}(x)=-0.125,P_{1}(x)=x,x=0.5]
+
+Although punctuation (a full stop) was specified in the settings,
+it has been ignored in the display of the result. Other settings would
+also have been ignored with the exception of the \verb`do` key which
+is required to know exactly which term to calculate. The star option
+produces a purely numerical answer without any trimmings.
+
+\section{Settings}
+
+The settings option is a comma-separated list of items of the form
+\emph{key~=~value}.
+
+\subsection{Inherited settings}
+
+Because recurrence terms are necessarily multi-token, the multi-token
+key is hard-coded in \verb`\recur` to \texttt{xx=1}.
+
+\subsubsection{Multi-line formatting of result}
+
+When the \verb`\recur` command wraps around math delimiters, the
+\texttt{vvd} setting is available to split display of the result over
+two or more lines. For example, \texttt{vvd=\{,\}\textbackslash\textbackslash (vv)}pushes
+the vv-list and sequence of calculated values to a second line; or,
+\texttt{vvd=\{,\}\textbackslash qquad(vv)\textbackslash\textbackslash}
+pushes only the sequence of calculated values to a second line; or
+\texttt{vvd=\{,\}\textbackslash\textbackslash (vv)\textbackslash\textbackslash}
+pushes the vv-list, centred, to a second line and the sequence of
+values, right aligned, to a third line. The \texttt{{*}} setting is
+available to suppress equation numbering (by substituting \verb`multline*`
+for \verb`multline`).
+\begin{verbatim}
+ \nmcRecur[do=8,see1=8,...,vvd={,}\qquad(vv)\\,*]
+ {$ f_{n+2}=f_{n+1}+f_{n} $}
+ [f_{1}=1,f_{0}=1]
+\end{verbatim}
+$\Longrightarrow$ \nmcRecur[do=8,see1=8,...,vvd={,}\qquad(vv)\\,*]
+ {$ f_{n+2}=f_{n+1}+f_{n} $}
+ [f_{1}=1,f_{0}=1]
+
+\subsection{\texttt{\textbackslash recur}-specific settings}
+
+\label{subsec:recurSpecific-settings}
+
+\subsubsection{Number of terms to calculate}
+
+By entering
+\begin{lyxcode}
+do~=~<integer>
+\end{lyxcode}
+in the settings option you can specify how many terms of a recurrence
+to calculate. The default is set to $7$ (largely to show a sufficient
+number of terms of the Fibonacci series to begin to be interesting).
+Note that \texttt{<integer>} will generally not correspond to the
+subscript on the last term calculated since that also depends on the
+value of the subscript of the lowest order term in the vv-list.
+
+\subsubsection{Number of terms to display}
+
+\begin{table}
+\centering{}\caption{\protect\label{tab:solveSettings-1}Settings for \texttt{\textbackslash nmcRecur}}
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{do}} & {\small int$\ge0$} & {\small number of terms to calculate} & {\small\texttt{7}}\tabularnewline
+{\small\texttt{see1}} & {\small int$\ge0$} & {\small number of initial terms to display} & {\small\texttt{3}}\tabularnewline
+{\small\texttt{see2}} & {\small int$\ge0$} & {\small number of final terms to display} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{...}} & {\small chars} & {\small follow display of values with an ellipsis} & \tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}/\mathtt{2}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+By entering
+\begin{lyxcode}
+see1~=~<integer1>,~see2=<integer2>
+\end{lyxcode}
+in the settings option, you can specify how many initial terms of
+the recurrence and how many of the final terms calculated you want
+to view. If the sum of these settings is less than the \texttt{do}
+setting, then the terms are displayed with an intervening ellipsis.
+If the sum is greater than the \texttt{do} setting, then the values
+are adjusted so that their sum equals the \texttt{do} setting and
+all terms are displayed.
+
+The adjustment is preferentially to \texttt{see1}. Suppose \texttt{do=}7,
+\texttt{see1=5}, \texttt{see2=4}. Then \texttt{see2} is left unchanged
+but \texttt{see1} is reduced to \texttt{7-4=3}. If, say, \texttt{do=}7,
+\texttt{see1=5}, \texttt{see2=8}, then \texttt{see2} is reduced to
+\texttt{7} and \texttt{see1} to \texttt{-1} (rather than zero, for
+technical reasons). The reason for preserving \texttt{see2} over \texttt{see1}
+is for the functioning of the \texttt{reuse} setting (see above).
+
+The default value for \texttt{see1} is $3$; the default value for
+\texttt{see2} is $2$.
+
+\subsubsection{Ellipsis}
+
+Including three dots in the settings option
+\begin{lyxcode}
+...
+\end{lyxcode}
+ensures that a (proper) ellipsis is inserted after the final term
+is displayed. An example is provided by the display of the Fibonacci
+sequence at the start of this chapter. By default this option is turned
+off.
+
+\subsubsection{Form of result saved by \texttt{\textbackslash reuse}}
+
+By entering
+\begin{lyxcode}
+reuse~=~<integer>
+\end{lyxcode}
+it is possible to specify the form of result that is saved when using
+\verb`\nmcReuse`. (This setting has no effect when the star option
+is used with \verb`\nmcRecur`. In that case only the numerical result
+of the final term calculated is saved.) There are three different
+outputs possible:
+\begin{itemize}
+\item \texttt{int=0} (or any integer $\mathtt{\ne1,2}$) saves the full
+display (the default);
+\item \texttt{int=1} saves a comma-separated list of braced pairs of the
+form: \texttt{\{$k$, value-of-term-$k$\}} for the last \texttt{see2}
+terms calculated;
+\item \texttt{int=2} saves a comma-separated list of the values of the last
+\texttt{see2} terms calculated.
+\end{itemize}
+As an example, using \texttt{reuse=1},
+\begin{verbatim}
+ \recur[reuse=1,p=.,vvmode=1,do=11,see1=4,see2=2,
+ vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(kxP_{n+1}(x)-(n+1)P_{n}(x)\Bigr)
+ \]}[k=2n+3,n=123,P_{1}(x)=x,P_{0}(x)=1,x=0.5]
+ \reuse[legendre]
+\end{verbatim}
+$\Longrightarrow$ \recur[reuse=1,p=.,vvmode=1,do=11,see1=4,see2=2,
+ vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(kxP_{n+1}(x)-(n+1)P_{n}(x)\Bigr)
+ \]}[k=2n+3,n=123,P_{1}(x)=x,P_{0}(x)=1,x=0.5]
+ \reuse[legendre]
+
+\noindent Now check to see what has been saved:
+\begin{centred}
+\verb`$\legendre$` $\Longrightarrow$ $ \legendre$.
+\end{centred}
+As you can see, the final two (because of \texttt{see2=2}) of the
+$12$ Legendre polynomials calculated have been saved, each value
+preceded by its index value. If the setting had been \texttt{reuse=2},
+only the two values would have been saved. The \verb`\legendre` control
+sequence contains the values as comma-separated braced pairs, as can
+be seen by using \TeX 's \verb`\meaning` command:
+\begin{centred}
+\verb`\meaning\legendre` $\Longrightarrow$ \meaning\legendre
+\end{centred}
+
+\subsection{Changing default values}
+
+\begin{wraptable}[12]{o}{0.5\columnwidth}%
+\centering{}\caption{\protect\label{tab:recurSettingsDefaults}Defaults for \texttt{\textbackslash nmcRecur}}
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small recur-do} & {\small\texttt{7}}\tabularnewline
+{\small recur-see-first} & {\small\texttt{3}}\tabularnewline
+{\small recur-see-last} & {\small\texttt{2}}\tabularnewline
+{\small recur-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}\end{wraptable}%
+If you wish to change the default values of the various settings for
+\verb`\nmcRecur` this can be done by entering new values in a configuration
+file \texttt{numerica.cfg} as described in the chapter on settings
+in the associated document \texttt{numerica-basics.pdf}. The relevant
+keys are listed in Table~\ref{tab:recurSettingsDefaults}, corresponding
+to the \verb`do`, \verb`see1`, \verb`see2` and \verb`reuse` settings
+of the \verb`\recur` command.
+
+\subsection{Orthogonal polynomials}
+
+I've used Legendre polynomials in examples above, but orthogonal polynomials
+generally lend themselves to the \verb`\recur` treatment. Quoting
+from \emph{HMF} 22.7, orthogonal polynomials $f_{n}$ satisfy recurrence
+relations of the form
+\[
+a_{1n}f_{n+1}(x)=(a_{2n}+a_{3n}x)f_{n}(x)-a_{4n}f_{n-1}(x),
+\]
+or in the standard form required by \verb`\recur`,
+\[
+f_{n+1}(x)=\frac{a_{2n}+a_{3n}x}{a_{1n}}f_{n}(x)-\frac{a_{4n}}{a_{1n}}f_{n-1}(x).
+\]
+\emph{HMF} 22.7 provides a listing of the coefficients $a_{in}$ for
+the polynomials of Jacobi, Chebyshev, Legendre, Laguerre, Hermite
+and others, and tables for these polynomials.
+
+For example, Laguerre polynomials satisfy the recurrence
+\[
+L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-\frac{n}{n+1}L_{n-1}(x).
+\]
+with initial values $L_{0}(x)=1$ and $L_{1}(x)=1-x$. So let's calculate
+the first $13$ Laguerre polynomials for, say, $x=0.5$:
+\begin{verbatim}
+ \recur[do=13,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)
+ \]}[L_{1}(x)=1-x,L_{0}(x)=1,x=0.5]
+\end{verbatim}
+$\Longrightarrow$ \recur[do=13,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)
+ \]}[L_{1}(x)=1-x,L_{0}(x)=1,x=0.5]
+
+\noindent and for $x=5$:
+\begin{verbatim}
+ \recur[p=.,do=13,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)
+ \]}[L_{1}(x)=1-x,L_{0}(x)=1,x=5]
+\end{verbatim}
+$\Longrightarrow$ \recur[p=.,do=13,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)
+ \]}[L_{1}(x)=1-x,L_{0}(x)=1,x=5]
+
+\noindent The results (reassuringly) coincide with those provided
+in \emph{HMF }Table 22.11.
+
+\subsection{Nesting}
+
+It is possible to use the \verb`\recur*` command (but only the starred
+form) in the \verb`\eval`, \verb`\iter`, and \verb`\solve` commands,
+and indeed in \verb`\recur` itself, but with this caveat: if \verb`\recur*`
+is nested within another command, the initial terms of the recurrence
+\textendash{} e.g., $f_{1}=1,f_{0}=1$, for the Fibonacci series,
+or $L_{1}(x)=1-x,L_{0}(x)=1$ for the Laguerre polynomials \textendash{}
+\emph{must be located in the vv-list of that inner }\verb`\recur*`\emph{
+command}. Other shared variables can often be shifted to the vv-list
+of the outer command, but not these initial terms.
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+The terms of a recurrence relation are multi-token variables but \texttt{numerica}
+requires single tokens for its calculations. The problem for \verb`\recur`
+is that the terms in the recurrence relation in the main (mandatory)
+argument differ from the terms in the vv-list: for instance $f_{n}$
+in the main argument, $f_{0}$ in the vv-list. If left like that,
+when \texttt{numerica} does its conversion from multi-token to single
+token variables, $f_{n}$ would not be found since it differs from
+$f_{0}$. Hence a crucial first step for \verb`\recur` is to reconcile
+the different forms, which it does by converting the forms in the
+vv-list to the forms in the recurrence in the main argument. To be
+available for this form change, they must reside in the \emph{inner}
+vv-list. In the outer vv-list they would be inaccessible to the inner
+command.
+
+{*}{*}{*}
+
+This suggests an alternative way of proceeding: write the inital values
+of the recurrence terms in the \emph{same} form in which they occur
+in the recurrence relation, together with an initial value for the
+recurrence variable: $f_{n+1}=1,f_{n}=1,n=0$, say. This is not how
+mathematicians write the initial values in recurrence relations, which
+is why I did not pursue it, but it neatly sidesteps what is otherwise
+an initial awkwardness. \end{shaded}%
+\end{minipage}
+
+In the following example I multiply together (rather futilely) the
+third and fourth members of the sequence of Laguerre polynomials for
+$x=5$ (the answer expected is \verb`$ \eval{3.5\times2.666667} $`
+$\Longrightarrow$ $ \eval{3.5\times2.666667} $). Note that although
+it is tempting to shift the shared vv-lists of the inner \verb`\recur*`
+commands to the vv-list of the outer \verb`\eval` command, in fact
+only the \verb`x=5` entry has been transferred:
+\begin{verbatim}
+ \eval[p=.]{$
+ \recur*[do=3]
+ { L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)}
+ [L_{1}(x)=1-x,L_{0}(x)=1]
+ \times
+ \recur*[do=4]
+ { L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)}
+ [L_{1}(x)=1-x,L_{0}(x)=1]
+ $}[x=5]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]{$
+ \recur*[do=3]
+ { L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)}
+ [L_{1}(x)=1-x,L_{0}(x)=1,x=5]
+ \times
+ \recur*[do=4]
+ { L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)}
+ [L_{1}(x)=1-x,L_{0}(x)=1,x=5]
+ $}
+
+\chapter{Reference summary}
+
+\section{Commands defined in \texttt{numerica-plus}}
+\begin{enumerate}
+\item \texttt{\textbackslash nmcIterate, \textbackslash iter}
+\item \texttt{\textbackslash nmcSolve, \textbackslash solve}
+\item \textbackslash\texttt{nmcRecur, \textbackslash recur}
+\end{enumerate}
+
+\section{Settings for the three commands}
+
+\subsection{Settings for \texttt{\textbackslash nmcIterate}}
+
+Settings option of \verb`\nmcIterate`:
+
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{var}} & {\small token(s)} & {\small iteration variable} & \tabularnewline
+{\small\texttt{+}} & {\small int} & {\small fixed point extra rounding} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{max}} & {\small int > 0} & {\small max. iteration count (fixed points)} & {\small\texttt{100}}\tabularnewline
+{\small\texttt{do}} & {\small int > 0} & {\small number of iterations to perform} & {\small\texttt{5}}\tabularnewline
+{\small\texttt{see}} & {\small int > 0} & {\small number of final iterations to view} & {\small\texttt{4}}\tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}/\mathtt{2}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+Configuration settings for \verb`\nmcIterate`:
+
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small iter-extra-rounding} & {\small\texttt{0}}\tabularnewline
+{\small iter-max-iterations} & {\small\texttt{100}}\tabularnewline
+{\small iter-do} & {\small\texttt{5}}\tabularnewline
+{\small iter-see-last} & {\small\texttt{4}}\tabularnewline
+{\small iter-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+\subsection{Settings for \texttt{\textbackslash nmcSolve}}
+
+Settings option of \verb`\nmcSolve`:
+
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{var}} & {\small token(s)} & {\small equation variable} & \tabularnewline
+{\small\texttt{dvar}} & {\small real $\ne0$} & {\small initial step size} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{+}} & {\small int} & {\small extra rounding} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{max}} & {\small int > 0} & {\small max. number of steps before cut off} & {\small\texttt{100}}\tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+Configuration settings for \verb`\nmcSolve`:
+
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small solve-first-step} & 1\tabularnewline
+{\small solve-extra-rounding} & {\small\texttt{0}}\tabularnewline
+{\small solve-max-steps} & {\small\texttt{100}}\tabularnewline
+{\small solve-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+\subsection{Settings for \texttt{\textbackslash nmcRecur}}
+
+Settings option of \verb`\nmcRecur`:
+
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{do}} & {\small int$\ge0$} & {\small number of terms to calculate} & {\small\texttt{7}}\tabularnewline
+{\small\texttt{see1}} & {\small int$\ge0$} & {\small number of initial terms to display} & {\small\texttt{3}}\tabularnewline
+{\small\texttt{see2}} & {\small int$\ge0$} & {\small number of final terms to display} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{...}} & {\small chars} & {\small follow display of values with an ellipsis} & \tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}/\mathtt{2}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+Configuration settings for \verb`\nmcRecur`:
+
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small recur-do} & {\small\texttt{7}}\tabularnewline
+{\small recur-see-first} & {\small\texttt{3}}\tabularnewline
+{\small recur-see-last} & {\small\texttt{2}}\tabularnewline
+{\small recur-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{document}
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+%% LyX 2.4.0-alpha1 created this file. For more info, see https://www.lyx.org/.
+%% Do not edit unless you really know what you are doing.
+\documentclass[english,tableposition=top]{report}
+\usepackage{lmodern}
+\renewcommand{\sfdefault}{lmss}
+\renewcommand{\ttdefault}{lmtt}
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\usepackage[latin9]{inputenc}
+\setcounter{secnumdepth}{3}
+\usepackage{color}
+\definecolor{shadecolor}{rgb}{0.667969, 1, 1}
+\usepackage{babel}
+\usepackage{array}
+\usepackage{float}
+\usepackage{booktabs}
+\usepackage{framed}
+\usepackage{url}
+\usepackage{multirow}
+\usepackage{amsmath}
+\usepackage[unicode=true,pdfusetitle,
+ bookmarks=true,bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=2,
+ breaklinks=false,pdfborder={0 0 1},backref=section,colorlinks=true]
+ {hyperref}
+
+\makeatletter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+%% Because html converters don't know tabularnewline
+\providecommand{\tabularnewline}{\\}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
+\newenvironment{centred}%
+ {\begin{center}\baselineskip=13pt\parskip=1pt}{\end{center}}
+\newenvironment{lyxcode}
+ {\par\begin{list}{}{
+ \setlength{\rightmargin}{\leftmargin}
+ \setlength{\listparindent}{0pt}% needed for AMS classes
+ \raggedright
+ \setlength{\itemsep}{0pt}
+ \setlength{\parsep}{0pt}
+ \normalfont\ttfamily}%
+ \item[]}
+ {\end{list}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
+\usepackage[tables]{numerica}
+
+\newcommand\rel{\,\varrho\;}
+\DeclareMathOperator{\erf}{erf}
+\DeclareMathOperator{\gd}{gd}
+
+\reuse
+
+\usepackage{upquote}
+
+\makeatother
+
+\begin{document}
+\title{\texttt{numerica-tables}~\\
+}
+\author{Andrew Parsloe\\
+(\url{ajparsloe at gmail.com})}
+\maketitle
+\begin{abstract}
+In this module of the \verb`numerica` package a command is defined
+which enables the creation of multi-column tables of function values
+in a wide variety of table styles. \\
+\\
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\paragraph*{Note:}
+\begin{itemize}
+\item {\normalsize This document applies to version 1.0.0 of }{\normalsize\texttt{numerica-tables.def}}{\normalsize .}{\small\par}
+\item {\normalsize Reasonably recent versions of the \LaTeX 3 bundles }{\normalsize\texttt{l3kernel}}{\normalsize{}
+and }{\normalsize\texttt{l3packages}}{\normalsize{} are required.}{\small\par}
+\item {\normalsize The }{\normalsize\texttt{booktabs}}{\normalsize{} package
+is required.}{\small\par}
+\item {\normalsize I refer many times in this document to }{\normalsize\emph{Handbook
+of Mathematical Functions}}{\normalsize , edited by Milton Abramowitz
+and Irene A. Segun, Dover, 1965. This is abbreviated to }{\normalsize\emph{HMF}}{\normalsize ,
+often followed by a reference to a specific table like Table 1.2.}{\small\par}
+\end{itemize}
+\end{shaded}%
+\end{minipage}
+
+\tableofcontents{}
+\end{abstract}
+
+\chapter{Introduction}
+
+Calling \texttt{numerica} with the \texttt{tables} package option
+in the preamble,
+\begin{lyxcode}
+~\textbackslash usepackage{[}tables{]}\{numerica\}
+\end{lyxcode}
+\noindent gives access to a command \verb`\nmcTabulate` for creating
+tables of function values. This command is defined in the package
+\texttt{numerica-tables.def} which is loaded with \texttt{numerica.sty}
+when the \texttt{tables} option is used. Note that \verb`\nmcTabulate`
+uses the \texttt{booktabs} package for the construction of its tables.
+The \texttt{booktabs} package is loaded automatically (provided it
+is available in your \TeX{} system) when \texttt{numerica} is loaded
+with the \texttt{tables} option.
+
+\section{Shared syntax}
+
+The \verb`\nmcTabulate` command (short-name form \verb`\tabulate`)
+shares the syntax of \verb`\nmcEvaluate` (see \texttt{numerica-basics.pdf})
+and of \verb`\nmcIterate`, \verb`\nmcSolve` and \verb`\nmcRecur`
+(see \texttt{numerica-plus.pdf}). When all options are used the command
+looks like
+\begin{lyxcode}
+\noindent \textbackslash nmcTabulate{*}{[}settings{]}\{expr.\}{[}vv-list{]}{[}num.~format{]}
+\end{lyxcode}
+\begin{enumerate}
+\item \verb`*` optional switch; if present ensures a single number output
+with no formatting or an appropriate error message if the single number
+cannot be produced;
+\item \verb`[settings]` comma-separated list of \emph{key=value }settings;
+\item \verb`{expr.}` mandatory argument specifying the mathematical expression
+in \LaTeX{} form to be tabulated;
+\item \verb`[vv-list]` comma-separated list of \emph{variable=value }items;
+\item \verb`[num. format]` optional format specification for presentation
+of the numerical result (rounding, padding with zeros, scientific
+notation)
+\end{enumerate}
+Unlike \verb`\nmcEvaluate` and the other commands, for \verb`\nmcTabulate`
+\begin{itemize}
+\item it makes no difference to the display of the result whether the command
+wraps around math delimiters, is wrapped within math delimters, or
+if there are no math delimiters involved whatever;
+\item the two apparently optional arguments straddling the main argument
+(\verb`settings` and \verb`vv-list`) are \emph{essential}; although
+neither is mandatory in the \LaTeX{} sense, each contains items necessary
+for the construction of any table of function values.
+\end{itemize}
+
+\subsection{Inherited settings}
+
+\label{subsec:Inherited-settings}
+\begin{table}
+\noindent \centering{}\caption{\protect\label{tab:introSettingsInherited}Settings options inherited
+from \texttt{\textbackslash nmcEvaluate}}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{xx}} & {\small int (}{\small\texttt{0}}{\small /}{\small\texttt{1}}{\small )} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{()}} & {\small int (}{\small\texttt{0}}{\small /}{\small\texttt{1}}{\small /}{\small\texttt{2}}{\small )} & {\small trig. function arg. parsing} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{o}} & & {\small degree switch for trig. functions} & \tabularnewline
+{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{vvmode}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvd}} & {\small tokens} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{vvi}} & {\small tokens} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{*} & & {\small suppress equation numbering if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
+in }{\small\texttt{vvd}} & \tabularnewline
+{\small\texttt{S+}} & {\small int} & {\small extra rounding for stopping criterion for sums} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{S?}} & {\small$\text{int}\ge0$} & {\small stopping criterion query terms for sums} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{P+}} & {\small int} & {\small extra rounding for stopping criterion for products} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{P?}} & {\small$\text{int}\ge0$} & {\small stopping criterion query terms for products} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+To create a table we need to specify a function to tabulate \textendash{}
+this fills the main (mandatory) argument. We also need to decide how
+the function values are to be displayed \textendash{} to how many
+decimal places and whether padded with zeros. These matters are determined
+by what is entered in the trailing optional argument, just as with
+\verb`\eval`, \verb`\iter`, \verb`\solve` and \verb`\recur`, although
+there are some addtional complications peculiar to tables which are
+more appropriately treated in the next chapter.
+
+Many of the settings available to the \verb`\eval` command are also
+available to \verb`\nmcTabulate`. To save switching between documents
+I reproduce the table of options found in \texttt{numerica-basics.pdf}
+(with only the punctuation \texttt{p} setting missing), although for
+discussion of the options you will need to refer to that document.
+But note that the \texttt{dbg} key is of limited usefulness for tables:
+\texttt{dbg=5} and \texttt{dbg=7} display floating point values in
+their `internal' format, and for \texttt{dbg=7}, no result is displayed.
+
+\chapter{\texttt{\textbackslash nmcTabulate}-specific settings}
+
+In addition to the shared settings, \verb`\nmcTabulate` has many
+settings specific to it. They are discussed in groups in subsequent
+sections, some in more than one place.
+
+\section{Row variable settings}
+
+\label{sec:Row-variable-settings}Deciding on a function to tabulate
+(entered in the main or mandatory argument of \verb`\nmcTablate`)
+will inevitably also mean deciding on the tabulation variable (the
+\emph{row} variable \verb`rvar`), and then what value to start tabulating
+from (specified in the vv-list), what value to tabulate to (\verb`rstop`),
+and how fine-grained the tabulation is to be, the step size (\verb`rstep`).
+
+\begin{table}[b]
+\centering{}\caption{Row variable specification}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{3cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small comment}\tabularnewline
+\midrule
+{\small\texttt{rvar}} & {\small token(s)} & {\small row variable} & \tabularnewline
+{\small\texttt{rstep}} & {\small real num.} & {\small step size} & \tabularnewline
+{\small\texttt{rstop}} & {\small real num.} & {\small stop value} & \multirow{2}{3cm}{either {\small\texttt{rstop}}{\small{} or }{\small\texttt{rows}}}\tabularnewline
+{\small\texttt{rows}} & {\small int} & {\small number of rows} & \tabularnewline
+{\small\texttt{rspec}} & {\small comma list} & {\small\texttt{\{start}}{\small , }{\small\texttt{step}}{\small , }{\small\texttt{stop\}}}{\small{}
+or }{\small\texttt{\{start}}{\small , }{\small\texttt{step}}{\small ,
+}{\small\texttt{(rows)\}}} & {\small short form spec.}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+The two tables in the first example below tabulate $\sin x$ and $\cos x$
+between $0$ and $1$ in increments of $0.2$. Note the start value
+of the tabulation variable in the vv-list. The reason for this placement
+is that for more complicated functions other parameters in the function
+and therefore in the vv-list may depend on the row variable. Therefore
+it is always placed in the vv-list.
+
+The difference in appearance of the tables results from padding with
+zeros in the second (the asterisk in the trailing optional argument).
+As you can see, padding applies not only to the values of the function
+but also to the values of the row variable \textendash{} and makes
+an obvious improvement to the table's appearance.
+\begin{verbatim}
+ \tabulate[rvar=x,rstep=0.2,rstop=1]
+ { \sin x }[x=0]\qquad
+ \tabulate[rvar=x,rstep=0.2,rstop=1]
+ { \cos x }[x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rvar=x,rstep=0.2,rstop=1]
+ { \sin x }[x=0] \qquad
+ \tabulate[rvar=x,rstep=0.2,rstop=1]
+ { \cos x }[x=0][*]\medskip{}
+
+Sometimes (perhaps often) it may prove more convenient to specify
+the number of rows (\texttt{rows}) rather than a stop value. Only
+one of \texttt{rows} and \texttt{rstop} should be given, but if both
+(inadvertently) are present, it is the value of \texttt{rows} that
+prevails.
+
+The second and third tables in the next example use an abbreviated
+form of the row variable specification (\texttt{rspec}). This is a
+$3$-element comma list of the form \verb`{rvar,rstep,rstop}` or
+\verb`{rvar,rstep,(rows)}`. Parentheses around the third item signify
+that it is \verb`rows` rather than \verb`rstop` that is being specified.
+In the example below, the second table specifies \texttt{rstop} (value
+\texttt{1}), the third \texttt{rows} (value \texttt{6}).
+
+It is worth noting that \verb`rstep` and \verb`rstop` can be \LaTeX{}
+expressions \textendash{} for instance \texttt{rstop=\textbackslash sin
+\textbackslash pi/2} (just as the expression for the initial value
+in the vv-list can be). However \verb`rows` can only be a simple
+integer expression \textendash{} an integer or integers linked by
+some or all of \verb`+ - * / ( )`.
+\begin{verbatim}
+ \tabulate[rvar=x,rstep=0.2,rows=6]
+ { \sin x/\cos x }[x=0][*] \qquad
+ \tabulate[rspec={x,0.2,1}]
+ { \tan x }[x=0][*] \qquad
+ \tabulate[rspec={x,0.2,(6)}]
+ { \sqrt{\sec^2 x - 1} }[x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rvar=x,rstep=0.2,rows=6]
+ { \sin x/\cos x }[x=0][*] \qquad
+ \tabulate[rspec={x,0.2,1}]
+ { \tan x }[x=0][*] \qquad
+ \tabulate[rspec={x,0.2,(6)}]
+ { \sqrt{\sec^2 x - 1} }[x=0][*]
+
+\subsection{Row-variable column formatting}
+
+\label{subsec:Row-var-col-formatting}Various settings are available
+to format the row variable column in addition to the padding option
+({*}) of the trailing optional argument.
+\begin{table}[H]
+\centering{}\caption{Row-variable column formatting}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{rround}} & {\small int} & {\small rounding} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{ralign}} & {\small char (}{\small\texttt{r/c/l}}{\small )} & {\small horizontal alignment} & {\small\texttt{r}}\tabularnewline
+{\small\texttt{rfont}} & {\small chars} & {\small font (}{\small\verb`\math<chars>`}{\small )} & \tabularnewline
+{\small\texttt{rhead}} & {\small tokens} & {\small header} & {\small\texttt{rvar}}\tabularnewline
+{\small\texttt{rhnudge}} & int & {\small nudge header }{\small{\small\verb`<int>`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small column position(s) } & {\small\texttt{1}}\tabularnewline
+{\small\texttt{rvar'}} & {\small tokens} & {\small 2nd row variable col. spec. } & {\small\texttt{rvar}}\tabularnewline
+{\small\texttt{rhead'}} & {\small tokens} & {\small header of 2nd rv col. (if it exists)} & {\small\texttt{rvar'}}\tabularnewline
+{\small\texttt{rhnudge'}} & int & {\small nudge 2nd rv col. header }{\small{\small\verb`<int>`}}{\small{}
+mu} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+
+
+\subsubsection{Rounding: \texttt{rround}}
+
+After studying the previous tables, we might decide to adjust the
+step size, say from $0.2$ to $0.25$. But changing \texttt{rstep}
+to the new value gives a disconcerting and \emph{prima facie} false
+result (the first table below). \texttt{numerica} uses a default rounding
+value of $1$ for the row variable and has rounded $0.25$ down to
+$0.2$ and $0.75$ up to $0.8$ accordingly. The second table corrects
+matters by adjusting the row variable rounding (\texttt{rround}) to
+\texttt{2}.
+\begin{verbatim}
+ \tabulate[rvar=x,rstep=0.25,rstop=1]
+ { \sin x }[x=0][*]\qquad
+ \tabulate[rvar=x,rstep=0.25,rstop=1,rround=2]
+ { \sin x }[x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rvar=x,rstep=0.25,rstop=1]
+ { \sin x }[x=0][*] \qquad
+ \tabulate[rvar=x,rstep=0.25,rstop=1,rround=2]
+ { \sin x }[x=0][*]
+
+\subsubsection{Alignment: \texttt{ralign}}
+
+By default, the alignment of all columns is to the right, as in the
+previous examples. This lends itself to neat output when padding with
+zeros is activated (the \verb`*` in the trailing argument) and when
+some values are negative \textendash{} the minus signs can interfere
+with neat output in left or centred alignments. But in a case like
+the second table in the last example, you might prefer to centre the
+headers for both the row and function value columns. These alignments
+are independently set. For the row variable column the default alignment
+is to the right \texttt{ralign=r}; \texttt{ralign=l} (lowercase L)
+aligns entries in the row variable column to the left, and \texttt{ralign=c}
+centres entries in the row variable column. The tables of the next
+example use a \texttt{c} alignment to centre the row variable column
+header. The third of those tables shows how minus signs spoil the
+effect.
+
+\subsubsection{Font: \texttt{rfont}}
+
+In the second table bolding (\texttt{rfont=bf}) has been applied to
+emphasize the distinction between the row variable values and the
+function values. Possible values for this key are those characters
+that can be adjoined to \verb`\math` to give a meaningful result.
+Thus other valid values are \verb`it` (italic), \verb`sf` (sans
+serif), \verb`tt` (typewriter); \verb`frak` (Fraktur); also \verb`rm`
+(roman) is available, but that is the default.
+
+\subsubsection{Header: \texttt{rhead}}
+
+In the second and third tables, the header for the row variable column
+has also been bolded. The default header is the row variable. That
+can be replaced by giving a value to the key \texttt{rhead}. I have
+used \texttt{rhead=\textbackslash boldsymbol\{x\}} (rather than \verb`\mathbf{x}`)
+in order to get an italicized bold symbol.
+\begin{verbatim}
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=1,
+ rround=2,ralign=c]
+ { \sin x }[x=0][*]\qquad
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=1,rround=2,
+ ralign=c,rfont=bf,rhead=\boldsymbol{x}]
+ { \sin x }[x=0][*]
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ ralign=c,rfont=bf,rhead=\boldsymbol{x}]
+ { \sin x }[x=-0.5][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rvar=x,rstep=0.25,rstop=1,
+ rround=2,ralign=c]
+ { \sin x }[x=0][*]\qquad
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=1,
+ rround=2,ralign=c,rfont=bf,rhead=\boldsymbol{x}]
+ { \sin x }[x=0][*]\qquad
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ ralign=c,rfont=bf,rhead=\boldsymbol{x}]
+ { \sin x }[x=-0.5][*]\medskip{}
+
+In these tables the row variable column has been given a centred alignment.
+The third table shows what goes wrong when \emph{some} values are
+negative. Better then is to use padding, a right alignment (the default),
+and to use a phantom in the header. The first table below does this.
+The second table incorporates kerning into the header to achieve the
+same effect:
+\begin{verbatim}
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x}\hphantom{0}]
+ { \sin x }[x=-0.5][*]\qquad
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x}\mkern 9 mu]
+ { \sin x }[x=-0.5][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x}\hphantom{0}]
+ { \sin x }[x=-0.5][*]\qquad
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x}\mkern 9 mu]
+ { \sin x }[x=-0.5][*]\medskip{}
+
+(To my eye, aligning the $\boldsymbol{x}$ above the first column
+of digits after the decimal point gives a better result than truly
+centring it in the column; compare these examples with the first two
+tables of the previous example.)
+
+\subsubsection{Nudging~the~header: \texttt{rhnudge}}
+
+However, you might prefer to avoid inserting positioning commands
+into the actual row variable header, obscuring its true content. You
+can avoid doing this by using \texttt{numerica}'s nudge setting \texttt{rhnudge}.
+
+The first table below reverts to the default right alignment, avoids
+any positioning commands in the row variable header, but instead nudges
+it into position with the setting \texttt{rhnudge=9}. For positive
+nudge values, nudging works in the opposite sense to the alignment.
+The units for nudging are mu (math units, 18 to a quad), but only
+a number \textendash{} generally an integer \textendash{} should be
+specified; the mu is supplied by \texttt{numerica}.
+
+In the second table below the row variable takes single digit integer
+values, while the row variable name now occupies more than one character.
+With a right alignment the header would protrude out to the left.
+Giving \texttt{rhnudge} a \emph{negative} value (\texttt{rhnudge=-12}
+in the example) brings it back to a centred position in the row variable
+column.
+\begin{verbatim}
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x},rhnudge=9]
+ { \sin x }[x=-0.5][4*]\qquad
+ \tabulate
+ [rvar=x_{\text{int}},rstep=1,rstop=4,
+ rround=0,rfont=bf,rhnudge=-12,
+ rhead=\boldsymbol{x_{\text{int}}}]
+ { \sin x_{\text{int}} }[x_{\text{int}}=0][4*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x},rhnudge=9]
+ { \sin x }[x=-0.5][4*]\qquad
+ \tabulate
+ [rvar=x_{\text{int}},rstep=1,rstop=4,
+ rround=0,rfont=bf,rhnudge=-12,
+ rhead=\boldsymbol{x_{\text{int}}}]
+ { \sin x_{\text{int}} }[x_{\text{int}}=0][4*]
+
+\subsubsection{Position in the table: \texttt{rpos}}
+
+\label{subsec:Row-var-col-pos}By default, the row variable column
+is the \emph{first} column of the table. Its position is determined
+by the value of the key \texttt{rpos}:
+\begin{itemize}
+\item \texttt{rpos=0}, suppressed (no row variable column);
+\item \texttt{rpos=1}, first column (the default);
+\item \texttt{rpos=2}, last column;
+\item \texttt{rpos=3}, first and last columns;\texttt{ }
+\item \texttt{rpos=4}, first and last columns, with the values in the last
+column a user-defined function of the first; see \xA7\ref{subsec:Second-row-var-col};
+\item Any other integer acts like \texttt{rpos=1}.
+\end{itemize}
+An example with \texttt{rpos=3} is given shortly below, \xA7\ref{subsec:Multiple-function-tables}.
+
+\subsubsection{\texttt{rvar'}, \texttt{rhead'}, \texttt{rhnudge'}}
+
+These settings become relevant only when \texttt{rpos=4}; see \xA7\ref{subsec:Second-row-var-col}.
+
+\subsection{Multiple function tables}
+
+How might one tabulate multiple functions simultaneously? \emph{HMF}
+has many, many examples where multiple functions (like the trigonometric
+or the hyperbolic functions) are tabulated in separate columns of
+the same table.
+
+\subsubsection{By adjoining tables}
+
+\label{subsec:Adjoining-tables} With the settings described so far,
+one way is to adjoin single column tables. In the tables below, which
+display as a single multi-columned table, I have used three different
+\texttt{rpos} settings (\texttt{rpos=1} is implicit in the first).
+This is one way to build a table that displays as multi-column. If
+you use this method, note that the \texttt{\%} comment characters
+are essential at the end of the last argument of the \verb`\tabulate`
+commands if you want the tables to abut exactly. Omitting them results
+in a space between the tables.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rhnudge=9]
+ { \sin x }[x=0][*]%
+ \tabulate
+ [rpos=0,rspec={x,0.2,(6)},rround=2]
+ { \cos x }[x=0][*]%
+ \tabulate
+ [rpos=2,rspec={x,0.2,(6)},rhnudge=9]
+ { \tan x }[x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rhnudge=9]
+ { \sin x }[x=0][*]%
+ \tabulate
+ [rpos=0,rspec={x,0.2,(6)},rround=2]
+ { \cos x }[x=0][*]%
+ \tabulate
+ [rpos=2,rspec={x,0.2,(6)},rhnudge=9]
+ { \tan x }[x=0][*]
+
+\subsubsection{Multiple functions in a single table}
+
+\label{subsec:Multiple-function-tables}However, tabulating more than
+one function at a time is too common a need to have to resort to adjoining
+tables. Instead, it suffices to enter the functions in the main argument
+separated by commas. The critical thing to do is to \emph{precede
+the first function with a comma}. That initial comma is the signal
+\texttt{numerica} needs to make the internal adjustments for a multi-function
+table (and note the \verb`o` setting, to indicate the arguments of
+$\sin$ and $\cos$ are in degrees):
+\begin{verbatim}
+ \tabulate[o,rpos=3.rvar=\theta,rstep=10,rstop=90,
+ rround=0,rules=ThB]
+ { ,\sin \theta,\cos \theta }[\theta=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[o,rpos=3,rvar=\theta,rstep=10,rstop=90,
+ rround=0,rules=ThB]
+ { ,\sin \theta,\cos \theta }[\theta=0][*]\medskip{}
+
+This table also suggests a space saving possibility: since $\sin$
+and $\cos$ are complementary functions ($\cos\theta=\sin(90-\theta)$),
+the values in the bottom half of the table duplicate values in the
+top half, only with the columns reversed. This is the reason for the
+space saving \texttt{rpos=4} setting (\xA7\ref{subsec:Second-row-var-col})
+which enables complementary functions to be tabulated in `half tables';
+see \emph{HMF} Tables 4.10\textendash 4.12 for the trigonometric functions.
+
+\section{Column variable settings}
+
+\label{sec:Column-variable-settings}When a function of \emph{two}
+variables is being tabulated, we generally think of one variable as
+the primary variable and the other as a parameter. To tabulate such
+a function, one way to proceed, would be to create and adjoin separate
+tables, one per parameter value. But this is clumsy. A more systematic
+procedure is to specify, in addition to the row variable, a \emph{column}
+variable and its start, step and stop values.
+
+\begin{table}[b]
+\caption{Column variable specification}
+
+\centering{}\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{cvar}} & {\small token(s)} & {\small column variable} & \tabularnewline
+{\small\texttt{cstep}} & {\small real num.} & {\small step size} & \tabularnewline
+{\small\texttt{cstop}} & {\small real num.} & {\small stop value} & {\small either }{\small\texttt{cstop}}\tabularnewline
+{\small\texttt{cols}} & {\small int} & {\small number of columns} & {\small or }{\small\texttt{cols}}\tabularnewline
+{\small\texttt{cspec}} & {\small comma list} & {\small\texttt{\{cvar,cstep}}{\small , }{\small\texttt{cstop\}}}{\small{}
+or}{\small\texttt{ \{cvar,cstep}}{\small ,}{\small\texttt{(cols)\}}} & {\small short form spec.}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+In the following example \verb`cvar=k` is the column variable. I
+have chosen a step size (\texttt{cstep}) of \texttt{2} and a stop
+value (\texttt{cstop}) of \texttt{9}. As with the row variable, the
+start value (\texttt{k=3}) of the column variable is specified in
+the vv-list. Although in the example these values are numbers, all
+three values could be \LaTeX{} expressions that evaluate to numbers.
+Note also the setting for \texttt{rhead} which shows the reader of
+the table that the numerical values displayed in the column headers
+are values of \verb`k`. This usage occurs throughout \emph{HMF}.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ rhead=x\backslash k,
+ cvar=k,cstep=2,cstop=9]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x\backslash k,
+ cvar=k,cstep=2,cstop=9]
+ { \sin kx }[k=3,x=0][*] \medskip{}
+
+Again, as with the row variable, rather than using an explicit stop
+value (\texttt{cstop}), you might prefer to specify the number of
+columns (\texttt{cols}) explicitly. I could have replaced \texttt{cstop=9}
+with \texttt{cols=4} to get the same result. Note that the number
+of columns specified here is the number of \emph{function value} columns;
+the row variable column is ignored for this count.
+
+And again, as with the row variable, it is possible to condense the
+specification into a comma list (\texttt{cspec}). This is a $3$-element
+comma list of the form \verb`{cvar,cstep,cstop}` or \verb`{cvar,cstep,(cols)}`
+where the parentheses distinguish \texttt{cols} from \texttt{cstop}
+in the third item. Thus, for the preceding table I could have written
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x\backslash k,
+ cvar=k,cstep=2,cols=4]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+or
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x\backslash k,
+ cspec={k,2,(4)}]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+or
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x\backslash k,
+ cspec={k,2,9}]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+and produced the same table.
+
+Although \verb`cstep` and \verb`cstop` can be \LaTeX{} expressions
+\textendash{} for instance \texttt{cstop=\textbackslash pi/2} \textendash{}
+\verb`cols` can only be a simple integer expression \textendash{}
+an integer, or integers linked by some or all of \verb`+ - * / ( )`.
+
+\subsection{Column header formatting}
+
+\label{subsec:Column-header-formatting}
+\begin{table}
+\begin{centering}
+\caption{Column variable header formatting}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{chstyle}} & {\small int (0/1/2/3)} & {\small header style} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{ctitle}} & {\small tokens} & {\small single col. alternative header} & \tabularnewline
+{\small\texttt{chead}} & {\small tokens} & {\small user-defined header} & \tabularnewline
+{\small\texttt{calign}} & {\small char (r/c/l)} & {\small column alignment} & {\small\texttt{r}}\tabularnewline
+{\small\texttt{chnudge}} & {\small int} & {\small nudge header }{\small{\small\verb`int`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{chround}} & {\small int} & {\small column header rounding} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{centering}
+\end{table}
+There are four built-in style settings for the header to the column
+variable (or function value) columns (the `ch' prefix evoking `column
+header'). If these don't meet your needs or otherwise satisfy, then
+it is possible to define your own header to the function value columns
+using the key \texttt{chead}. First I discuss the built-in styles.
+
+\subsubsection{Header style: single-column case}
+
+When there is only one column of function values, the function being
+tabulated is by default set as the header to the column. This corresponds
+to setting \texttt{ctitle={*}} (see \xA7\ref{subsec:Title:-ctitle-setting}
+below). You may want some other header. Then give \texttt{ctitle}
+some other value (although note that giving it the value \texttt{{*}{*}}
+will set both the function and the vv-list as the header; again see
+\xA7\ref{subsec:Title:-ctitle-setting}). Whatever value you set, it
+will be typeset between math delimiters (\verb`$` signs) and can
+be nudged (see \xA7\ref{subsec:Nudgingtheheaders:-chnudge}) left or
+right to fine-tune its position in the column. (If you want an asterisk
+as the header, you will need to place it between two pairs of braces,
+\texttt{ctitle=\{\{{*}\}\}} to prevent it being misinterpreted as
+the default setting.)
+
+If you want some more complicated header, perhaps not constrained
+by the \verb`$` delimiters, then give \texttt{chead} a value. This
+key I discuss below in \xA7\ref{subsec:chead}. \texttt{chead} is entirely
+up to the user to specify, including any math environment and positioning.
+
+If both \texttt{ctitle} and \texttt{chead} are given, the \texttt{chead}
+value prevails.
+
+\subsubsection{Header~style: multi-column case}
+
+\texttt{chstyle=0} which is the default gives a header of the form
+displayed in the last example, with only the column-variable value
+at the head of each column. This style generally requires the row-variable
+header to indicate what the values denote, as in \texttt{rhead=x\textbackslash backslash
+k} where the backslash separates row from column variable. \emph{HMF}
+contains a multitude of instances; see Tables~9.7, 17.5, 21.1, 24.3,
+27.4, etc. for examples.
+
+\texttt{chstyle=1} changes the header of the \emph{first} function
+value column to the form \emph{variable=value} \textendash{} in the
+example below, to $k=3$. This may be an apt choice when the columns
+are narrow, perhaps from using a small rounding value. I can find
+only one real instance in \emph{HMF}, Table~26.7. Note that the row
+variable setting \texttt{rhead} no longer needs the `\texttt{\textbackslash backslash
+k}' part since the column variable is now explicitly indicated.
+
+\texttt{chstyle=2} changes the header of all function value columns
+to the form \emph{variable=value}.\emph{ }In \emph{HMF} examples are
+Tables~7.4, 7.9, 10.10, 16.6, etc. Note that the row variable setting
+\texttt{rhead} no longer needs the `\texttt{\textbackslash backslash
+k}' part since the column variable is now explicitly indicated (the
+right-hand table).
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(2)},chstyle=1]
+ { \sin kx }[k=3,x=0][3*]\quad
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(3)},chstyle=2]
+ { \sin kx }[k=3,x=0][3*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(3)},chstyle=1]
+ { \sin kx }[k=3,x=0][3*]\quad
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(3)},chstyle=2]
+ { \sin kx }[k=3,x=0][3*]\medskip{}
+
+Finally, \texttt{chstyle=3} fills each column-variable header with
+the expression being tabulated but with the column variable replaced
+by its respective values. See \emph{HMF} Tables~5.4, 8.1, 9.1, 19.1,
+etc. for examples.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(3)},chstyle=3]
+ { \sin kx }[k=3,x=0][4*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(3)},chstyle=3]
+ { \sin kx }[k=3,x=0][4*]
+
+\subsubsection{User-defined header: \texttt{chead}}
+
+\label{subsec:chead}Perhaps none of the built-in styles appeal? By
+assigning content to the key \texttt{chead}, users can create their
+own header to the function value columns. \texttt{chead} must contain
+the correct number of tab characters (\verb`&`), allowing for any
+\verb`\multicolumn`-s used, but ignoring the row variable column
+or columns. It is a header only to the function value columns. The
+user will need to insert \verb`$` signs as appropriate.
+
+Non-empty content for the \texttt{chead} key overrides any \texttt{chstyle}
+setting and in the case of a single function value column, overrides
+any \texttt{ctitle} setting.
+
+\subsubsection{Alignment: \texttt{calign}}
+
+The function value columns are aligned right (\texttt{calign=r}) by
+default. Also available are \texttt{calign=c} for centred alignment
+and \texttt{calign=l} (lowercase L) for left alignment. Using centred
+alignment with {\ttfamily\verb`chstyle=2`} in the now familiar
+example table gives
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,ralign=c,
+ cspec={k,2,(3)},chstyle=2,calign=c]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x,ralign=c,
+ cspec={k,2,(3)},chstyle=2,calign=c]
+ { \sin kx }[k=3,x=0][*]\medskip{}
+
+\noindent The first column of function values looks better, but the
+minus signs spoil the effect in the others. Handling signs in tables
+is discussed below; see \xA7\ref{subsec:Signs}.
+
+\subsubsection{Nudging~the~headers: \texttt{chnudge}}
+
+\label{subsec:Nudgingtheheaders:-chnudge}In left or right alignment
+it is possible to nudge the headers in the opposite direction by giving
+a numerical value to the the key \texttt{chnudge}. The header is moved
+by the specified number of mu (math units; 18 to a quad). Note that
+the `mu' does not need to be written. \texttt{numerica} provides
+that. In the example I have chosen \texttt{chnudge=12} to nudge the
+column headers to the left to give a centred effect to the header
+but leaving the function values with their (potentially) awkward minus
+signs right aligned.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhnudge=9,
+ cspec={k,2,(3)},chstyle=2,chnudge=12]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhnudge=9,
+ cspec={k,2,(3)},chstyle=2,chnudge=12]
+ { \sin kx }[k=3,x=0][*]\medskip{}
+
+The \texttt{chnudge} value does not need to be positive. Negative
+nudges can be useful when a column header is \emph{longer} than the
+rounded function values. In the second example below, I've reduced
+the rounding value for function values to $3$, and chosen an initial
+$k$ value of $100$ to ensure this circumstance. To centre the column
+headers I have used \texttt{chnudge=-9}.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhnudge=9,
+ cspec={k,2,(3)},chstyle=2,chnudge=-9]
+ { \sin kx }[k=100,x=0][3*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhnudge=9,
+ cspec={k,2,(3)},chstyle=2,chnudge=-9]
+ { \sin kx }[k=100,x=0][3*]
+
+\subsubsection{Rounding: \texttt{chround}}
+
+In the examples so far, the column variable has incremented in integer
+steps. The default rounding value for the column variable is $0$
+(for the row variable it is $1$), so if it increments by some non-integer
+amount, the result will be confusing \textendash{} if $k$ incremented
+by, say, $0.25$, starting from $k=3$, then the next column would
+also have a header $k=3$ (since $3.25$ with a rounding value $0$
+rounds to $3$). The appropriate key to remedy this state of affairs
+is \texttt{chround}. For a step size of $0.25$ the appropriate setting
+is \texttt{chround=2}.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2]
+ { \sin kx }[k=3,x=0][*]
+
+\section{Whole-of-table formatting}
+
+\label{sec:Whole-of-table-formatting}There are a number of settings
+pertaining to the appearance of the table as a whole, things like
+the position of the row variable column, division of the function
+values into blocks to aid readability, the presence of horizontal
+rules or of a collective column title or of a footer row. I discuss
+these here.
+
+\begin{table}[H]
+\noindent \centering{}\caption{\protect\label{tab:Table-formatting-settings}Table formatting}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{ctitle}} & {\small tokens} & {\small collective title for function-value columns} & \tabularnewline
+{\small\texttt{rules}} & {\small chars} & {\small horizontal rules template} & {\small\texttt{ThB}}\tabularnewline
+{\small\texttt{foot}} & {\small tokens} & {\small content of footer line} & \tabularnewline
+{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small row-variable column position(s)} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{rbloc}} & {\small comma list} & {\small division of rows into blocks} & \tabularnewline
+{\small\texttt{rblocsep}} & {\small length} & {\small extra spacing between blocks of rows} & {\small\texttt{1 ex}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+
+
+\subsection{Title for function-value columns: \texttt{ctitle}}
+
+\label{subsec:Title:-ctitle-setting}The function value columns have
+individual headers, formatted in the various ways provided by the
+settings discussed above, but it can also be helpful to have a collective
+title for these columns. This is provided by the \texttt{ctitle} key.
+This can be set to whatever you like (e.g. \texttt{ctitle=\textbackslash text\{Fred\}}),
+but for more relevant titles there are two in-built settings: \texttt{ctitle={*}},
+which makes the formula the title, and \texttt{ctitle={*}{*}}, which
+makes a title of the formula and vv-list. In the example below, it
+makes no sense to display the vv-list since it contains only the initial
+values of the row and column variables, already obvious in the table;
+hence \texttt{ctitle={*}}.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2,ctitle=*]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2,ctitle=*]
+ { \sin kx }[k=3,x=0][*]\medskip{}
+
+For an example of \texttt{ctitle={*}{*}} see \xA7\ref{subsec:Rules:-rules-setting}.
+
+\subsection{Rules: \texttt{rules} setting}
+
+\label{subsec:Rules:-rules-setting}The \texttt{booktabs} package
+which \texttt{numerica} uses is most emphatic that one should `1.
+Never, ever use vertical rules. 2. Never use double rules.' Most
+of the tables proper in \emph{HMF} lack rules of any kind although
+closer inspection shows smaller tables within the text generally \emph{are}
+delimited by horizontal rules (often also with vertical rules).\footnote{The tables in \emph{HMF} are often inelegantly typeset, and sometimes
+ugly. For all that, I have used it as a source of table types, of
+the variety of structures that the editors found necessary or at least
+useful for presenting a multitude of different kinds of numerical
+data.} I have used horizontal rules in the various examples in the present
+document because these too are tables within text. Some form of delineation
+seems necessary. Also, to my eye, the table title, $\sin kx$, in
+the last example seems `naked' without a rule to emphasize its domain.
+
+The \texttt{rules} key enables precisely which rules are used to be
+specified. The value of the key is a `word' \textendash{} a sequence
+of letters \textendash{} where the characters have the significance
+and default thicknesses (from \texttt{booktabs}) shown in Table~\ref{tab:Rules}.
+The default setting is \texttt{rules=ThB}. To insert a rule beneath
+the title, for example, change this to \texttt{rules=TthB}. If in
+addition you are using a footer row and want a rule above it, then
+the specification is \texttt{rules=TthfB}.
+
+\begin{table}[H]
+\noindent \centering{}\caption{\protect\label{tab:Rules}Rules}
+\noindent \begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small char} & {\small rule} & {\small position} & {\small default rule thickness}\tabularnewline
+\midrule
+{\small\texttt{T}} & {\small top rule} & {\small above table} & {\small\texttt{\textbackslash heavyrulewidth=.08em}}\tabularnewline
+{\small\texttt{t}} & {\small title rule} & {\small below title} & {\small\texttt{\textbackslash cmidrulewidth =.03em}}\tabularnewline
+{\small\texttt{h}} & {\small header rule} & {\small below header} & {\small\texttt{\textbackslash lightrulewidth=.05em}}\tabularnewline
+{\small\texttt{f}} & {\small footer rule} & {\small above footer} & {\small\texttt{\textbackslash cmidrulewidth =.03em}}\tabularnewline
+{\small\texttt{B}} & {\small bottom rule} & {\small below table} & {\small\texttt{\textbackslash heavyrulewidth=.08em}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+If you wish to change the thickness of a rule from its default, then
+enter new values for any or all of {\small\texttt{\textbackslash heavyrulewidth}},
+{\small\texttt{\textbackslash lightrulewidth}}, {\small\texttt{\textbackslash cmidrulewidth}}
+in the preamble. The values listed in Table~\ref{tab:Rules} are
+the default values in the \texttt{booktabs} package (except for the
+footer rule, which \texttt{booktabs} does not cover; in \texttt{numerica}
+the footer rule is assigned a thickness of {\small\texttt{\textbackslash cmidrulewidth}}).
+
+In the example table below, a rule for the column title has been specified
+(the \verb`t` in the setting \verb`rules=TthB`). Also note the use
+of \texttt{ctitle={*}{*}}. The fornula contains an extra parameter
+$a$, assigned a value in the vv-list. It now makes sense to display
+the vv-list in the column title (but note the braces around \texttt{k}
+and \texttt{x} so that they don't show in the vv-list).
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2,
+ ctitle=**,rules=TthB]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2,
+ ctitle=**,rules=TthB]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*]
+
+\subsection{Footer row: \texttt{foot} setting}
+
+Some tables have a footer row and \texttt{numerica} allows such a
+row to be inserted, but its entire content, with one exception, is
+the responsibility of the user, including insertion of the necessary
+number of tab characters \verb`&`. This will be $1$ less than the
+total number of columns (including row variable columns) in the table
+\textendash{} or some adjustment thereof if you use \verb`\multicol`.
+You can put into the footer what you wish.
+\begin{lyxcode}
+foot=<tokens>
+\end{lyxcode}
+\emph{HMF} uses the footer mainly for cryptic descriptions of the
+accuracy and needs of interpolation methods.
+
+The one exception is when \texttt{foot={*}}. This will fill the footer
+with the header, but \emph{reversed}. This is useful for tabulating
+complementary functions like the sine and cosine or, more generally,
+$f(x)$ and $g(x)$ where $g(x)=f(k-x)$ for some constant $k$. Values
+for the complementary function are read from the bottom up and require
+a reversed row variable column on the right of the table; see \xA7\ref{subsec:Second-row-var-col}.
+
+\subsubsection{Footer functions}
+
+It is also possible to use the footer for displaying the values of
+certain column functions. \texttt{numerica} provides five of these.
+They can be used in the footer (and only in the footer): \verb`SUM`,
+\verb`AVE` (average), \verb`MAX`, \verb`MIN` and \verb`DEL` (for
+$\Delta$=\verb`MAX-MIN`). These functions act on the function values
+of the column they are in. They \emph{do not} combine mathematically:
+entering \verb`MAX-MIN` in the footer of a given column will produce
+a footer entry containing two values (those of \verb`MAX` and \verb`MIN`)
+separated by a minus sign, not the value of \verb`DEL`. The numerical
+output from each function is automatically wrapped in math delimiters
+(\verb`$`) so that minus signs display correctly.
+
+In the following example, I have chosen a column variable step size
+of zero. This is possible because in the column spec., I have also
+specified the exact number of columns. Zeroing the step size means
+the same set of figures can be used for the five footer functions
+to act on.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0,(5)},chstyle=2,
+ chround=2,calign=r,ctitle=**,rules=TthfB,
+ foot={\small Func:} & SUM & AVE & MAX & MIN & DEL ]
+ { a\sin kx }[a=2/\pi,{k}=3.5,{x}=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0,(5)},chstyle=2,
+ chround=2,calign=r,ctitle=**,rules=TthfB,
+ foot={\small Func:}&SUM&AVE&MAX&MIN&DEL]
+ { a\sin kx }[a=2/\pi,{k}=3.5,{x}=0][*]
+
+\subsection{Second row variable column}
+
+\label{subsec:Second-row-var-col}In \xA7\ref{subsec:Row-var-col-pos}
+I discussed \texttt{rpos=0,1,2,3}. There is another value available
+for this key, \texttt{rpos=}4. Like \texttt{rpos=3} this adds the
+row variable column to both left and right sides of the table, but
+for the right column the values are functions of those in the left
+column.
+
+For example, the sine and cosine are complementary functions; when
+working in degrees $\cos\theta=\sin(90-\theta)$. We can exploit this
+fact to halve the table size needed to tabulate the two functions.
+\begin{verbatim}
+ \tabulate[o,rpos=4,rspec={\theta,5,45},rround=0,
+ chnudge=14,rvar'=90-\theta,rhead'=\theta',
+ rules=ThfB,foot=*]
+ { ,\sin\theta,\cos\theta }[\theta=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[o,rspec={\theta,5,45},rround=0,rpos=4,
+ chnudge=14,rvar'=90-\theta,rhead'=\theta',
+ rules=ThfB,foot=*]
+ { ,\sin\theta,\cos\theta }[\theta=0][*]\ \medskip{}
+
+\noindent where $\theta'=90-\theta$. The first half of the tabulation
+is read normally, down and from the left. The second half of the tabulation
+is read up and from the right. Note the degree setting \verb`o` in
+the settings option and
+\begin{itemize}
+\item the use of \verb`rvar'` to specify the values tabulated on the right
+(\verb`rvar'` defaults to \verb`rvar`);
+\item the use of \verb`rhead'` to specify the content of the header for
+the right-hand row variable column (\verb`rhead'` defaults to \verb`rvar'`);
+\item the footer setting \texttt{foot={*}} to obtain the header reversed
+in the footer;
+\item a rule \emph{above} the footer row specified by the \verb`f` added
+to the \verb`rules` setting, \verb`rules=ThfB`.
+\end{itemize}
+Although there is a significant space saving with tables of this kind
+(see \emph{HMF} Tables 4.10, 4.11, 4.12), they are not `kind to the
+reader'. They require a certain concentration to read and in my view
+should be avoided unless space is seriously constrained.
+
+\emph{HMF} Tables 6.1 and 6.2 are tables of the gamma function and
+its relatives where $y=x-1$ is used (stemming from $y!=\Gamma(x-1)$)
+in the row variable column on the right; Table 6.5 in effect uses
+$\langle1/x\rangle$ (the nearest integer to $1/x$) for the row variable
+on the right.
+
+\subsection{Separating blocks of rows: \texttt{rbloc}}
+
+Readability of long columns of figures can be aided by breaking the
+columns into blocks with extra white space between blocks of rows.
+This is achieved with the \texttt{rbloc} key:
+\begin{lyxcode}
+rbloc~=~<comma~list~of~positive~integers>
+\end{lyxcode}
+specifies how many rows belong to each block. For example, \texttt{rbloc=\{5,5,6\}}
+breaks the table into blocks of $5$ rows, $5$ rows, then $6$ rows.
+If the number of rows in the table is greater than the sum of the
+entries in the comma list, then division into blocks continues as
+specified by the last entry in the comma list. Thus \texttt{rbloc=5}
+(strictly \texttt{rbloc=\{5\}} but the braces can be omitted in this
+case since no comma is enclosed) divides a table into blocks of $5$
+rows; \texttt{rbloc=\{1,5\}} divides a table into $1$ row followed
+by blocks of $5$ rows. A division of this kind may be appropriate
+when, say, the row variable runs from $0$ to $1$ in increments of
+$0.1$ \textendash{} there are $11$ rows of which the first (when
+the row variable is zero) may have distinctive values.
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\subsubsection*{The pull of the nice round number}
+
+However, this is not how \emph{HMF} sets out its tables.\emph{ }The
+dominant practice in \emph{HMF} is division into blocks of (generally)
+$5$ rows, many of which start with a zero value for the row variable.
+Rather than isolate this initial value, they include it in the first
+block of $5$, then continue with blocks of $5$ until a single isolated
+row is left at the bottom of the page or the table. There seems to
+be a psychological need to finish a page or table with the row variable
+set to a nice round number. Thus: tabulate from $0$ to $10$ rather
+than $0$ to $9$, from $0$ to $1$ rather than $0$ to $0.9$, and
+even from $0$ to $30$ or $0$ to $2$ rather than $0$ to $29$
+or $0$ to $1.9$. Using blocks of $5$ the consequence is that there
+is always an isolated line at the end \textendash{} a kind of punctuation,
+marking the end of the page or the table.\end{shaded}%
+\end{minipage}
+
+\medskip{}
+In the next example I have divided the columns into blocks of $5$
+rows by means of the setting \texttt{rbloc=5}.
+\begin{verbatim}
+ \tabulate[o,rspec={\theta,10,90},rround=0,rbloc=5]
+ { ,\sin \theta, \cos \theta}[\theta=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[o,rspec={\theta,10,90},rround=0,rbloc=5]
+ { ,\sin \theta, \cos \theta}[\theta=0][*]
+
+\subsubsection{Adjusting the extra space\texttt{ rblocsep} }
+
+By default \texttt{numerica} sets the extra space between blocks of
+rows at \verb`1 ex`. This value can easily by changed with the setting
+\texttt{rblocsep=<length>}. The units need to be included in the specification.
+The default is $1$ \texttt{ex.}
+
+\section{Function value formatting}
+
+\label{sec:Function-value-formatting}
+\begin{table}
+
+\noindent \begin{centering}
+\caption{\protect\label{tab:Function-value-formatting}Function value formatting}
+\noindent \begin{center}
+\begin{tabular}{cc>{\raggedright}p{4cm}c}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{(pad)}} & {\small int} & {\small (t-notation) phantom padding} & \tabularnewline
+{\small\texttt{signs}} & {\small int} & {\small sign handling for function values} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{diffs}} & {\small int } & {\small insert differences \& pre-pad with zeros} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{Q?}} & {\small tokens} & {\small special cell conditional} & \tabularnewline
+{\small\texttt{A!}} & {\small tokens} & {\small special cell formatting } & \tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{centering}
+\end{table}
+In the examples used so far, function values have been limited to
+a narrow range, generally $[-1,1]$. What happens when function values
+span orders of magnitude?
+
+\subsection{Trailing optional argument}
+
+The primary tool for function value formatting is the trailing optional
+argument of the \verb`\tabulate` command where the rounding value
+is specified, padding with zeros is set or not (generally \emph{set}),
+and scientific notation is set or not. Elegant scientific notation,
+set with an \texttt{x} in the trailing optional argument, is generally
+not appropriate for use in tables; see the first table below. Adding
+a prime to the \texttt{x} in the trailing optional argument (the second
+table) so that scientific notation extends to numbers in the range
+$[1,10)$ helps, particularly with the \emph{left} alignment chosen
+for the function value column, but the result is wasteful of space
+and the repetition of the `$\times10$' is borderline distracting
+and would certainly be so for a larger table. The \texttt{x} specification
+should be used in tables, if at all, only for small tables \textendash{}
+a few function values at most.
+\begin{verbatim}
+ \tabulate[rspec={x,1,3},rround=0]
+ { e^x}[x=-5][*x]\qquad
+ \tabulate[rspec={x,1,3},rround=0,calign=l]
+ { e^x}[x=-3][*x']\qquad
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,1,3},rround=0]
+ { e^x }[x=-3][*x]\qquad
+ \tabulate[rspec={x,1,3},rround=0,calign=l]
+ { e^x}[x=-3][*x']\qquad
+
+\subsubsection{The \texttt{t} option}
+
+\emph{HMF} uses a special notation for coping with function values
+spanning orders of magnitude. This notation can be invoked by inserting
+\texttt{t} in the trailing optional argument. Repeating the previous
+two tables, and adding a \texttt{chnudge} value, gives a more compact
+and visually appealing result:
+\begin{verbatim}
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24]
+ { e^x}[x=-3][*t]\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24]
+ { e^x}[x=-3][*t']
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,1,3},rround=0,chnudge=24]
+ { e^x}[x=-3][*t]\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24]
+ { e^x}[x=-3][*t']
+
+\subsection{Padding the exponent: \texttt{(pad)}}
+
+In the second table of the last example some might quibble at the
+lack of alignment of the left parentheses. \emph{HMF} tends to align
+these and \texttt{numerica} offers the setting
+\begin{lyxcode}
+(pad)~=~<integer>
+\end{lyxcode}
+to achieve the effect. \texttt{<integer>} is the number of digits/characters
+to pad to. In the last example, the setting is \texttt{(pad)=2}. Here
+it is repeated with this setting:
+\begin{verbatim}
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24,(pad)=2]
+ { e^x}[x=-3][*t]\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24,(pad)=2]
+ { e^x}[x=-3][*t']
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,1,3},rround=0,chnudge=24,(pad)=2]
+ { e^x}[x=-3][*t]\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24,(pad)=2]
+ { e^x}[x=-3][*t']\medskip{}
+
+Note that this setting is relevant only when the \texttt{t} option
+is used in the trailing number-formatting argument of the \verb`\tabulate`
+command. Examples in \emph{HMF} of the style exemplified by the first
+table are, among others, Tables 8.6, 9.2, 20.1, and of the style exemplified
+by the second table, among many, Tables 9.9, 10.5, 13.1, 14.1, 19.1.
+
+\subsection{Accommodating signs: \texttt{signs}}
+
+\label{subsec:Signs}Instead of $e^{x}$ as the test function, use
+$e^{x}-1.$ Now there are positive, zero and negative function values
+to contend with. Recall that in the `t-notation' the \emph{exponent}
+is the parenthesized integer part of a number, the \emph{significand}
+the following decimal figures. \texttt{numerica} offers the \texttt{signs}
+key to align (or not) the exponents. The setting is
+\begin{lyxcode}
+signs~=~<integer>
+\end{lyxcode}
+There are four effective values for \texttt{<integer>} and the do-nothing
+default (\texttt{signs=0}):
+\begin{itemize}
+\item \texttt{signs=2 }inserts a $+$ sign between exponent and significand
+of every non-negative number;
+\item \texttt{signs=1 }inserts a $+$ sign between exponent and significand
+of every non-negative number immediately preceding or following a
+negative number;
+\item \texttt{signs=-1 }inserts a $+$ sign between exponent and significand
+of any non-negative number immediately preceding or following a negative
+number, and a phantom $-$ sign between exponent and significand of
+every other non-negative number;
+\item \texttt{signs=-2 }inserts a phantom $-$ sign between exponent and
+significand of every non-negative number;
+\end{itemize}
+With the `t-notation', the negative values and \verb`signs=2` seem
+the appropriate ones to use :
+\begin{verbatim}
+ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=-2]
+ { e^x-1}[x=-3][4*t']\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=-1]
+ { e^x-1}[x=-3][4*t']\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=2]
+ { e^x-1}[x=-3][4*t']
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=-2]
+ { e^x-1}[x=-3][4*t']\quad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=-1]
+ { e^x-1}[x=-3][4*t']\quad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=2]
+ { e^x-1}[x=-3][4*t']\medskip{}
+
+\noindent In \emph{HMF} Table 23.2 illustrates \verb`signs=-2`; Tables
+10.1, 13.1, 14.1, 19.1 among many others illustrate \verb`signs=-1`;
+and Tables 9.4, 10.6, 20.2, 22.11 among others illustrate \verb`signs=2`.
+
+However the \texttt{signs} key is not limited to the `t-notation'.
+In the following tables where the notation is not used, positive values
+for the key give appropriate results (I've included also the default
+setting):
+\begin{verbatim}
+ \tabulate[rspec={x,0.1,0.4},(pad)=2,signs=2]
+ { 10\sin 5x}[x=-0.4][*4]\qquad
+ \tabulate[rspec={x,0.1,0.4},(pad)=2,signs=1]
+ { 10\sin 5x}[x=-0.4][*4]\qquad
+ \tabulate[rspec={x,0.1,0.4},(pad)=2]
+ { 10\sin 5x}[x=-0.4][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,0.1,0.4},(pad)=2,signs=2]
+ { 10\sin 5x}[x=-0.4][*4]\qquad
+ \tabulate[rspec={x,0.1,0.4},(pad)=2,signs=1]
+ { 10\sin 5x}[x=-0.4][*4]\qquad
+ \tabulate[rspec={x,0.1,0.4},(pad)=2]
+ { 10\sin 5x}[x=-0.4][*4]\medskip{}
+
+\emph{HMF} seems to use \verb`signs=2` when the sign of function
+values changes every few entries and \verb`signs=1` when there are
+runs of entries of the same sign. They would use the middle table
+of the three here.
+
+\subsection{Differences: \texttt{diffs}}
+
+In fine-grained tables where function values change only slowly from
+entry to entry it can be helpful to include a difference entry between
+function value entries as an aid to interpolation (and a test of eyesight).
+By entering
+\begin{lyxcode}
+diffs~=~<non-negative~integer>
+\end{lyxcode}
+the \verb`tabulate` command will include differences in a table.
+The \texttt{<non-negative integer>} is the maximum number of digits
+in a difference.
+\begin{verbatim}
+ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=3]
+ { \sinh x }[x=1][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=3]
+ { \sinh x }[x=1][*4]\medskip{}
+
+I have deliberately chosen the function and settings here \textendash{}
+particularly \texttt{diffs=3} \textendash{} to give a good result.
+It is easy to get this wrong. The evidence will be in the misalignment
+of the first row of function values or unnecessary padding of differences
+with leading zeros. It is a good idea to create your table first,
+see how function values change between successive rows and judge how
+many digits there will be in a difference. In the following examples
+I have deliberately put \texttt{diffs=2} and \texttt{diffs=4} to show
+the effect of a misjudgement. In the second table the function is
+\emph{decreasing}, $-\sinh x$, to show how it is the absolute value
+of the difference between successive function values that is tabulated.
+A difference is always a non-negative value.
+\begin{verbatim}
+ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=2]
+ { \sinh x }[x=1][*4]\qquad
+ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=4]
+ { -\sinh x }[x=1][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=2]
+ { \sinh x }[x=1][*4]\qquad
+ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=4]
+ { -\sinh x }[x=1][*4]\medskip{}
+
+\noindent When the \texttt{diffs} setting is too small, function values
+in the first row are misaligned, the amount depending on how much
+too small. When the \texttt{diffs} setting is too big, alignment is
+fine but differences are padded with unnecessary leading zeros, meaning
+the column header will need a bigger nudge to bring \emph{it} into
+alignment.
+
+\subsection{Formatting special values: \texttt{Q?} and \texttt{A!}}
+
+You may wish to highlight or display in some special way a particular
+function value or values. \verb`\nmcTabulate` has two related settings
+that enable this: \texttt{Q?=<tokens>} and \texttt{A!=<tokens>}. As
+the names suggest: Question? and Answer!
+
+The question should be an expression that \texttt{l3fp} can digest
+and produce a boolean answer to (1 for `true' or 0 for `false').
+\texttt{numerica} uses \texttt{@} to denote the current function value,
+so queries like \texttt{Q?=@<0} (Is the current function value negative?)
+or \texttt{Q?=\{}@\texttt{>=pi\}} (Is the current function value greater
+than or equal to $\pi$?) are valid questions. (Note the braces in
+the second question, used to hide the equality sign.) Other possible
+useful components of such questions are \texttt{exp(1)} for the number
+$e$, || for logical Or, \texttt{\&\&} for logical And, and \texttt{!}
+for logical Not,\texttt{ }as well as the familiar arithmetic symbols,
+\texttt{+ - {*} /} and \texttt{\textasciicircum}, relation symbols
+\texttt{< > =} and their combinations like \texttt{!= >= <=} etc.,
+and parentheses. For everything \texttt{l3fp} makes available see
+the relevant chapter in \texttt{Interface3.pdf}, part of the documentation
+that comes with the \LaTeX 3 package \texttt{l3kernel}. In addition
+to these components, \texttt{numerica} offers \texttt{MAX} and \texttt{MIN}
+which are the maximum and minimum function values tabulated, so that,
+e.g., \texttt{Q?=\{@=MIN\}} (note the braces) is the question: Is
+the current function value equal to the minimum function value for
+the whole table?
+
+The answer must be in the form of a \LaTeXe{} formatting statement,
+again using \texttt{@} to denote the current function value. Thus
+\texttt{A!=\textbackslash mathbf\{@\}} is a valid answer; so is \texttt{A!=\textbackslash color\{red\}\{@\}}
+(provided you have \texttt{\textbackslash usepackage\{color\}} in
+the preamble); and so is \texttt{A!=(@)}. Another valid answer is
+\texttt{A!=} , meaning that function values satisfying the \texttt{Q?}
+question are omitted from the output.
+
+For example, suppose we wish to focus on the values of $\cos(m\pi/n)$
+lying between $0$ and $\tfrac{1}{2}$ inclusive for certain values
+of $m$ and $n$. Rather than cluttering the table with values outside
+that interval, we suppress them (the two occurrences of `\texttt{1e-14}'
+in the query are there to prevent rounding errors confusing the result):
+\begin{verbatim}
+ \tabulate
+ [rspec={n,1,15},rround=0,rpos=2,rules=Tth,
+ cspec={m,1,5},ctitle=*,chstyle=2,
+ Q?={@<-1e-14||@>0.5+1e-14},A!=]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={n,1,15},rround=0,rpos=2,rules=Tth,
+ cspec={m,1,5},ctitle=*,chstyle=2,
+ Q?={@<-1e-14||@>0.5+1e-14},A!=]
+ { \cos(m\pi/n)}[n=4,m=2][*4]
+
+\section{Table placement}
+
+\label{sec:Table-placement}There is only one setting in this category
+that is part of \texttt{numerica} as such, the tabular vertical alignment
+option; see \xA7\ref{subsec:Verticalalignment}. But \LaTeX{} allows one
+to insert vertical space with its \verb`\bigskip`, \verb`\medskip`,
+\verb`\smallskip`, usually about one line space, a half line space,
+and a quarter line space (with stretch and shrink), and \verb`booktabs`
+provides \verb`\abovetopsep` and \verb`\belowbottomsep`, both set
+by default to \verb`0ex` (or any other unit you care to use) and
+easily changed by writing, e.g., \verb`\abovetopsep=1.25ex` if you
+want to insert \verb`1.25ex` of space above the table (perhaps to
+fit captions).
+
+\subsection{Vertical~alignment}
+
+\label{subsec:Verticalalignment}By writing\texttt{ }
+\begin{lyxcode}
+valign~=~<char>~
+\end{lyxcode}
+where \texttt{<char>} is one of \texttt{t} or \texttt{b} the vertical
+alignment of the table can be set relative to the text baseline.\texttt{ valign=t
+}aligns the top of the table with the text baseline, \texttt{valign=b}
+the bottom of the table with the text baseline. By default (no \texttt{valign}
+setting or \texttt{valign} equated to some character other than \texttt{t}
+or \texttt{b}) the middle of the table is aligned with the text baseline.
+Repeating an example from earlier (\xA7\ref{sec:Row-variable-settings})
+I have added letters A, B, C to show where the baseline is. In the
+first table the top of the table aligns with the baseline; in the
+second table (default case) the middle of the table aligns with the
+baseline; in the third table, the bottom of the table aligns with
+the baseline.
+\begin{verbatim}
+ A \tabulate[valign=t,rvar=x,rstep=0.2,rows=6]
+ { \sin x/\cos x }[x=0][*] \qquad
+ B \tabulate[rspec={x,0.2,1}]
+ { \tan x }[x=0][*] \qquad
+ C \tabulate[valign=b,rspec={x,0.2,(6)}]
+ { \sqrt{\sec^2 x - 1} }[x=0][*]
+\end{verbatim}
+$\Longrightarrow$ A \tabulate[valign=t,rvar=x,rstep=0.2,rows=6,rstop=2]
+ { \sin x/\cos x }[x=0][*] \qquad
+ B \tabulate[rspec={x,0.2,1}]
+ { \tan x }[x=0][*] \qquad
+ C \tabulate[valign=b,rspec={x,0.2,(6)}]
+ { \sqrt{\sec^2 x - 1} }[x=0][*]\medskip{}
+
+As explained in \xA7\ref{subsec:Adjoining-tables}, tables can be adjoined
+to give the appearance of a single larger table. If tables with different
+numbers of rows are adjoined in this manner, then a middle alignment
+fails and either a top or bottom alignment is necessary.
+
+\section{Star option}
+
+If the \texttt{Q?} question is satisfied by at least one function
+value then adding a star (asterisk) to the \verb`\tabulate` command
+will display the first such instance. I.e., like the other starred
+commands, \verb`\eval*`, \verb`\iter*`, \verb`\solve*` and \verb`\recur*`,
+ \verb`\tabulate*` outputs a single number:
+\begin{verbatim}
+ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?={@<-1e-14||@>0.5+1e-14}]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?={@<-1e-14||@>0.5+1e-14}]
+ { \cos(m\pi/n) }[n=4,m=2][*4]. Ineed, if you omit the \texttt{Q?} and \texttt{A!} settings from
+the previous table then this is the value that follows $0.5000$ in
+the \texttt{m=2} column \textendash{} the first function value encountered
+satisfying the query \texttt{Q?}.
+
+If \emph{no} function value satisfies the query then a message is
+generated:
+\begin{verbatim}
+ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?=|@>1]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?=@>1]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+
+And if there is no query at all when the star option is chosen, another
+message is shown:
+\begin{verbatim}
+ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5}]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate*
+ [ rspec={n,1,15}, cspec={m,1,5}]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+
+\subsection{Scientific notation}
+
+If you want the number output in scientific notation when the star
+option is chosen, then enter the exponent mark in the trailing number-format
+option. This is straightforward for a letter like the commonly used
+\texttt{e}, but remember that if it is the \texttt{x} option that
+you enter, then you will need to place the \verb`\tabulate*` command
+between math delimiters, otherwise the \verb`\times` symbol resulting
+from the \texttt{x} option will generate a \LaTeX{} error (`Missing
+\$ inserted'):
+\begin{verbatim}
+ $
+ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?={@<-1e-14||@>0.5+1e-14},A!=]
+ { \cos(m\pi/n) }[n=4,m=2][*4x]
+ $
+\end{verbatim}
+$\Longrightarrow$ $
+ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?={@<-1e-14||@>0.5+1e-14},A!=]
+ { \cos(m\pi/n) }[n=4,m=2][*4x]
+ $.
+
+\section{The \texttt{reuse} setting}
+
+By entering
+\begin{lyxcode}
+reuse~=~<non-negative~integer>
+\end{lyxcode}
+it is possible to specify what is saved when the \verb`\tabulate`
+command is followed by a \verb`\reuse` command.
+\begin{itemize}
+\item \texttt{reuse=0} saves the table as displayed (the default);
+\item \texttt{reuse=n} saves the $n$-th \emph{column} of function values
+\begin{itemize}
+\item if \texttt{rpos} is non-zero: in a comma-separated list of braced
+pairs \texttt{\{row-variable value,function value}\};
+\item if \texttt{rpos=0}: in a comma-separated list of function values.
+\end{itemize}
+\item \texttt{reuse=-n} saves the $n$-th \emph{row} of function values
+in a comma-separated list that includes the row variable value in
+its appropriate position(s) if \texttt{rpos} is non-zero.
+\end{itemize}
+In the following example the third row is saved to the control sequence
+\verb`\rowiii` by using the setting \verb`reuse=-3`.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,
+ chround=2,calign=r,ctitle=**,
+ rules=TthB,reuse=-3]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] \reuse[rowiii]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.25,(5)},rround=2,
+ rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,
+ chround=2,calign=r,ctitle=**,rules=TthB,reuse=-3]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] \reuse[rowiii]\medskip{}
+
+\noindent Now test the content of the control sequence
+\begin{lyxcode}
+\noindent \verb`\rowiii`~$\Longrightarrow$~\rowiii ~
+\end{lyxcode}
+\noindent You can see that indeed the third row has been `captured
+for posterity'.
+
+Alternatively, we could save the second column in the control sequence
+\verb`\colii`:
+\begin{verbatim}
+ [rspec={x,0.25,(5)},rround=2,rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,
+ chround=2,calign=r,ctitle=**,
+ rules=TthB,reuse=2]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] \reuse[colii]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.25,(5)},rround=2,
+ rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,
+ chround=2,calign=r,ctitle=**,rules=TthB,reuse=2]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] \reuse[colii]\medskip{}
+
+To see what is saved in \verb`\colii` I use \TeX 's \verb`\meaning`
+command to show that it is indeed braced pairs:
+\begin{verbatim}
+ \meaning \colii
+\end{verbatim}
+$\Longrightarrow$ \meaning\colii
+
+\chapter{Nesting}
+
+\label{sec:tableNesting}Provided the starred form of any one of \verb`\eval`,
+\verb`\iter`, \verb`\solve`, \verb`recur` actually does produce
+a numerical result and not an error message then it can be nested
+within the main argument of any one of the other commands, including
+itself. The associated document \texttt{numerica-plus.pdf} has a number
+of illustrations of this. The starred form can also be used in the
+vv-list of any one of the commands, including itself. \texttt{numerica-plus.pdf}
+has an example of \verb`\solve*` being used in the vv-list of an
+\verb`\eval` command, and the document \texttt{numerica-basics.pdf}
+shows examples of \verb`\eval*` being used in the vv-list of an \verb`\eval`
+command.
+
+These nesting properties are also true of \verb`\tabulate`. It's
+starred form can be used in other commands. Other commands' starred
+forms can be used in \verb`\tabulate`.
+
+For the first, \verb`\tabulate*` inside \verb`\eval`, I have also
+used the example to show how formatting settings (shown for the first
+\verb`\tabulate*` in the example) can in fact be dispensed with (as
+shown for the second \verb`\tabulate*`). Only \emph{functional} settings
+are required. Hence there is no \verb`A!` setting in either. It is
+the \emph{question}, \verb`Q?`, not the answer, that isolates the
+single-value that is the result of \verb`\tabulate*`.
+\begin{verbatim}
+ \eval{$
+ (\tabulate*
+ [rspec={n,1,15},rround=0,rpos=2,rules=Tth,
+ cspec={m,1,5},ctitle=*,chstyle=2,
+ Q?={@=MAX}]
+ { \cos(m\pi/n) }[n=4,m=2][*4])\sinh t +
+ (\tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?={@=MIN}]
+ { \sin(m\pi/n) }[n=4,m=2][*4])\cosh t
+ $}[t=2][4]
+\end{verbatim}
+$\Longrightarrow$ \eval{$
+ (\tabulate*
+ [rspec={n,1,15},rround=0,rpos=2,rules=Tth,
+ cspec={m,1,5},ctitle=*,chstyle=2,
+ Q?={@=MAX}]
+ { \cos(m\pi/n) }[n=4,m=2][*4])\sinh t +
+ (\tabulate*
+ [rspec={n,1,15},rround=0,rpos=2,rules=Tth,
+ cspec={m,1,5},ctitle=*,chstyle=2,
+ Q?={@=MIN}]
+ { \sin(m\pi/n) }[n=4,m=2][*4])\cosh t
+ $}[t=2][4].
+
+For using the starred forms of other commands inside \verb`\tabulate`,
+it makes more sense to use one or more of \verb`\iter*`, \verb`\solve*`
+or \verb`\recur*` inside \verb`\tabulate` than it does \verb`\eval*`,
+but that assumes the reader is familiar with those com$\frac{}{}$mands
+and that \texttt{numerica-plus.def} is loaded for this document, which
+it is not. So, here is a rather pointless illustration of nesting
+\verb`\eval*` in \verb`\tabulate`. (At least it is reassuring that
+the values of $\Delta(N)$ are identical in the two columns.)
+\begin{verbatim}
+ \tabulate[rspec={N,100,500},rround=0,ctitle=\Delta(N),
+ chead=eval*&direct]
+ { ,\eval*{ \sum_{n=1}^N 1/n-\ln N-\gamma },
+ \sum_{n=1}^N 1/n-\ln N-\gamma }[N=100][4*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={N,100,500},rround=0,ctitle=\Delta(N),
+ chead=eval*&direct]
+ { ,\eval*{ \sum_{n=1}^N 1/n-\ln N-\gamma },
+ \sum_{n=1}^N 1/n-\ln N-\gamma }[N=100][4*]
+
+\chapter{Reference summary}
+
+\section{Commands defined in \texttt{numerica-tables}}
+
+\texttt{\textbackslash nmcTabulate, \textbackslash tabulate}
+
+\section{Settings for \texttt{\textbackslash nmcTabulate}}
+
+\subsubsection*{Row variable specification \xA7\ref{sec:Row-variable-settings}}
+\begin{center}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{3cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small comment}\tabularnewline
+\midrule
+{\small\texttt{rvar}} & {\small token(s)} & {\small row variable} & \tabularnewline
+{\small\texttt{rstep}} & {\small real num.} & {\small step size} & \tabularnewline
+{\small\texttt{rstop}} & {\small real num.} & {\small stop value} & \multirow{2}{3cm}{either {\small\texttt{rstop}}{\small{} or }{\small\texttt{rows}}{\small ,
+not both}}\tabularnewline
+{\small\texttt{rows}} & {\small int} & {\small number of rows} & \tabularnewline
+{\small\texttt{rspec}} & {\small comma list} & {\small\texttt{\{start}}{\small , }{\small\texttt{step}}{\small , }{\small\texttt{stop\}}}{\small{}
+or }{\small\texttt{\{start}}{\small , }{\small\texttt{step}}{\small ,
+}{\small\texttt{(rows)\}}} & {\small short form spec.}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+
+\subsubsection*{Row variable column formatting \xA7\ref{subsec:Row-var-col-formatting}}
+\begin{center}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{rround}} & {\small int} & {\small rounding} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{ralign}} & {\small char (}{\small\texttt{r/c/l}}{\small )} & {\small horizontal alignment} & {\small\texttt{r}}\tabularnewline
+{\small\texttt{rfont}} & {\small chars} & {\small font (}{\small\verb`\math<chars>`}{\small )} & \tabularnewline
+{\small\texttt{rhead}} & {\small tokens} & {\small header} & {\small\texttt{rvar}}\tabularnewline
+{\small\texttt{rhnudge}} & int & {\small nudge header }{\small{\small\verb`<int>`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small column position(s)} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{rvar'}} & {\small tokens} & {\small 2nd row variable col. spec.} & {\small\texttt{rvar}}\tabularnewline
+{\small\texttt{rhead'}} & {\small tokens} & {\small header of 2nd rv col. (if it exists)} & {\small\texttt{rvar'}}\tabularnewline
+{\small\texttt{rhnudge'}} & int & {\small nudge 2nd rv col. header }{\small{\small\verb`<int>`}}{\small{}
+mu} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+
+\subsubsection*{Column variable specification \xA7\ref{sec:Column-variable-settings}.}
+\begin{center}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{cvar}} & {\small token(s)} & {\small column variable} & \tabularnewline
+{\small\texttt{cstep}} & {\small real num.} & {\small step size} & \tabularnewline
+{\small\texttt{cstop}} & {\small real num.} & {\small stop value} & {\small either }{\small\texttt{cstop}}\tabularnewline
+{\small\texttt{cols}} & {\small int} & {\small number of columns} & {\small or }{\small\texttt{cols}}{\small , not both}\tabularnewline
+{\small\texttt{cspec}} & {\small comma list} & {\small\texttt{\{cvar,cstep}}{\small , }{\small\texttt{cstop\}}}{\small{}
+or}{\small\texttt{ \{cvar,cstep}}{\small ,}{\small\texttt{(cols)\}}} & {\small short form spec.}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+
+\subsubsection*{Column variable header formatting \xA7\ref{subsec:Column-header-formatting}.}
+\begin{center}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{chstyle}} & {\small int (0/1/2/3)} & {\small header style} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{ctitle}} & {\small tokens} & {\small single col. alternative header} & \tabularnewline
+{\small\texttt{chead}} & {\small tokens} & {\small user-defined header} & \tabularnewline
+{\small\texttt{calign}} & {\small char (r/c/l)} & {\small column alignment} & {\small\texttt{r}}\tabularnewline
+{\small\texttt{chnudge}} & {\small int} & {\small nudge header }{\small{\small\verb`int`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{chround}} & {\small int} & {\small column header rounding} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}\newpage{}
+\par\end{center}
+
+\subsubsection*{Function value formatting \xA7\ref{sec:Function-value-formatting}.}
+\noindent \begin{center}
+\noindent \begin{center}
+\begin{tabular}{cc>{\raggedright}p{4cm}c}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{(pad)}} & {\small int} & {\small (t-notation) phantom padding} & \tabularnewline
+{\small\texttt{signs}} & {\small int} & {\small sign handling for function values} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{diffs}} & {\small int} & {\small insert differences \& pre-pad with zeros} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{Q?}} & {\small tokens} & {\small special cell conditional} & \tabularnewline
+{\small\texttt{A!}} & {\small tokens} & {\small special cell formatting} & \tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+
+\subsubsection*{Whole-of-table formatting \xA7\ref{sec:Whole-of-table-formatting}.}
+\noindent \begin{center}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{ctitle}} & {\small tokens} & {\small collective title for function-value columns} & \tabularnewline
+{\small\texttt{rules}} & {\small chars} & {\small horizontal rules template} & {\small\texttt{ThB}}\tabularnewline
+{\small\texttt{foot}} & {\small tokens} & {\small content of footer line} & \tabularnewline
+{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small row-variable column position(s)} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{rbloc}} & {\small comma list} & {\small division of rows into blocks} & \tabularnewline
+{\small\texttt{rblocsep}} & {\small length} & {\small extra spacing between blocks of rows} & {\small\texttt{1 ex}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+
+\subsubsection*{Table placement \xA7\ref{subsec:Verticalalignment}.}
+\noindent \begin{center}
+\noindent \begin{center}
+\begin{tabular}{cc>{\raggedright}p{4cm}c}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{valign}} & {\small char (t/b)} & {\small vertical alignment} & \tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+\end{document}
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--- trunk/Master/texmf-dist/tex/latex/numerica/numerica-lyx.def (rev 0)
+++ trunk/Master/texmf-dist/tex/latex/numerica/numerica-lyx.def 2021-02-15 22:22:05 UTC (rev 57759)
@@ -0,0 +1,44 @@
+% This is file `numerica-lyx.def',
+% part of the `numerica' package
+%
+% This work may be distributed and/or modified under the conditions
+% of the LaTeX Project Public License, either version 1.3c of this
+% license or any later version; see
+% http://www.latex-project.org/lppl.txt
+%
+% Andrew Parsloe (ajparsloe at gmail.com)
+%
+\ProvidesExplFile
+ {numerica-lyx.def}
+ {2021/02/15}
+ {1.0.0}
+ { Enable reuse command to preview in LyX }
+%----------------------------------------------------------
+% copy (only once) .nmc file from doc to temp
+% directory; use nmcDummy.txt to mark success
+\tl_set:NV \l_tmpb_tl \jobname
+\tl_put_right:Nn \l_tmpb_tl {.nmc}
+\file_if_exist:nTF { nmcDummy.txt }
+ { \tl_gset_eq:NN \g__nmc_reuse_filename_tl \l_tmpb_tl }
+ {
+ \iow_open:Nn \g__nmc_iow { nmcDummy.txt }
+ \iow_close:N \g__nmc_iow
+ \makeatletter
+ \tl_set:Nx \l_tmpa_tl \input at path
+ \makeatother
+ \exp_last_unbraced:NNV
+ \tl_gset:Nn \g__nmc_reuse_filename_tl \l_tmpa_tl
+ \tl_gput_right:NV \g__nmc_reuse_filename_tl \l_tmpb_tl
+ \file_get:VnNTF \g__nmc_reuse_filename_tl {} \l__nmc_reuse_retrieved_tl
+ {
+ \tl_gset_eq:NN \g__nmc_reuse_filename_tl \l_tmpb_tl
+ \iow_open:NV \g__nmc_iow \g__nmc_reuse_filename_tl
+ \bool_lazy_or:nnF
+ { \tl_if_blank_p:V \l__nmc_reuse_retrieved_tl }
+ { \tl_if_head_eq_meaning_p:VN \l__nmc_reuse_retrieved_tl \par }
+ { \iow_now:NV \g__nmc_iow \l__nmc_reuse_retrieved_tl }
+ \iow_close:N \g__nmc_iow
+ }
+ { \tl_gset_eq:NN \g__nmc_reuse_filename_tl \l_tmpb_tl }
+ }
+% end of `numerica-lyx.def'
\ No newline at end of file
Property changes on: trunk/Master/texmf-dist/tex/latex/numerica/numerica-lyx.def
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===================================================================
--- trunk/Master/texmf-dist/tex/latex/numerica/numerica-plus.def (rev 0)
+++ trunk/Master/texmf-dist/tex/latex/numerica/numerica-plus.def 2021-02-15 22:22:05 UTC (rev 57759)
@@ -0,0 +1,876 @@
+% This is file `numerica-plus.def',
+% part of the `numerica' package.
+%
+% This work may be distributed and/or modified under the conditions
+% of the LaTeX Project Public License, either version 1.3c of this
+% license or any later version; see
+% http://www.latex-project.org/lppl.txt
+%
+% Andrew Parsloe (ajparsloe at gmail.com)
+%
+\ProvidesExplFile
+ {numerica-plus.def}
+ {2021/02/15}
+ {1.0.0}
+ { Iterate functions, find fixed points, zeros, extrema and recurrence terms }
+%----------------------------------------------------------
+\cs_new_protected:Npn \__nmc_plus_reuse:nNN #1#2#3
+ {
+ \int_case:nnF { #1 }
+ {
+ { 1 }
+ {
+ \seq_pop:NN #2 \l_tmpa_tl
+ \seq_pop:NN #3 \l_tmpb_tl
+ \tl_gset:Nx \g__nmc_reuse_tl { { \l_tmpa_tl , \l_tmpb_tl } }
+ \seq_mapthread_function:NNN #2 #3 \__nmc_plus_reuse_display:nn
+ }
+ { 2 }
+ {
+ \tl_gset:Nx \g__nmc_reuse_tl
+ { \seq_use:Nn #3 { , } }
+ }
+ }
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ }
+\cs_new_protected:Npn \__nmc_plus_reuse_display:nn #1#2
+ { \tl_gput_right:Nx \g__nmc_reuse_tl { , { #1 , #2 } } }
+%--------------------------------------
+% \nmcIterate
+\int_new:N \l__nmc_iter_total_int
+\int_new:N \l__nmc_iter_view_int
+\int_new:N \l__nmc_iter_round_int
+\tl_new:N \l__nmc_iter_var_tl
+\fp_new:N \l__nmc_iter_first_fp
+\seq_new:N \l__nmc_iter_result_seq
+\seq_new:N \l__nmc_iter_index_seq
+
+\fp_new:N \l__nmc_iter_fixedpti_fp
+\fp_new:N \l__nmc_iter_fixedptii_fp
+
+\tl_new:N \l__nmc_iter_begin_tl
+\tl_new:N \l__nmc_iter_end_tl
+\tl_new:N \g__nmc_iter_info_tl
+
+\nmc_define:nnN { \nmcIterate } { iter } \iter
+
+\cs_new_protected:Npn \__nmc_iter_settings_digest:
+ {
+ \bool_set_false:N \l__nmc_allow_TF_out_bool
+ \int_set:Nn \l__nmc_iter_total_int
+ { \int_max:nn { \l__nmc_iter_total_int } { 1 } }
+ \int_set:Nn \l__nmc_iter_view_int
+ { \int_min:nn { \int_max:nn { \l__nmc_iter_view_int }
+ { 1 } } { \l__nmc_iter_total_int } }
+ }
+\cs_new_protected:Npn \__nmc_iter_vv_digest:N #1
+ {
+ \tl_if_empty:NTF \l__nmc_iter_var_tl
+ { \__nmc_vv_digest:N #1 }
+ {
+ \clist_push:NV \l__nmc_formula_tl \l__nmc_iter_var_tl
+ \__nmc_vv_digest:N #1
+ \clist_pop:NN \l__nmc_formula_tl \l__nmc_iter_var_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_process:
+ {
+ \int_set:Nn \l__nmc_iter_round_int
+ { \l__nmc_round_int + \l__nmc_iter_extra_int }
+ % first iterate
+ \__nmc_error_where:n { formula }
+ \tl_clear:N \l__nmc_fp_expr_tl
+ \__nmc_fpify:VN \l__nmc_formula_tl \l__nmc_fp_expr_tl
+ \fp_set:Nn \l__nmc_result_fp { \l__nmc_fp_expr_tl }
+ \__nmc_error_fpflag:
+
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \fp_set_eq:NN \l__nmc_iter_first_fp \l__nmc_result_fp
+ \tl_if_empty:NT \l__nmc_iter_var_tl
+ { % get iter. var if unspecified
+ \seq_get:NN \l__nmc_vv_all_seq \l_tmpa_tl
+ \__nmc_vv_split_item:V \l_tmpa_tl
+ \tl_set_eq:NN \l__nmc_iter_var_tl \l__nmc_eq_var_tl
+ }
+ \__nmc_iter_do:
+ \tl_set_eq:NN \l__nmc_fp_expr_tl \l__nmc_fp_exprn_tl
+ }
+ \bool_if:nF { \g__nmc_error_bool || \l__nmc_num_only_bool }
+ {
+ \__nmc_iter_environ:
+ \__nmc_iter_write:
+ }
+ }
+%--------------------------------------
+\cs_new_protected:Npn \__nmc_iter_do:
+ {
+ \bool_if:NTF \l__nmc_num_only_bool
+ {
+ \__nmc_iter_fixed_pt:
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_num_format:nNnN { \l__nmc_iter_fixedpti_fp }
+ \l_nmc_result_tl { \l__nmc_round_int }
+ \l__nmc_sci_num_out_bool
+ \int_decr:N \l__nmc_iter_total_int
+ \tl_gset:Nx \g__nmc_iter_info_tl
+ { \int_use:N \l__nmc_iter_total_int }
+ }
+ }
+ { % don't print initial iterations
+ \int_step_function:nnnN { 1 } { 1 }
+ { \l__nmc_iter_total_int - \l__nmc_iter_view_int - 1 }
+ \__nmc_iter_current:n
+ \bool_if:NF \g__nmc_error_bool
+ { % store then print these ones
+ \seq_clear:N \l__nmc_iter_result_seq
+ \int_step_function:nnnN { 1 } { 1 }
+ { \int_min:nn { \l__nmc_iter_view_int }
+ { \l__nmc_iter_total_int - 1 } } \__nmc_iter_current_store:n
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_current:n #1
+ { % stepping function
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_iter_var_tl \l__nmc_formula_tl
+ { \l__nmc_result_fp } \l__nmc_result_fp
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_current_store:n #1
+ { % stepping function
+ \__nmc_calc_fn_val:VNnN \l__nmc_iter_var_tl \l__nmc_formula_tl
+ { \l__nmc_result_fp } \l__nmc_result_fp
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_num_format:nNnN { \l__nmc_result_fp } \l_nmc_result_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \seq_put_right:NV \l__nmc_iter_result_seq \l_nmc_result_tl
+ \int_set:Nn \l_tmpa_int
+ { \l__nmc_iter_total_int - \l__nmc_iter_view_int + #1 }
+ \seq_put_right:Nx \l__nmc_iter_index_seq { \int_use:N \l_tmpa_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_fixed_pt:
+ { % already 1 iteration
+ \int_set:Nn \l__nmc_iter_total_int { 1 }
+ \fp_set_eq:NN \l__nmc_iter_fixedpti_fp \l__nmc_iter_first_fp
+ \fp_set:Nn \l__nmc_iter_fixedptii_fp { \l__nmc_iter_fixedpti_fp + 1 }
+ \bool_until_do:nn
+ {
+ \fp_compare_p:nNn { 0 } =
+ { round( \l__nmc_iter_fixedpti_fp - \l__nmc_iter_fixedptii_fp,
+ \l__nmc_iter_round_int ) }
+ || \g__nmc_error_bool
+ }
+ {
+ \int_incr:N \l__nmc_iter_total_int
+ \fp_set_eq:NN \l__nmc_iter_fixedptii_fp \l__nmc_iter_fixedpti_fp
+ \__nmc_calc_fn_val:VNnN \l__nmc_iter_var_tl \l__nmc_formula_tl
+ { \l__nmc_result_fp } \l__nmc_result_fp
+ \fp_set_eq:NN \l__nmc_iter_fixedpti_fp \l__nmc_result_fp
+ \int_compare:nNnF { \l__nmc_iter_total_int } <
+ { \l__nmc_iter_max_int }
+ {
+ \bool_gset_true:N \g__nmc_error_bool
+ \__nmc_error_what:n
+ {
+ No~fixed~point~attained~after~$\int_use:N
+ \l__nmc_iter_total_int$~iterations~of
+ }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_environ:
+ {
+ \tl_set:Nx \l__nmc_iter_begin_tl
+ {
+ \mode_if_math:F { \exp_not:o \l__nmc_math_delimi_tl }
+ \exp_not:N \begin{array}{r@{}l}
+ \bool_if:NTF \l__nmc_wraps_math_bool
+ { \exp_not:o \l__nmc_formula_dup_tl &{}= }
+ { &{} }
+ }
+ \tl_set:Nx \l__nmc_iter_end_tl
+ {
+ \exp_not:N \end{array}
+ \mode_if_math:F { \exp_not:o \l__nmc_math_delimii_tl }
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_write:
+ {
+ \__nmc_num_format:nNnN { \l__nmc_iter_first_fp } \l_tmpa_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \tl_gset:Nx \g__nmc_iter_info_tl { \int_use:N \l__nmc_iter_total_int }
+ \__nmc_if_mod_zero:nnTF { \l__nmc_debug_int } { 7 }
+ { \seq_get_right:NN \l__nmc_iter_result_seq \l_nmc_result_tl }
+ {
+ \tl_set:Nx \l_nmc_result_tl
+ {
+ \exp_not:o \l__nmc_iter_begin_tl
+ \l_tmpa_tl \l__nmc_vv_display_tl
+ \int_compare:nNnT
+ { \l__nmc_iter_total_int } > { \l__nmc_iter_view_int + 1 }
+ {
+ \\ & \exp_not:N \ldots
+ \ \exp_not:N\mbox{final\
+ \int_use:N \l__nmc_iter_view_int\ of\ \g__nmc_iter_info_tl :}
+ }
+ \int_compare:nNnT { \l__nmc_iter_total_int } > { 1 }
+ { \\ & \exp_not:N \hookrightarrow }
+ \seq_use:Nn \l__nmc_iter_result_seq { \\ & \hookrightarrow }
+ \exp_not:o \l__nmc_iter_end_tl
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_display:
+ {
+ \bool_if:NTF \l__nmc_num_only_bool
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ {
+ \__nmc_plus_reuse:nNN { \l__nmc_iter_reuse_int }
+ \l__nmc_iter_index_seq \l__nmc_iter_result_seq
+ }
+ \l_nmc_result_tl
+ }
+%--------------------------------------
+%\nmcSolve
+\int_new:N \l__nmc_solve_round_int
+\int_new:N \l__nmc_solve_steps_int
+\int_new:N \l__nmc_solve_slope_int
+\int_new:N \l__nmc_solve_slopei_int
+\int_new:N \l__nmc_solve_signs_int
+\fp_new:N \l__nmc_solvea_fp
+\fp_new:N \l__nmc_solveb_fp
+\fp_new:N \l__nmc_solvec_fp
+\fp_new:N \l__nmc_solved_fp
+
+\fp_new:N \l__nmc_solvefa_fp
+\fp_new:N \l__nmc_solvefb_fp
+\fp_new:N \l__nmc_solvefc_fp
+\bool_new:N \l__nmc_solve_stop_bool
+\tl_new:N \g__nmc_sove_info_tl
+
+\nmc_define:nnN { \nmcSolve } { solve } \solve
+
+\cs_new_protected:Npn \__nmc_solve_settings_digest:
+ {
+ \bool_set_false:N \l__nmc_allow_TF_out_bool
+ \tl_if_empty:NF \l__nmc_solve_step_tl
+ {
+ \__nmc_fpify_set:NN \l__nmc_solve_step_fp \l__nmc_solve_step_tl
+ \fp_compare:nNnT \l__nmc_solve_step_fp = { 0 }
+ { \__nmc_error_what:n { Non-zero~initial~step~required~in } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_solve_vv_digest:N #1
+ {
+ \tl_if_empty:NTF \l__nmc_solve_var_tl
+ { \__nmc_vv_digest:N #1 }
+ {
+ \clist_push:NV \l__nmc_formula_tl \l__nmc_solve_var_tl
+ \__nmc_vv_digest:N #1
+ \clist_pop:NN \l__nmc_formula_tl \l__nmc_solve_var_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_solve_process:
+ {
+ \int_set:Nn \l__nmc_solve_round_int
+ { \l__nmc_round_int + \l__nmc_solve_extra_int }
+ % get equation var if unspecified
+ \tl_if_empty:NT \l__nmc_solve_var_tl
+ {
+ \seq_get:NN \l__nmc_vv_all_seq \l_tmpa_tl
+ \__nmc_vv_split_item:V \l_tmpa_tl
+ \tl_set_eq:NN \l__nmc_solve_var_tl \l__nmc_eq_var_tl
+ }
+ \bool_if:NF \g__nmc_error_bool
+ { \__nmc_solve_get_trial_vals: }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_error_where:n { function }
+ \__nmc_solve_do:
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_num_format:nNnN { \l__nmc_solvea_fp } \l_nmc_result_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \tl_gset:Nx \g__nmc_solve_info_tl
+ { \clist_use:Nn \g__nmc_solve_info_tl { + } }
+ \tl_set_eq:NN \l__nmc_fp_expr_tl \l__nmc_fp_exprn_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_solve_get_trial_vals:
+ {
+ \prop_get:NVN \g__nmc_subst_var_prop
+ \l__nmc_solve_var_tl \l__nmc_subst_tl
+ % ensure a < b
+ \int_case:nn { \fp_sign:n { \l__nmc_solve_step_fp } }
+ {
+ { 1 }
+ {
+ \fp_set:Nn \l__nmc_solvea_fp { \l__nmc_subst_tl }
+ \fp_set:Nn \l__nmc_solveb_fp
+ { \l__nmc_solvea_fp + \l__nmc_solve_step_fp }
+ }
+ { -1 }
+ {
+ \fp_set:Nn \l__nmc_solveb_fp { \l__nmc_subst_tl }
+ \fp_set:Nn \l__nmc_solvea_fp
+ { \l__nmc_solveb_fp + \l__nmc_solve_step_fp }
+ }
+ }
+ }
+% find opp. signs, zero, or fn min.
+% a b = var vals; fa fb = fn vals
+\cs_new_protected:Npn \__nmc_solve_do:
+ {
+ \__nmc_solve_calc_values:
+ \int_zero:N \l__nmc_solve_steps_int
+ \tl_gclear:N \g__nmc_solve_info_tl
+ \bool_do_until:nn
+ {
+ \g__nmc_error_bool || \l__nmc_solve_stop_bool
+ || \int_compare_p:nNn
+ { \l__nmc_solve_steps_int } > { \l__nmc_solve_max_int }
+ || \fp_compare_p:nNn { 0 } =
+ { round(\l__nmc_solveb_fp - \l__nmc_solvea_fp,
+ \l__nmc_solve_round_int) }
+ }
+ {
+ \int_incr:N \l__nmc_solve_steps_int
+ \int_case:nn { \l__nmc_solve_signs_int }
+ {
+ { 0 } { \__nmc_solve_do_bingo: }
+ { -1 } { \__nmc_solve_do_bisect: }
+ { 1 } { \__nmc_solve_do_slope: }
+ }
+ }
+ \bool_if:nF { \g__nmc_error_bool || \l__nmc_solve_stop_bool }
+ {
+ \__nmc_error_where:n { $\l__nmc_formula_tl$ }
+ \__nmc_error_what:n
+ {
+ No~zero/extremum~found~after~$\int_use:N
+ \l__nmc_solve_max_int$~steps~for~function
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_solve_do_bingo:
+ { % fn = 0 to 16 figures
+ \fp_compare:nNnTF { \l__nmc_solvefb_fp } = { 0 }
+ {
+ \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solveb_fp
+ \fp_set_eq:NN \l__nmc_solvefa_fp \l__nmc_solvefb_fp
+ }
+ {
+ \fp_set_eq:NN \l__nmc_solveb_fp \l__nmc_solvea_fp
+ \fp_set_eq:NN \l__nmc_solvefb_fp \l__nmc_solvefa_fp
+ }
+ \bool_set_true:N \l__nmc_solve_stop_bool
+ }
+\cs_new_protected:Npn \__nmc_solve_do_bisect:
+ {
+ \tl_gset:Nx \g__nmc_solve_info_tl { \int_use:N \l__nmc_solve_steps_int }
+ \int_zero:N \l__nmc_solve_steps_int
+ \fp_set:Nn \l__nmc_solvec_fp { ( \l__nmc_solvea_fp + \l__nmc_solveb_fp ) / 2 }
+ \__nmc_calc_fn_val:VNnN \l__nmc_solve_var_tl \l__nmc_formula_tl
+ { \l__nmc_solvec_fp } \l__nmc_solvefc_fp
+ \fp_set:Nn \l__nmc_solved_fp { \l__nmc_solvec_fp + 1 }
+ \bool_until_do:nn
+ {
+ \g__nmc_error_bool ||
+ \fp_compare_p:nNn { 0 } =
+ { round( \l__nmc_solvec_fp - \l__nmc_solved_fp ,
+ \l__nmc_solve_round_int ) }
+ }
+ {
+ \int_incr:N \l__nmc_solve_steps_int
+ \fp_set_eq:NN \l__nmc_solved_fp \l__nmc_solvec_fp
+ \fp_compare:nNnTF { 0 } = { \l__nmc_solvefc_fp }
+ {
+ \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solvec_fp
+ \fp_set_eq:NN \l__nmc_solvefa_fp \l__nmc_solvefc_fp
+ }
+ {
+ \fp_compare:nNnTF
+ { sign(\l__nmc_solvefa_fp)sign(\l__nmc_solvefc_fp) } = { 1 }
+ {
+ \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solvec_fp
+ \fp_set_eq:NN \l__nmc_solvefa_fp \l__nmc_solvefc_fp
+ }
+ {
+ \fp_set_eq:NN \l__nmc_solveb_fp \l__nmc_solvec_fp
+ \fp_set_eq:NN \l__nmc_solvefb_fp \l__nmc_solvefc_fp
+ }
+ \fp_set:Nn \l__nmc_solvec_fp
+ { ( \l__nmc_solvea_fp + \l__nmc_solveb_fp ) / 2 }
+ \__nmc_calc_fn_val:VNnN \l__nmc_solve_var_tl
+ \l__nmc_formula_tl { \l__nmc_solvec_fp } \l__nmc_solvefc_fp
+ }
+ }
+ \bool_set_true:N \l__nmc_solve_stop_bool
+ \clist_gput_right:Nx \g__nmc_solve_info_tl { \int_use:N \l__nmc_solve_steps_int }
+ }
+\cs_new_protected:Npn \__nmc_solve_do_slope:
+ {
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_if_zero:nTF { \l__nmc_solve_slope_int }
+ { % contract
+ \fp_add:Nn \l__nmc_solvea_fp
+ { ( \l__nmc_solveb_fp - \l__nmc_solvea_fp ) / 4 }
+ \fp_sub:Nn \l__nmc_solveb_fp
+ { ( \l__nmc_solveb_fp - \l__nmc_solvea_fp ) / 4 }
+ }
+ { % always towards x-axis
+ \fp_compare:nNnTF { 0 } <
+ { \l__nmc_solvefa_fp * \l__nmc_solve_slope_int }
+ { \__nmc_solve_do_slope_left: }
+ { \__nmc_solve_do_slope_right: }
+ }
+ \fp_set_eq:NN \l__nmc_solved_fp \l__nmc_solvea_fp
+ \int_set_eq:NN \l__nmc_solve_slopei_int \l__nmc_solve_slope_int
+ \__nmc_solve_calc_values:
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_compare:nNnF { \l__nmc_solve_slope_int } =
+ { \l__nmc_solve_slopei_int }
+ {
+ \fp_set:Nn \l__nmc_solvec_fp { ( \l__nmc_solvea_fp + \l__nmc_solveb_fp ) / 2 }
+ \int_case:nn { \l__nmc_solve_slopei_int }
+ {
+ { 1 } { \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solvec_fp }
+ { -1 } { \fp_set_eq:NN \l__nmc_solveb_fp \l__nmc_solvec_fp }
+ }
+ \__nmc_solve_calc_values:
+ }
+ }
+ \fp_compare:nNnT { 0 } =
+ { round(\l__nmc_solveb_fp - \l__nmc_solvea_fp,
+ \l__nmc_solve_round_int) }
+ { \bool_set_true:N \l__nmc_solve_stop_bool }
+ }
+\cs_new_protected:Npn \__nmc_solve_do_slope_left:
+ {
+ \fp_set:Nn \l__nmc_solvec_fp { 2 \l__nmc_solvea_fp - \l__nmc_solveb_fp }
+ \fp_set_eq:NN \l__nmc_solveb_fp \l__nmc_solvea_fp
+ \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solvec_fp
+ }
+\cs_new_protected:Npn \__nmc_solve_do_slope_right:
+ {
+ \fp_set:Nn \l__nmc_solvec_fp { 2 \l__nmc_solveb_fp - \l__nmc_solvea_fp }
+ \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solveb_fp
+ \fp_set_eq:NN \l__nmc_solveb_fp \l__nmc_solvec_fp
+ }
+\cs_new_protected:Npn \__nmc_solve_calc_values:
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_solve_var_tl \l__nmc_formula_tl
+ { \l__nmc_solvea_fp } \l__nmc_solvefa_fp
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_solve_var_tl \l__nmc_formula_tl
+ { \l__nmc_solveb_fp } \l__nmc_solvefb_fp
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_set:Nn \l__nmc_solve_slope_int
+ { \fp_eval:n { sign(\l__nmc_solvefb_fp - \l__nmc_solvefa_fp) } }
+ \int_set:Nn \l__nmc_solve_signs_int
+ { \fp_eval:n { sign(\l__nmc_solvefa_fp) sign(\l__nmc_solvefb_fp) } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_solve_display:
+ {
+ \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl
+ \bool_if:NF \l__nmc_num_only_bool
+ {
+ \__nmc_num_format:nNnN { \l__nmc_solvefa_fp } \l_tmpa_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \tl_set:Nx \l_nmc_result_tl
+ {
+ \exp_not:o \l__nmc_math_delimi_tl
+ \bool_if:NTF \l__nmc_wraps_math_bool
+ {
+ \exp_not:o \l__nmc_formula_tl = \l_tmpa_tl
+ \bool_if:NTF \l__nmc_vv_multline_bool
+ { \exp_not:o \l__nmc_vv_display_tl }
+ {
+ \mathchoice{ \exp_not:o \l__nmc_vv_display_tl }
+ { \exp_not:o \l__nmc_vv_inline_tl }{}{}
+ }
+ \rightarrow \exp_not:N \quad
+ \l__nmc_solve_var_tl = \l_nmc_result_tl
+ }
+ {
+ \l__nmc_solve_var_tl = \l_nmc_result_tl
+ \mathchoice{ \exp_not:o \l__nmc_vv_display_tl }
+ { \exp_not:o \l__nmc_vv_inline_tl }{}{}
+ }
+ \l__nmc_punc_tl
+ \exp_not:o \l__nmc_math_delimii_tl
+ }
+ \int_if_zero:nT { \l__nmc_solve_reuse_int }
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ }
+ \l_nmc_result_tl
+ }
+%--------------------------------------
+% \nmcRecur
+\bool_new:N \l__nmc_recur_ellipsis_bool
+\int_new:N \l__nmc_recur_last_int
+\fp_new:N \l__nmc_recur_result_fp
+
+\int_new:N \l__nmc_recur_subscr_ini_int
+\int_new:N \l__nmc_recur_subscr_val_int
+\int_new:N \l__nmc_recur_order_int
+\int_new:N \l__nmc_recur_var_int
+
+\tl_new:N \l__nmc_recurrence_tl
+\tl_new:N \l__nmc_recur_base_var_tl
+\tl_new:N \l__nmc_recur_subscr_var_tl
+
+\seq_new:N \l__nmc_recur_result_seq
+\seq_new:N \l__nmc_recur_index_seq
+\seq_new:N \l__nmc_recur_vars_seq
+
+\nmc_define:nnN { \nmcRecur } { recur } \recur
+
+\cs_new_protected:Npn \__nmc_recur_settings_digest:
+ {
+ \bool_set_false:N \l__nmc_allow_TF_out_bool
+ \int_set:Nn \l__nmc_recur_total_int
+ { \int_max:nn { \l__nmc_recur_total_int } { 1 } }
+ \int_set:Nn \l__nmc_recur_last_int
+ { \int_max:nn { 0 } { \int_min:nn
+ { \l__nmc_recur_last_int } { \l__nmc_recur_total_int } } }
+ \int_set:Nn \l__nmc_recur_first_int
+ { \int_max:nn { 0 } { \int_min:nn { \l__nmc_recur_first_int }
+ { \l__nmc_recur_total_int - \l__nmc_recur_last_int } } }
+ \int_if_zero:nT { \l__nmc_recur_first_int }
+ { \int_decr:N \l__nmc_recur_first_int }
+ }
+\cs_new_protected:Npn \__nmc_recur_vv_digest:N #1
+ { % #1 = reversed vv clist
+ \bool_set_true:N \l__nmc_multitok_bool
+ \__nmc_recur_elements:
+ \__nmc_recur_vars_change:N #1
+ \__nmc_vv_digest:N #1
+ \__nmc_recur_vv_post:
+ \tl_set_eq:NN \l__nmc_formula_tl \l__nmc_recurrence_tl
+ }
+% \l__nmc_recurrence_tl, \l__nmc_recur_base_var_tl,
+% \l__nmc_recur_subscr_var_tl, \l__nmc_recur_subscr_val_int
+\cs_new_protected:Npn \__nmc_recur_elements:
+ {
+ \tl_clear:N \l_tmpa_tl
+ \bool_set_false:N \l_tmpa_bool
+ \tl_map_inline:Nn \l__nmc_formula_tl
+ {
+ \bool_if:NTF \l_tmpa_bool
+ {
+ \tl_set:Nn \l_tmpa_tl { ##1 }
+ \tl_map_break:
+ }
+ {
+ \token_if_math_subscript:NTF ##1
+ {
+ \tl_set:NV \l__nmc_recur_base_var_tl \l_tmpa_tl
+ \bool_set_true:N \l_tmpa_bool
+ }
+ { \tl_put_right:Nn \l_tmpa_tl { ##1 } }
+ }
+ }
+ \__nmc_recur_parse_subscr:N \l_tmpa_tl
+ \exp_last_unbraced:NV\__nmc_split_eq:w \l__nmc_formula_tl \q_stop
+ \tl_set:NV \l__nmc_recurrence_tl \l__nmc_eq_val_tl
+ \tl_set_rescan:Nno \l__nmc_recurrence_tl { \ExplSyntaxOn } \l__nmc_recurrence_tl
+ }
+\cs_new_protected:Npn \__nmc_recur_parse_subscr:N #1
+ {
+ \tl_clear:N \l__nmc_recur_subscr_var_tl
+ \tl_set:Nn \l_tmpb_tl { 0 }
+ \int_zero:N \l__nmc_recur_subscr_val_int
+ \bool_set_false:N \l_tmpa_bool
+ \tl_map_inline:Nn #1
+ {
+ \bool_if:NTF \l_tmpa_bool
+ { \tl_put_right:Nn \l_tmpb_tl { ##1 } }
+ {
+ \tl_if_in:nnTF { +- } { ##1 }
+ {
+ \tl_put_right:Nn \l_tmpb_tl { ##1 }
+ \bool_set_true:N \l_tmpa_bool
+ }
+ { \tl_put_right:Nn \l__nmc_recur_subscr_var_tl { ##1 } }
+ }
+ }
+ \int_set:Nn \l__nmc_recur_subscr_val_int { \l_tmpb_tl }
+ }
+\cs_new_protected:Npn \__nmc_recur_vars_change:N #1
+ { % f_{1} etc ==> f_{n-1} etc in #1 (reverse order vv-list)
+ \clist_reverse:N #1
+ \int_zero:N \l__nmc_recur_order_int
+ \clist_clear:N \l_tmpa_clist % --> \l__nmc_recur_vars_seq
+ \clist_clear:N \l_tmpb_clist % --> #1
+ \int_set:Nn \l_tmpb_int { \l__nmc_recur_subscr_val_int - 1 }
+ \tl_set_rescan:Nno #1 { \ExplSyntaxOn } #1
+ \clist_map_inline:Nn #1
+ {
+ \seq_set_split:Nnn \l_tmpa_seq {_} { ##1 }
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl
+ \seq_if_empty:NTF \l_tmpa_seq
+ { \clist_put_left:NV \l_tmpb_clist \l_tmpa_tl }
+ {
+ \tl_if_eq:NNTF \l_tmpa_tl \l__nmc_recur_base_var_tl
+ { % change e.g. f_{1}(x) to f_{n-1}(x)
+ \int_incr:N \l__nmc_recur_order_int
+ \tl_put_right:Nn \l_tmpa_tl { _ }
+ \tl_set_eq:NN \l_tmpb_tl \l__nmc_recur_subscr_var_tl
+ \int_case:nn { \int_sign:n { \l_tmpb_int } }
+ {
+ { -1 }
+ {
+ \tl_put_right:Nx \l_tmpb_tl
+ { \int_use:N \l_tmpb_int }
+ }
+ { 0 } { \prg_do_nothing: }
+ { 1 }
+ {
+ \tl_put_right:Nn \l_tmpb_tl { + }
+ \tl_put_right:Nx \l_tmpb_tl
+ { \int_use:N \l_tmpb_int }
+ }
+ }
+ \tl_put_right:Nx \l_tmpa_tl { { \l_tmpb_tl } }
+ \int_decr:N \l_tmpb_int
+ \seq_pop:NN \l_tmpa_seq \l_tmpb_tl
+ \int_set:Nn \l__nmc_recur_subscr_ini_int
+ { \tl_head:N \l_tmpb_tl }
+ \tl_put_right:Nx \l_tmpa_tl
+ { \tl_range:Nnn \l_tmpb_tl { 2 } { -1 } }
+ \clist_put_left:NV \l_tmpb_clist \l_tmpa_tl
+ \seq_set_split:NnV \l_tmpb_seq { = } \l_tmpa_tl
+ \seq_pop:NN \l_tmpb_seq \l_tmpa_tl
+ \clist_put_left:NV \l_tmpa_clist \l_tmpa_tl
+ }
+ { \clist_put_left:Nn \l_tmpb_clist { ##1 } }
+ }
+ }
+ \int_set:Nn \l__nmc_recur_var_int
+ { \l__nmc_recur_subscr_ini_int + \l__nmc_recur_order_int
+ - \l__nmc_recur_subscr_val_int }
+ \clist_set_eq:NN #1 \l_tmpb_clist
+ \clist_put_left:NV \l_tmpa_clist \l__nmc_recur_subscr_var_tl
+ \clist_concat:NNN \l__nmc_formula_tl \l__nmc_recurrence_tl \l_tmpa_clist
+ }
+\cs_new_protected:Npn \__nmc_recur_vv_post:
+ {
+ \clist_pop:NN \l__nmc_formula_tl \l__nmc_recurrence_tl
+ \clist_pop:NN \l__nmc_formula_tl \l__nmc_recur_subscr_var_tl
+ \tl_set:Nx \l_tmpa_tl { \int_use:N \l__nmc_recur_var_int }
+ \__nmc_vv_record:NVN \l__nmc_recur_subscr_var_tl \l_tmpa_tl \c_empty_prop
+ \tl_set_eq:NN \l_tmpa_tl \l__nmc_recur_subscr_var_tl
+ \tl_put_right:Nn \l_tmpa_tl { =0 } % formal value
+ \seq_put_left:NV \l__nmc_calc_fn_seq \l_tmpa_tl
+ \seq_set_from_clist:NN \l__nmc_recur_vars_seq \l__nmc_formula_tl
+ }
+%%%%%%%%%%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_recur_process:
+ { % store initial vals; generate later vals
+ \__nmc_recur_store_ini:
+ \__nmc_error_where:n { recurrence~formula }
+ \__nmc_recur_generate:
+ \seq_get_right:NN \l__nmc_recur_result_seq \l_nmc_result_tl
+ \tl_set_eq:NN \l__nmc_fp_expr_tl \l__nmc_fp_exprn_tl
+ }
+\cs_new_protected:Npn \__nmc_recur_store_ini:
+ {
+ \seq_set_eq:NN \l_tmpa_seq \l__nmc_recur_vars_seq
+ \int_step_inline:nnnn { 1 } { 1 } { \l__nmc_recur_order_int }
+ {
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl
+ \prop_get:NVN \g__nmc_subst_var_prop \l_tmpa_tl \l__nmc_subst_tl
+ \__nmc_num_format:nNnN { \l__nmc_subst_tl } \l_tmpa_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \seq_put_right:NV \l__nmc_recur_result_seq \l_tmpa_tl
+ \tl_set:Nx \l_tmpb_tl { \int_eval:n
+ { \l__nmc_recur_subscr_ini_int + ##1 -1 } }
+ \seq_put_right:NV \l__nmc_recur_index_seq \l_tmpb_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_recur_generate:
+ {
+ \prop_get:NVN \g__nmc_subst_var_prop
+ \l__nmc_recur_subscr_var_tl \l__nmc_subst_tl
+ \int_set:Nn \l__nmc_recur_var_int { \l__nmc_subst_tl }
+ \int_set:Nn \l_tmpa_int { \l__nmc_recur_var_int +
+ \l__nmc_recur_total_int - \l__nmc_recur_order_int - 1 }
+ \__nmc_error_where:n { recurrence~relation }
+ \__nmc_if_mod_zero:nnT { \l__nmc_debug_int } { 7 }
+ { \__nmc_fpify:VN \l__nmc_recurrence_tl \l__nmc_fp_expr_tl }
+ \int_step_function:nnnN { \l__nmc_recur_var_int } { 1 }
+ {\l_tmpa_int } \__nmc_recur_generate_loop:n
+ }
+\cs_new_protected:Npn \__nmc_recur_generate_loop:n #1
+ {
+ \bool_if:NF \g__nmc_error_bool
+ { % calc. the next term
+ \fp_set:Nn \l_tmpa_fp { #1 }
+ \__nmc_calc_fn_val:VNnN \l__nmc_recur_subscr_var_tl
+ \l__nmc_recurrence_tl { \l_tmpa_fp } \l__nmc_recur_result_fp
+ }
+ \bool_if:NF \g__nmc_error_bool
+ { % store the result
+ \__nmc_num_format:nNnN { \l__nmc_recur_result_fp }
+ \l_tmpa_tl { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \seq_put_right:NV \l__nmc_recur_result_seq \l_tmpa_tl
+ \seq_put_right:Nn \l__nmc_recur_index_seq { #1 }
+ % shift vals "down variable"; tmpa above, tmpb below
+ \seq_set_eq:NN \l_tmpa_seq \l__nmc_recur_vars_seq
+ \seq_pop:NN \l_tmpa_seq \l_tmpb_tl % low var
+ \int_step_inline:nnnn {2} { 1 } { \l__nmc_recur_order_int }
+ {
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl % hi var
+ \prop_get:NVN \g__nmc_subst_var_prop \l_tmpa_tl \l__nmc_subst_tl
+ \prop_put:NVV \g__nmc_subst_var_prop \l_tmpb_tl \l__nmc_subst_tl
+ \prop_put:NVV \l__nmc_vv_change_prop \l_tmpb_tl \l__nmc_subst_tl
+ \tl_set_eq:NN \l_tmpb_tl \l_tmpa_tl
+ }
+ % use tmpb, not tmpa, in case order = 1
+ \prop_put:NVx \g__nmc_subst_var_prop \l_tmpb_tl
+ { \fp_use:N \l__nmc_recur_result_fp }
+ \prop_put:NVx \l__nmc_vv_change_prop \l_tmpb_tl
+ { \fp_use:N \l__nmc_recur_result_fp }
+ }
+ }
+\cs_new_protected:Npn \__nmc_recur_display:
+ {
+ \bool_if:NTF \l__nmc_num_only_bool
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ {
+ \seq_clear:N \l_tmpa_seq
+ \seq_clear:N \l_tmpb_seq
+ \seq_clear:N \l_tmpc_seq
+ \__nmc_recur_result:NN \l_tmpa_seq \l_tmpb_seq
+ \tl_set:Nx \l_nmc_result_tl
+ {
+ \exp_not:o \l__nmc_math_delimi_tl
+ \bool_if:NT \l__nmc_wraps_math_bool
+ {
+ \exp_not:o \l__nmc_formula_dup_tl
+ \bool_if:NTF \l__nmc_vv_multline_bool
+ { \exp_not:o \l__nmc_vv_display_tl }
+ {
+ \mathchoice{ \exp_not:o \l__nmc_vv_display_tl }
+ { \exp_not:o \l__nmc_vv_inline_tl }{}{}
+ }
+ \rightarrow \exp_not:N \quad
+ }
+ \seq_use:Nn \l_tmpa_seq { ,\ }
+ \l__nmc_punc_tl
+ \exp_not:o \l__nmc_math_delimii_tl
+ }
+ \__nmc_plus_reuse:nNN { \l__nmc_recur_reuse_int } \l_tmpc_seq \l_tmpb_seq
+ }
+ \l_nmc_result_tl
+ }
+\cs_new_protected:Npn \__nmc_recur_result:NN #1#2
+ {
+ \int_zero:N \l_tmpa_int
+ \seq_map_inline:Nn \l__nmc_recur_result_seq
+ {
+ \seq_pop:NN \l__nmc_recur_index_seq \l__nmc_toss_tl
+ \int_compare:nNnTF { \l_tmpa_int } < { \l__nmc_recur_first_int }
+ { \seq_put_right:Nn #1 { ##1 } }
+ {
+ \int_compare:nTF { \l_tmpa_int = \l__nmc_recur_first_int
+ < \l__nmc_recur_total_int - \l__nmc_recur_last_int }
+ { \seq_put_right:Nn #1 { \ldots } }
+ {
+ \int_compare:nNnT { 1 + \l_tmpa_int } >
+ { \l__nmc_recur_total_int - \l__nmc_recur_last_int }
+ {
+ \seq_put_right:Nn #1 { ##1 }
+ \seq_put_right:Nn #2 { ##1 }
+ \seq_put_right:Nx \l_tmpc_seq { \int_eval:n { \l_tmpa_int + \l__nmc_recur_subscr_val_int } }
+ }
+ }
+ }
+ \int_incr:N \l_tmpa_int
+ }
+ \bool_if:NT \l__nmc_recur_ellipsis_bool
+ { \seq_put_right:Nn #1{ \ldots } }
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\keys_define:nn { numerica }
+ {
+ iter-do .int_set:N = \l__nmc_iter_total_int,
+ iter-see-last .int_set:N = \l__nmc_iter_view_int,
+ iter-max-iterations .int_set:N = \l__nmc_iter_max_int,
+ iter-extra-rounding .int_set:N = \l__nmc_iter_extra_int,
+ iter-reuse .int_set:N = \l__nmc_iter_reuse_int,
+
+ iter-do .default:n = 5,
+ iter-see-last .default:n = 4,
+ iter-max-iterations .default:n = 100,
+ iter-extra-rounding .default:n = 0,
+ iter-reuse .default:n = 0,
+%
+ solve-max-steps .int_set:N = \l__nmc_solve_max_int,
+ solve-extra-rounding .int_set:N = \l__nmc_solve_extra_int,
+ solve-first-step .tl_set:N = \l__nmc_solve_step_tl,
+ solve-reuse .int_set:N = \l__nmc_solve_reuse_int,
+
+ solve-max-steps .default:n = 100,
+ solve-extra-rounding .default:n = 0,
+ solve-first-step .default:n = 1,
+ solve-reuse .default:n = 0,
+%
+ recur-do .int_set:N = \l__nmc_recur_total_int,
+ recur-see-first .int_set:N = \l__nmc_recur_first_int,
+ recur-see-last .int_set:N = \l__nmc_recur_last_int,
+ recur-reuse .int_set:N = \l__nmc_recur_reuse_int,
+
+ recur-do .default:n = 7,
+ recur-see-first .default:n = 3,
+ recur-see-last .default:n = 2,
+ recur-reuse .default:n = 0
+ }
+\keys_set_known:nn { numerica }
+ {
+ iter-do,iter-see-last,iter-max-iterations,
+ iter-extra-rounding,iter-reuse,
+ solve-variable,solve-first-step,solve-max-steps,
+ solve-extra-rounding,solve-reuse,
+ recur-do,recur-see-first,recur-see-last,recur-reuse
+ }
+\keys_define:nn { numerica/iter }
+ {
+ var .tl_set:N = \l__nmc_iter_var_tl,
+ + .int_set:N = \l__nmc_iter_extra_int,
+ max .int_set:N = \l__nmc_iter_max_int,
+ do .int_set:N = \l__nmc_iter_total_int,
+ see .int_set:N = \l__nmc_iter_view_int,
+ reuse .int_set:N = \l__nmc_iter_reuse_int
+ }
+\keys_define:nn { numerica/solve }
+ {
+ var .tl_set:N = \l__nmc_solve_var_tl,
+ + .int_set:N = \l__nmc_solve_extra_int,
+ max .int_set:N = \l__nmc_solve_max_int,
+ dvar .tl_set:N = \l__nmc_solve_step_tl,
+ reuse .int_set:N = \l__nmc_solve_reuse_int
+ }
+\keys_define:nn { numerica/recur }
+ {
+ do .int_set:N = \l__nmc_recur_total_int,
+ see1 .int_set:N = \l__nmc_recur_first_int,
+ see2 .int_set:N = \l__nmc_recur_last_int,
+ ... .code:n = \bool_set_true:N \l__nmc_recur_ellipsis_bool,
+ reuse .int_set:N = \l__nmc_recur_reuse_int
+ }
+% end of `numerica-plus.def'
\ No newline at end of file
Property changes on: trunk/Master/texmf-dist/tex/latex/numerica/numerica-plus.def
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Added: trunk/Master/texmf-dist/tex/latex/numerica/numerica-tables.def
===================================================================
--- trunk/Master/texmf-dist/tex/latex/numerica/numerica-tables.def (rev 0)
+++ trunk/Master/texmf-dist/tex/latex/numerica/numerica-tables.def 2021-02-15 22:22:05 UTC (rev 57759)
@@ -0,0 +1,1138 @@
+% This is file `numerica-tables.def',
+% part of the `numerica' package.
+%
+% This work may be distributed and/or modified under the conditions
+% of the LaTeX Project Public License, either version 1.3c of this
+% license or any later version; see
+% http://www.latex-project.org/lppl.txt
+%
+% Andrew Parsloe (ajparsloe at gmail.com)
+%
+\RequirePackage{booktabs}
+%
+\ProvidesExplFile
+ {numerica-tables.def}
+ {2021/02/15}
+ {1.0.0}
+ {Create tables of function values}
+%----------------------------------------------------------
+\cs_new_protected:Npn \__nmc_table_prepad:Nnn #1#2#3
+ {
+ \tl_put_left:Nx #1 { \prg_replicate:nn
+ { \int_max:nn { #2 - \tl_count:N #1 } { 0 } } { #3 } }
+ }
+\cs_new:Npn \__nmc_sci_num_table_plus:nn #1#2
+ { % #1 significand, #2 exponent
+ \fp_compare:nTF { #1 >= 0 }
+ { ( #2 )\,{ \__nmc_sci_num_table:n { #1 } } }
+ { ( #2 )\,{ #1 } }
+ }
+\cs_new:Npn \__nmc_sci_num_table:n #1
+ {
+ \int_compare:nNnF { \l__nmc_num_sgn_int } < { 0 }
+ {
+ \int_case:nn { \l__nmc_table_signs_int }
+ {
+ {-2 } { \exp_not:N \hphantom{-} }
+ {-1 } { \bool_if:NTF \l__nmc_table_signs_bool
+ { + }{ \exp_not:N \hphantom{-} } }
+ { 1 } { \bool_if:NT \l__nmc_table_signs_bool { + } }
+ { 2 } { + }
+ }
+ }
+ #1
+ }
+%--------------------------------------
+\tl_new:N \l__nmc_x_tl
+\seq_new:N \l_tmpd_seq
+\tl_const:Nn \c__nmc_r_tl { r }
+\tl_const:Nn \c__nmc_l_tl { l }
+\bool_new:N \l__nmc_table_clist_bool
+\clist_new:N \l__nmc_formula_clist
+\tl_new:N \l__nmc_table_valign_tl
+\seq_new:N \l_nmc_table_reuse_seq
+
+\tl_new:N \l__nmc_table_rvar_tl
+\tl_new:N \l__nmc_table_rvari_tl
+\tl_new:N \l__nmc_table_rstart_tl
+\tl_new:N \l__nmc_table_rstop_tl
+\tl_new:N \l__nmc_table_rstep_tl
+\fp_new:N \l__nmc_table_rstart_fp
+\fp_new:N \l__nmc_table_rstop_fp
+\fp_new:N \l__nmc_table_rstep_fp
+\fp_new:N \l__nmc_table_r_fp
+\fp_new:N \l__nmc_table_c_fp
+\int_new:N \l__nmc_table_rnum_int
+\clist_new:N \l__nmc_table_rspec_clist
+\bool_new:N \l__nmc_table_rspec_bool
+
+\tl_new:N \l__nmc_table_cvar_tl
+\tl_new:N \l__nmc_table_cstart_tl
+\tl_new:N \l__nmc_table_cstop_tl
+\tl_new:N \l__nmc_table_cstep_tl
+\fp_new:N \l__nmc_table_cstart_fp
+\fp_new:N \l__nmc_table_cstop_fp
+\fp_new:N \l__nmc_table_cstep_fp
+\int_new:N \l__nmc_table_cnum_int
+\clist_new:N \l__nmc_table_cspec_clist
+\bool_new:N \l__nmc_table_cspec_bool
+
+\seq_new:N \l__nmc_table_rvraw_seq
+\seq_new:N \l__nmc_table_rvcol_seq
+\seq_new:N \l__nmc_table_rvcoli_seq
+\seq_new:N \l__nmc_table_this_col_seq
+\seq_new:N \l__nmc_table_body_cols_seq
+\seq_new:N \l__nmc_table_body_rows_seq
+
+\seq_new:N \l__nmc_table_header_seq
+\tl_new:N \l__nmc_table_header_tl
+\tl_new:N \l__nmc_table_rfont_tl
+\tl_new:N \l__nmc_table_rhead_tl
+\tl_new:N \l__nmc_table_rheadi_tl
+\tl_new:N \l__nmc_table_rhnudge_tl
+\tl_new:N \l__nmc_table_rhnudgei_tl
+\int_new:N \l__nmc_table_chround_int
+\int_new:N \l__nmc_table_chstyle_int
+\tl_new:N \l__nmc_table_chnudge_tl
+
+\tl_new:N \l__nmc_table_ctitle_tl
+\bool_new:N \l__nmc_table_ctitle_bool
+\bool_new:N \l__nmc_table_ctitle_vv_bool
+\int_new:N \l__nmc_table_rules_int
+\tl_new:N \l__nmc_table_footer_tl
+\bool_new:N \l__nmc_table_foot_bool
+\bool_new:N \l__nmc_table_footops_bool
+\tl_new:N \l__nmc_table_csum_tl
+\tl_new:N \l__nmc_table_cave_tl
+\tl_new:N \l__nmc_table_cmax_tl
+\tl_new:N \l__nmc_table_cmin_tl
+\tl_new:N \l__nmc_table_cdel_tl
+\tl_new:N \l__nmc_table_rbloc_tl
+\tl_new:N \l__nmc_table_rblock_tl
+\int_new:N \l__nmc_table_rbloc_int
+\clist_new:N \l__nmc_table_rbloc_clist
+
+\int_new:N \l__nmc_table_signs_int
+\int_new:N \l__nmc_table_tsigns_int
+\int_new:N \l__nmc_table_c_int
+\int_new:N \l__nmc_table_b_int
+\int_new:N \l__nmc_table_a_int
+\bool_new:N \l__nmc_table_signs_bool
+\bool_new:N \l__nmc_table_diffs_bool
+\tl_new:N \l__nmc_table_diff_tl
+\tl_new:N \l__nmc_table_diff_null_tl
+
+\tl_new:N \l__nmc_table_Q_tl
+\tl_new:N \l__nmc_table_A_tl
+\fp_new:N \l__nmc_table_max_fp
+\fp_new:N \l__nmc_table_min_fp
+\bool_new:N \l__nmc_table_QA_max_bool
+\bool_new:N \l__nmc_table_QA_min_bool
+\bool_new:N \l__nmc_table_QA_bool
+\bool_new:N \l__nmc_table_num_only_bool
+\tl_new:N \l__nmc_table_num_only_tl
+\tl_new:N \l__nmc_table_query_tl
+%------------------------------------------------
+\nmc_define:nnN { \nmcTabulate } { table } \tabulate
+
+\cs_new_protected:Npn \__nmc_table_settings_digest:
+ {
+ \bool_set_false:N \l__nmc_allow_TF_out_bool
+ \int_compare:nTF { 0 <= \l__nmc_table_rpos_int <= 4 }
+ {
+ \int_compare:nNnT { \l__nmc_table_rpos_int } = { 4 }
+ { \int_incr:N \l__nmc_table_rpos_int }
+ }
+ { \int_set:Nn \l__nmc_table_rpos_int { 1 } }
+ \clist_if_empty:NF \l__nmc_table_rspec_clist
+ { \__nmc_table_settings_spec:N r }
+ \clist_if_empty:NF \l__nmc_table_cspec_clist
+ { \__nmc_table_settings_spec:N c }
+ \tl_if_empty:NT \l__nmc_table_rhead_tl
+ { \tl_set_eq:NN \l__nmc_table_rhead_tl \l__nmc_table_rvar_tl }
+ \tl_if_empty:NT \l__nmc_table_rheadi_tl
+ {
+ \tl_if_empty:NTF \l__nmc_table_rvari_tl
+ { \tl_set_eq:NN \l__nmc_table_rheadi_tl \l__nmc_table_rhead_tl }
+ { \tl_set_eq:NN \l__nmc_table_rheadi_tl \l__nmc_table_rvari_tl }
+ }
+ \tl_if_empty:NT \l__nmc_table_rhnudgei_tl
+ { \tl_set_eq:NN \l__nmc_table_rhnudgei_tl \l__nmc_table_rhnudge_tl }
+
+ \tl_if_empty:NF \l__nmc_table_ctitle_tl
+ { \l__nmc_table_settings_ctitle: }
+ \tl_if_empty:NTF \l__nmc_table_Q_tl
+ { \tl_set:Nn \l__nmc_table_query_tl { *~option~ } }
+ {
+ \tl_set:Nn \l__nmc_table_query_tl { table~value~satisfies~ }
+ \__nmc_table_settings_QA:
+ }
+ \clist_if_empty:NTF \l__nmc_table_rbloc_clist
+ { \tl_set:Nn \l__nmc_table_rbloc_tl { 10000 } }
+ { \clist_pop:NN \l__nmc_table_rbloc_clist \l__nmc_table_rbloc_tl }
+ \tl_set_eq:NN \l__nmc_table_rblock_tl \l__nmc_table_rbloc_tl
+ \int_set:Nn \l__nmc_table_rbloc_int { \l__nmc_table_rbloc_tl }
+
+ \__nmc_table_settings_rules:
+ \int_if_zero:nTF \l__nmc_table_diffs_int
+ { \bool_set_false:N \l__nmc_table_diffs_bool }
+ { \bool_set_true:N \l__nmc_table_diffs_bool }
+ \tl_if_empty:NF \l__nmc_table_chead_tl
+ { \int_set:Nn \l__nmc_table_chstyle_int { 4 } }
+ \tl_if_empty:NF \l__nmc_table_footer_tl
+ {
+ \tl_if_eq:VnTF \l__nmc_table_footer_tl { * }
+ { \bool_set_true:N \l__nmc_table_foot_bool }
+ { \__nmc_table_footer:N \l__nmc_table_footer_tl }
+ }
+ }
+\cs_new_protected:Npn \l__nmc_table_settings_ctitle:
+ {
+ \bool_set_true:N \l__nmc_table_ctitle_bool
+ \tl_if_head_eq_charcode:VNT \l__nmc_table_ctitle_tl *
+ {
+ \tl_if_eq:VnT \l__nmc_table_ctitle_tl { ** }
+ { \bool_set_true:N \l__nmc_table_ctitle_vv_bool }
+ \tl_if_head_eq_charcode:VNT \l__nmc_table_ctitle_tl *
+ {
+ \tl_set:Nn \l__nmc_table_ctitle_tl
+ { \exp_not:o \l__nmc_formula_dup_tl }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_settings_rules:
+ {
+ \tl_replace_once:Nnn \l__nmc_table_rules_tl { T } { *2 }
+ \tl_replace_once:Nnn \l__nmc_table_rules_tl { t } { *3 }
+ \tl_replace_once:Nnn \l__nmc_table_rules_tl { h } { *5 }
+ \tl_replace_once:Nnn \l__nmc_table_rules_tl { f } { *7 }
+ \tl_replace_once:Nnn \l__nmc_table_rules_tl { B } { *11 }
+ \int_set:Nn \l__nmc_table_rules_int { 1 \l__nmc_table_rules_tl }
+ }
+\cs_new_protected:Npn \__nmc_table_settings_QA:
+ {
+ \bool_set_true:N \l__nmc_table_QA_bool
+ \regex_replace_all:nnN { (\@) } { \c{l_tmpb_fp} } \l__nmc_table_Q_tl
+ \tl_set_rescan:Nno \l__nmc_table_Q_tl { \ExplSyntaxOn } \l__nmc_table_Q_tl
+ \regex_replace_all:nnN { (\cC.) } { \c{exp_not:N}\1 } \l__nmc_table_A_tl
+ \regex_replace_all:nnN { (\@) } { \c{l_tmpb_tl} } \l__nmc_table_A_tl
+ \tl_if_in:NnT \l__nmc_table_Q_tl { MAX }
+ {
+ \bool_set_true:N \l__nmc_table_QA_max_bool
+ \regex_replace_all:nnN { MAX } { \c{l__nmc_table_max_fp} }
+ \l__nmc_table_Q_tl
+ \fp_set:Nn \l__nmc_table_max_fp { -inf }
+ }
+ \tl_if_in:NnT \l__nmc_table_Q_tl { MIN }
+ {
+ \bool_set_true:N \l__nmc_table_QA_min_bool
+ \regex_replace_all:nnN { MIN } { \c{l__nmc_table_min_fp} }
+ \l__nmc_table_Q_tl
+ \fp_set:Nn \l__nmc_table_min_fp { inf }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_footer:N #1
+ {
+ \tl_set_eq:NN \l_tmpa_tl #1
+ \regex_replace_all:nnN { SUM } { \c{l__nmc_table_csum_tl} } #1
+ \regex_replace_all:nnN { AVE } { \c{l__nmc_table_cave_tl} } #1
+ \regex_replace_all:nnN { MAX } { \c{l__nmc_table_cmax_tl} } #1
+ \regex_replace_all:nnN { MIN } { \c{l__nmc_table_cmin_tl} } #1
+ \regex_replace_all:nnN { DEL } { \c{l__nmc_table_cdel_tl} } #1
+ \tl_if_eq:NNF \l_tmpa_tl #1
+ { \bool_set_true:N \l__nmc_table_footops_bool }
+ }
+\cs_new_protected:Npn \__nmc_table_settings_spec:N #1
+ {
+ \bool_set_true:c { l__nmc_table_ #1 spec_bool }
+ \clist_pop:cc { l__nmc_table_#1spec_clist } { l__nmc_table_ #1 var_tl }
+ \clist_pop:cc { l__nmc_table_ #1 spec_clist } { l__nmc_table_ #1 step_tl }
+ \clist_pop:cN { l__nmc_table_ #1 spec_clist } \l_tmpa_tl
+ \tl_if_head_eq_charcode:VNTF \l_tmpa_tl (
+ {
+ \int_set:cn { l__nmc_table_ #1 num_int }
+ { \exp_last_unbraced:NV \__nmc_paren_arg:w \l_tmpa_tl }
+ }
+ { \tl_set_eq:cN { l__nmc_table_ #1 stop_tl } \l_tmpa_tl }
+ }
+\cs_new:Npn \__nmc_paren_arg:w (#1) { #1 }
+
+\cs_new_protected:Npn \__nmc_table_vv_digest:N #1
+ {
+ \bool_if:NTF \l__nmc_multitok_bool
+ {
+ \clist_push:NV \l__nmc_formula_tl \l__nmc_table_cvar_tl
+ \clist_push:NV \l__nmc_formula_tl \l__nmc_table_rvar_tl
+ \__nmc_vv_digest:N #1
+ \clist_pop:NN \l__nmc_formula_tl \l__nmc_table_rvar_tl
+ \tl_if_empty:NF \l__nmc_table_cvar_tl
+ { \clist_pop:NN \l__nmc_formula_tl \l__nmc_table_cvar_tl }
+ }
+ { \__nmc_vv_digest:N #1 }
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ \__nmc_table_get_ini_vals:
+ \bool_if:nT
+ {
+ \int_if_zero_p:n { \l__nmc_table_rpos_int } &&
+ \int_if_zero_p:n { \l__nmc_table_cnum_int }
+ }
+ { \__nmc_error_what:n { Null~table~specified~in } }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \bool_if:NT \l__nmc_table_ctitle_vv_bool
+ {
+ \exp_args:NNnx\tl_replace_once:Nnn \l__nmc_vv_inline_tl { vv }
+ { \seq_use:Nn \l__nmc_vv_visible_seq { , } }
+ \tl_put_right:Nn \l__nmc_table_ctitle_tl
+ { \exp_not:o \l__nmc_vv_inline_tl }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_get_ini_vals:
+ {
+ \__nmc_error_where:n { settings }
+ \__nmc_table_get_ini_vals_aux:N r
+ \tl_if_empty:NTF \l__nmc_table_cvar_tl
+ {
+ \int_set:Nn \l__nmc_table_cnum_int
+ {
+ \int_compare:nNnTF { \l__nmc_table_cnum_int } = { -1 }
+ { 1 } { \int_min:nn { 1 }
+ { \int_abs:n { \l__nmc_table_cnum_int } } }
+ }
+ }
+ { \__nmc_table_get_ini_vals_aux:N c }
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ }
+\cs_new_protected:Npn \__nmc_table_get_ini_vals_aux:N #1
+ {
+ \prop_get:NvNTF \g__nmc_subst_var_prop { l__nmc_table_ #1 var_tl } \l_tmpa_tl
+ {
+ \fp_set:cn { l__nmc_table_ #1 start_fp } { \l_tmpa_tl }
+ \bool_if:cTF { l__nmc_table_ #1 spec_bool }
+ {
+ \__nmc_fpify_set:cc { l__nmc_table_ #1 step_fp }
+ { l__nmc_table_ #1 step_tl }
+ \int_compare:vNnT { l__nmc_table_ #1 num_int } = { -1 }
+ {
+ \__nmc_fpify_set:cc { l__nmc_table_ #1 stop_fp }
+ { l__nmc_table_ #1 stop_tl }
+ \l__nmc_table_calc_num:cnnn { l__nmc_table_ #1 num_int }
+ { l__nmc_table_ #1 stop_fp } { l__nmc_table_ #1 start_fp }
+ { l__nmc_table_ #1 step_fp }
+ }
+ }
+ { \__nmc_table_get_individual_ini_vals:N #1 }
+ }
+ { \__nmc_error_what:n { No~ \__nmc_table_rc:N #1 ~variable~specified~in } }
+ }
+\cs_new:Npn \__nmc_table_rc:N #1
+ { \tl_if_eq:NNTF #1 \c__nmc_r_tl { row } { column } }
+
+\cs_new_protected:Npn \l__nmc_table_calc_num:Nnnn #1#2#3#4
+ {
+ \int_set:Nn #1 { \fp_to_int:n { ( \exp_not:v { #2 } - \exp_not:v { #3 } )
+ / \exp_not:v { #4 } } + 1 }
+ }
+\cs_generate_variant:Nn \l__nmc_table_calc_num:Nnnn { c }
+
+\cs_new_protected:Npn \__nmc_table_get_individual_ini_vals:N #1
+ {
+ \int_case:vnTF { l__nmc_table_ #1 num_int }
+ {
+ { -1 }
+ {
+ \__nmc_fpify_set:cc { l__nmc_table_ #1 step_fp }
+ { l__nmc_table_ #1 step_tl }
+ \bool_if:NF \g__nmc_error_bool
+ { \__nmc_fpify_set:cc { l__nmc_table_ #1 stop_fp }
+ { l__nmc_table_ #1 stop_tl } }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \l__nmc_table_calc_num:cnnn { l__nmc_table_ #1 num_int }
+ { l__nmc_table_ #1 stop_fp } { l__nmc_table_ #1 start_fp }
+ { l__nmc_table_ #1 step_fp }
+ }
+ }
+ { 0 }
+ {
+ \fp_set:cn { l__nmc_table_ #1 stop_fp }
+ { \exp_not:v { l__nmc_table_ #1 start_fp } - 1 }
+ \fp_set:cn { l__nmc_table_ #1 step_fp } { 1 }
+ }
+ { 1 }
+ {
+ \fp_set_eq:cc { l__nmc_table_ #1 stop_fp }
+ { l__nmc_table_ #1 start_fp }
+ \fp_set:cn { l__nmc_table_ #1 step_fp } { 1 }
+ }
+ }
+ {
+ \bool_if:NT \g__nmc_error_bool
+ { \__nmc_error_what:n { Missing~
+ \__nmc_table_rc:N #1-related~variable~in } }
+ }
+ {
+ \int_compare:vNnTF { l__nmc_table_ #1 num_int } > { 1 }
+ { \__nmc_fpify_set:cc { l__nmc_table_ #1 step_fp }
+ { l__nmc_table_ #1 step_tl } }
+ { \__nmc_error_what:n { Check~number~of~
+ \__nmc_table_rc:N #1s~specified~in } }
+ }
+ }
+%----------------------------------------------------------
+\cs_new_protected:Npn \__nmc_table_process:
+ {
+ \__nmc_error_where:n { formula }
+ \__nmc_table_calc_rvar_col:
+ \seq_clear:N \l__nmc_table_header_seq
+ \tl_if_head_eq_charcode:VNT \l__nmc_formula_tl ,
+ { \__nmc_table_multifn: }
+ \__nmc_table_calc_cvar_cols:
+ \tl_clear:N \l_nmc_result_tl
+ \__nmc_table_form_body:N \l_nmc_result_tl
+ \tl_set_eq:NN \l__nmc_fp_expr_tl \l__nmc_fp_exprn_tl
+ \bool_if:NT \l__nmc_num_only_bool
+ {
+ \bool_if:NTF \l__nmc_table_num_only_bool
+ {
+ \tl_set_eq:NN \l_nmc_result_tl \l__nmc_table_num_only_tl
+ \tl_gset_eq:NN \g__nmc_reuse_tl \l__nmc_table_num_only_tl
+ }
+ {
+ \__nmc_error_where:n { settings }
+ \__nmc_error_what:n
+ { No~ \l__nmc_table_query_tl query~\__nmc_verb:n {Q?}~in }
+ \tl_clear:N \l_nmc_result_tl
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_multifn:
+ {
+ \bool_set_true:N \l__nmc_table_clist_bool
+ \clist_set:NV \l__nmc_formula_clist \l__nmc_formula_tl
+ \seq_set_from_clist:NN \l__nmc_table_header_seq \l__nmc_formula_clist
+ \int_set:Nn \l__nmc_table_cnum_int
+ { \clist_count:N \l__nmc_formula_clist }
+ \fp_set:Nn \l__nmc_table_cstep_fp { 0 }
+ \fp_set:Nn \l__nmc_table_cstart_fp { 0 }
+ \tl_set:Nn \l__nmc_table_cvar_tl { \l__nmc_x_tl }
+ \prop_put:Nnn \g__nmc_subst_var_prop { \l__nmc_x_tl } { 1 }
+ \int_set:Nn \l__nmc_table_chstyle_int { 4 }
+ }
+\cs_new_protected:Npn \__nmc_table_calc_rvar_col:
+ {
+ \fp_set_eq:NN \l__nmc_table_r_fp \l__nmc_table_rstart_fp
+ \int_step_inline:nnnn { 1 } { 1 } { \l__nmc_table_rnum_int }
+ {
+ \seq_put_right:NV \l__nmc_table_rvraw_seq \l__nmc_table_r_fp
+ \fp_add:Nn \l__nmc_table_r_fp { \l__nmc_table_rstep_fp }
+ }
+ }
+\cs_new:Npn \__nmc_table_calc_cvar_cols:
+ {
+ \fp_set_eq:NN \l__nmc_table_c_fp \l__nmc_table_cstart_fp
+ \int_step_inline:nn { \l__nmc_table_cnum_int }
+ {
+ \bool_if:NTF \l__nmc_table_clist_bool
+ { \clist_pop:NN \l__nmc_formula_clist \l__nmc_formula_tl }
+ { \seq_put_right:NV \l__nmc_table_header_seq { \l__nmc_table_c_fp } }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_clear:N \l__nmc_table_this_col_seq
+ \fp_set_eq:NN \l__nmc_table_r_fp \l__nmc_table_rstart_fp
+ \int_step_function:nN { \l__nmc_table_rnum_int }
+ \__nmc_table_rvar_step_fn:n
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_put_right:NV \l__nmc_table_body_cols_seq
+ \l__nmc_table_this_col_seq
+ \fp_add:Nn \l__nmc_table_c_fp \l__nmc_table_cstep_fp
+ \__nmc_table_prop_mode:n {\l__nmc_mode_int}
+ \l__nmc_table_rvar_tl \l__nmc_table_rstart_fp
+ \__nmc_calc_fn_val:VNnN \l__nmc_table_cvar_tl \l__nmc_formula_tl
+ { \l__nmc_table_c_fp } \l__nmc_table_fn_val_fp
+ }
+ }
+ }
+\cs_new:Npn \__nmc_table_prop_mode:n #1
+ {
+ \int_if_zero:nTF { #1 }
+ { \prop_put:NVV \g__nmc_subst_var_prop }
+ { \prop_put:NVV \l__nmc_vv_change_prop }
+ }
+\cs_new_protected:Npn \__nmc_table_rvar_step_fn:n #1
+ {
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_table_rvar_tl
+ \l__nmc_formula_tl { \l__nmc_table_r_fp } \l_tmpa_fp
+ \seq_put_right:NV \l__nmc_table_this_col_seq \l_tmpa_fp
+ \bool_if:NT \l__nmc_table_QA_bool
+ { \__nmc_table_maxmin_do:n { \l_tmpa_fp } }
+ \fp_add:Nn \l__nmc_table_r_fp { \l__nmc_table_rstep_fp }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_maxmin_do:n #1
+ {
+ \bool_if:NT \l__nmc_table_QA_max_bool
+ { \fp_set:Nn \l__nmc_table_max_fp
+ { \fp_max:nn { \l__nmc_table_max_fp } { #1} } }
+ \bool_if:NT \l__nmc_table_QA_min_bool
+ { \fp_set:Nn \l__nmc_table_min_fp
+ { \fp_min:nn { \l__nmc_table_min_fp } { #1 } } }
+ }
+%----------------------------------------------------------
+\cs_new_protected:Npn \__nmc_table_display:
+ {
+ \bool_if:NF \l__nmc_num_only_bool
+ {
+ \__nmc_table_headercnum:
+ \tl_set:Nx \l__nmc_table_header_tl { \__nmc_table_header: }
+ \bool_if:NT \l__nmc_table_foot_bool
+ {
+ \seq_set_split:NnV \l_tmpa_seq { & }\l__nmc_table_header_tl
+ \seq_reverse:N \l_tmpa_seq
+ \tl_set:Nx \l__nmc_table_footer_tl { \seq_use:Nn \l_tmpa_seq { & } }
+ }
+ \tl_put_left:Nx \l_nmc_result_tl
+ {
+ \__nmc_table_upper:
+ \exp_not:o \l__nmc_table_header_tl
+ \tabularnewline
+ \__nmc_if_mod_zero:nnT { \l__nmc_table_rules_int } { 5 }
+ { \exp_not:N \midrule }
+ }
+ \tl_put_right:Nx \l_nmc_result_tl { \__nmc_table_form_lower: }
+ \__nmc_table_reuse:
+ }
+ \l_nmc_result_tl
+ }
+\cs_new:Npn \__nmc_table_upper:
+ {
+ \exp_not:n { \begin{tabular} }
+ \exp_not:o \l__nmc_table_valign_tl
+ {
+ \int_if_odd:nT { \l__nmc_table_rpos_int }
+ { \l__nmc_table_rcalign_tl }
+ \prg_replicate:nn
+ { \l__nmc_table_cnum_int } \l__nmc_table_ccalign_tl
+ \int_compare:nNnT { \l__nmc_table_rpos_int } > { 1 }
+ { \l__nmc_table_rcalign_tl }
+ }
+ \__nmc_if_mod_zero:nnT { \l__nmc_table_rules_int } { 2 }
+ { \exp_not:N \toprule }
+ \bool_if:NT \l__nmc_table_ctitle_bool
+ { \__nmc_table_form_ctitle: }
+ }
+\cs_new:Npn \__nmc_table_form_ctitle:
+ {
+ \int_compare:nNnTF { \l__nmc_table_rpos_int } = { 0 }
+ {
+ \int_compare:nNnF { \l__nmc_table_cnum_int } = { 1 }
+ {
+ \__nmc_table_form_ctitle:nnnn {} {} { 1 }
+ { \l__nmc_table_cnum_int }
+ }
+ }
+ {
+ \int_compare:nNnT { \l__nmc_table_cnum_int } > { 1 }
+ {
+ \int_case:nnF { \l__nmc_table_rpos_int }
+ {
+ { 1 } { \__nmc_table_form_ctitle:nnnn { & } {} { 2 }
+ { \l__nmc_table_cnum_int + 1 } }
+ { 2 } { \__nmc_table_form_ctitle:nnnn {} { & } { 1 }
+ { \l__nmc_table_cnum_int } }
+ }
+ { % 3,4/5
+ \__nmc_table_form_ctitle:nnnn { & } { & } { 2 }
+ { \l__nmc_table_cnum_int + 1 }
+ }
+ }
+ }
+ }
+\cs_new:Npn \__nmc_table_form_ctitle:nnnn #1#2#3#4
+ {
+ #1 \exp_not:N \multicolumn
+ { \int_eval:n { \l__nmc_table_cnum_int } } { c }
+ { $ \l__nmc_table_ctitle_tl $ }
+ #2 \tabularnewline
+ \__nmc_if_mod_zero:nnT { \l__nmc_table_rules_int } { 3 }
+ {
+ \exp_not:N \cmidrule(lr)
+ { #3-\int_eval:n { #4 } }
+ }
+ }
+%%%%%%%%%%%%%%%%%%%%
+\cs_new:Npn \__nmc_table_header:
+ {
+ \int_if_zero:nTF { \l__nmc_table_cnum_int }
+ {
+ \__nmc_table_header_rvar:nn { \exp_not:o \l__nmc_table_rhead_tl }
+ \l__nmc_table_rhnudge_tl
+ }
+ {
+ \int_case:nnF { \l__nmc_table_rpos_int }
+ {
+ { 0 } { \exp_not:o \l__nmc_table_chead_tl }
+ { 1 }
+ {
+ \__nmc_table_header_rvar:nn
+ { \exp_not:o \l__nmc_table_rhead_tl }
+ \l__nmc_table_rhnudge_tl
+ & \exp_not:o \l__nmc_table_chead_tl
+ }
+ { 2 }
+ {
+ \exp_not:o \l__nmc_table_chead_tl &
+ \__nmc_table_header_rvar:nn
+ { \exp_not:o \l__nmc_table_rhead_tl }
+ \l__nmc_table_rhnudge_tl
+ }
+ }
+ { % 3,4/5
+ \__nmc_table_header_rvar:nn
+ { \exp_not:o \l__nmc_table_rhead_tl }
+ \l__nmc_table_rhnudge_tl
+ & \exp_not:o \l__nmc_table_chead_tl &
+ \__nmc_table_header_rvar:nn
+ { \exp_not:o \l__nmc_table_rheadi_tl }
+ \l__nmc_table_rhnudgei_tl
+ }
+ }
+ }
+\cs_new:Npn \__nmc_table_headercnum:
+ {
+ \int_compare:nNnTF { \l__nmc_table_cnum_int } = { 1 }
+ { \__nmc_table_headeri: }
+ { \__nmc_table_headern: }
+ }
+\cs_new:Npn \__nmc_table_header_rvar:nn #1#2
+ { \__nmc_table_header_nudge:Nnn r { #1 } { #2 } }
+
+\cs_new:Npn \__nmc_table_headeri:
+ {
+ \tl_if_empty:NT \l__nmc_table_chead_tl
+ {
+ \tl_set:Nx \l__nmc_table_chead_tl
+ {
+ \__nmc_table_header_nudge:Nnn c
+ {
+ \bool_if:NTF \l__nmc_table_ctitle_bool
+ { \exp_not:o \l__nmc_table_ctitle_tl }
+ { \exp_not:o \l__nmc_formula_dup_tl }
+ }
+ \l__nmc_table_chnudge_tl
+ }
+ }
+ }
+\cs_new:Npn \__nmc_table_headern:
+ {
+ \tl_if_empty:NT \l__nmc_table_chead_tl
+ {
+ \seq_clear:N \l_tmpa_seq
+ \bool_set_true:N \l__nmc_table_coli_only_bool
+ \bool_if:NTF \l__nmc_table_clist_bool
+ {
+ \seq_map_inline:Nn \l__nmc_table_header_seq
+ { \__nmc_table_header_auxi:Nn \l_tmpa_seq { ##1 } }
+ }
+ {
+ \seq_map_inline:Nn \l__nmc_table_header_seq
+ {
+ \__nmc_num_format:nNnN { ##1 } \l_tmpb_tl
+ { \l__nmc_table_chround_int } \l__nmc_sci_num_out_bool
+ \__nmc_table_chstyler:NN \l_tmpa_seq \l_tmpb_tl
+ }
+ }
+ \tl_set:Nx \l__nmc_table_chead_tl { \seq_use:Nn \l_tmpa_seq { & } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_chstyler:NN #1 #2
+ {
+ \int_case:nnF { \l__nmc_table_chstyle_int }
+ {
+ { 0 } { \__nmc_table_header_auxi:NV #1 #2 }
+ { 1 }
+ {
+ \bool_if:NTF \l__nmc_table_coli_only_bool
+ {
+ \__nmc_table_header_auxi:Nx #1 { \l__nmc_table_cvar_tl = #2 }
+ \bool_set_false:N \l__nmc_table_coli_only_bool
+ }
+ { \seq_put_right:NV #1 #2 }
+ }
+ { 2 } { \__nmc_table_header_auxi:Nx #1 { \l__nmc_table_cvar_tl = #2 } }
+ { 3 } { \__nmc_table_header_auxiii:NV #1 #2 }
+ { 4 } { \seq_set_split:NnV #1 { & } \l__nmc_table_chead_tl }
+ }
+ { \__nmc_table_header_auxi:NV #1 #2 }
+ }
+\cs_new_protected:Npn \__nmc_table_header_auxi:Nn #1#2
+ {
+ \seq_put_right:Nx #1 { \__nmc_table_header_nudge:Nnn c
+ { \exp_not:n { #2 } } \l__nmc_table_chnudge_tl }
+ }
+\cs_generate_variant:Nn \__nmc_table_header_auxi:Nn { NV,Nx }
+
+\cs_new_protected:Npn \__nmc_table_header_auxiii:Nn #1#2
+ {
+ \tl_set_eq:NN \l_tmpa_tl \l__nmc_formula_dup_tl
+ \regex_replace_all:nnNT { \u{l__nmc_table_cvar_tl} } { #2 } \l_tmpa_tl
+ { \__nmc_table_header_auxi:NV #1 \l_tmpa_tl }
+ }
+\cs_generate_variant:Nn \__nmc_table_header_auxiii:Nn { NV }
+
+\cs_new:Npn \__nmc_table_header_nudge:Nnn #1#2#3
+ { % #1=r(ow)/c(ol), #2=var, #3=nudge
+ \tl_if_eq:cNTF { l__nmc_table_ #1 calign_tl } \c__nmc_r_tl
+ { $ #2 #3 $ }
+ {
+ \tl_if_eq:cNTF { l__nmc_table_ #1 calign_tl } \c__nmc_l_tl
+ { $ #3 #2 $ } { $ #2 $ }
+ }
+ }
+%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_table_form_body:N #1
+ { % #1 \l_nmc_result_tl
+ \int_if_zero:nF { \l__nmc_table_rpos_int }
+ {
+ \__nmc_table_form_rvcol:NN
+ \l__nmc_table_rvraw_seq \l__nmc_table_rvcol_seq
+ }
+ \int_zero:N \l__nmc_table_rbloc_int
+ \int_if_zero:nTF { \l__nmc_table_cnum_int }
+ { \__nmc_table_form_body_rvcol_only:N #1 }
+ { \__nmc_table_form_body_cvcols:N #1 }
+ }
+\cs_new_protected:Npn \__nmc_table_form_rvcol:NN #1#2
+ {
+ \seq_clear:N \l_tmpb_seq
+ \seq_map_variable:NNn #1 \l_tmpa_tl
+ {
+ \__nmc_num_format:nNnN \l_tmpa_tl \l_tmpb_tl
+ { \l__nmc_table_rround_int } \c_false_bool
+ \seq_put_right:Nx \l_tmpb_seq
+ { \exp_not:o \l__nmc_table_rfont_tl { \l_tmpb_tl } }
+ }
+ \seq_set_eq:NN #2 \l_tmpb_seq
+ \int_compare:nNnTF { \l__nmc_table_rpos_int } > { 3 }
+ { \__nmc_table_form_rvcoli:NN #1 #2 } % reverse/pivot
+ { \seq_set_eq:NN \l__nmc_table_rvcoli_seq #2 }
+ }
+\cs_new_protected:Npn \__nmc_table_form_rvcoli:NN #1 #2
+ {
+ \seq_clear:N \l__nmc_table_rvcoli_seq
+ \seq_map_variable:NNn #1 \l_tmpa_tl
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_table_rvar_tl
+ \l__nmc_table_rvari_tl \l_tmpa_tl \l_tmpa_fp
+ \__nmc_num_format:nNnN { \l_tmpa_fp } \l_tmpb_tl
+ { \l__nmc_table_rround_int } \c_false_bool
+ \seq_put_right:Nx \l__nmc_table_rvcoli_seq
+ { \exp_not:o \l__nmc_table_rfont_tl { \l_tmpb_tl } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_form_body_rvcol_only:N #1
+ { % #1 = \l_nmc_result_tl
+ \seq_clear:N \l_tmpb_seq
+ \seq_map_variable:NNn \l__nmc_table_rvcol_seq \l_tmpa_tl
+ {
+ \tl_put_right:Nx #1
+ { \exp_not:o \l_tmpa_tl \tabularnewline }
+ \__nmc_table_rbloc_spc:NN \l__nmc_table_rbloc_int \l_tmpa_bool
+ \bool_if:NT \l_tmpa_bool
+ {
+ \tl_put_right:Nx #1
+ { \exp_not:o \addlinespace[\l__nmc_table_blocsep_tl] }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_form_body_cvcols:N #1
+ { % #1 \l_nmc_result_tl
+ \seq_clear:N \l_tmpd_seq
+ \seq_map_variable:NNn \l__nmc_table_body_cols_seq \l_tmpa_seq
+ {
+ \seq_clear:N \l_tmpb_seq
+ \__nmc_table_form_cols:NN \l_tmpa_seq \l_tmpb_seq
+ \seq_put_right:NV \l_tmpd_seq \l_tmpb_seq
+ }
+ \bool_if:NT \l__nmc_table_footops_bool
+ { \__nmc_table_footops:N \l_tmpd_seq }
+ \__nmc_table_transpose:NN \l_tmpd_seq \l__nmc_table_body_rows_seq
+
+ \seq_map_variable:NNn \l__nmc_table_body_rows_seq \l_tmpa_seq
+ {
+ \__nmc_table_rbloc_spc:NN \l__nmc_table_rbloc_int \l_tmpa_bool
+ \tl_put_right:Nx #1
+ {
+ \seq_use:Nn \l_tmpa_seq { & }
+ \tabularnewline
+ \bool_if:NT \l_tmpa_bool
+ { \exp_not:o \addlinespace[\l__nmc_table_blocsep_tl] }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_footops:N #1
+ {
+ \group_begin:
+ \seq_clear:N \l_tmpb_seq
+ \seq_set_split:NnV \l_tmpa_seq { & } \l__nmc_table_footer_tl
+ \int_if_odd:nT { \l__nmc_table_rpos_int }
+ {
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl
+ \seq_put_left:NV \l_tmpb_seq \l_tmpa_tl
+ }
+ \seq_mapthread_function:NNN #1 \l_tmpa_seq \__nmc_table_footops:nn
+ \int_compare:nNnT { \l__nmc_table_rpos_int } > { 1 }
+ {
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl
+ \seq_put_right:NV \l_tmpb_seq \l_tmpa_tl
+ }
+ \tl_set:Nx \l__nmc_table_footer_tl { \seq_use:Nn \l_tmpb_seq { & } }
+ \exp_args:NNNV
+ \group_end:
+ \tl_set:Nn \l__nmc_table_footer_tl \l__nmc_table_footer_tl
+ }
+\cs_new_protected:Npn \__nmc_table_footops:nn #1#2
+ {
+ \tl_set:Nn \l_tmpc_seq { #1 }
+ \tl_set:Nx \l_tmpa_tl { \seq_use:Nn \l_tmpc_seq { + } }
+ \fp_set:Nn \l_tmpa_fp { \l_tmpa_tl }
+ \fp_set:Nn \l_tmpb_fp { \l_tmpa_fp / \l__nmc_table_rnum_int }
+ \__nmc_num_format:nNnN \l_tmpa_fp \l__nmc_table_csum_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_num_format:nNnN \l_tmpb_fp \l__nmc_table_cave_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \tl_set:Nx \l_tmpa_tl { max(\seq_use:Nn \l_tmpc_seq { , }) }
+ \fp_set:Nn \l_tmpa_fp { \l_tmpa_tl }
+ \__nmc_num_format:nNnN \l_tmpa_fp \l__nmc_table_cmax_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \tl_set:Nx \l_tmpa_tl { min(\seq_use:Nn \l_tmpc_seq { , }) }
+ \fp_set:Nn \l_tmpb_fp { \l_tmpa_tl }
+ \__nmc_num_format:nNnN \l_tmpb_fp \l__nmc_table_cmin_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_num_format:nNnN { \l_tmpa_fp - \l_tmpb_fp } \l__nmc_table_cdel_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \seq_put_right:Nx \l_tmpb_seq { $ #2 $ }
+ }
+\cs_new_protected:Npn \__nmc_table_transpose:NN #1#2
+ { % #1 seq of seq in #2 seq of seq out
+ \int_if_zero:nF { \l__nmc_table_rpos_int }
+ {
+ \int_compare:nNnF { \l__nmc_table_rpos_int } = { 2 }
+ { \seq_put_left:NV #1 \l__nmc_table_rvcol_seq }
+ \int_compare:nNnT { \l__nmc_table_rpos_int } > { 1 }
+ { \seq_put_right:NV #1 \l__nmc_table_rvcoli_seq }
+ }
+ \int_step_inline:nn { \l__nmc_table_rnum_int }
+ {
+ \seq_clear:N \l_tmpb_seq
+ \seq_clear:N \l_tmpc_seq
+ \seq_map_variable:NNn #1 \l_tmpa_seq
+ {
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl
+ \seq_put_right:Nx \l_tmpb_seq { $\exp_not:o\l_tmpa_tl $ }
+ \seq_put_right:NV \l_tmpc_seq \l_tmpa_seq
+ }
+ \seq_put_right:NV \l__nmc_table_body_rows_seq \l_tmpb_seq
+ \seq_set_eq:NN #1 \l_tmpc_seq
+ }
+ }
+\cs_new:Npn \__nmc_table_rbloc_spc:NN #1#2
+ { %
+ \int_incr:N #1
+ \bool_set_false:N #2
+ \int_compare:nNnT { #1 } = { \l__nmc_table_rblock_tl }
+ {
+ \int_compare:nNnF { #1 } = { \l__nmc_table_rnum_int }
+ { \bool_set_true:N #2 }
+ \clist_if_empty:NF \l__nmc_table_rbloc_clist
+ { \clist_pop:NN \l__nmc_table_rbloc_clist \l__nmc_table_rbloc_tl }
+ \tl_set:Nx \l__nmc_table_rblock_tl
+ { \int_eval:n { \l__nmc_table_rblock_tl + \l__nmc_table_rbloc_tl } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_form_cols:NN #1#2
+ { % #1 raw cseq, #2 <--formatted cseq
+ \int_case:nnF { \l__nmc_table_rnum_int }
+ {
+ { 0 } { \prg_do_nothing: }
+ { 1 } { \__nmc_table_form_cols_ri:NN #1#2 }
+ { 2 } { \__nmc_table_form_cols_rii:NN #1#2 }
+ }
+ { \__nmc_table_form_cols_riii:NN #1#2 }
+ }
+\cs_new_protected:Npn \__nmc_table_form_cols_ri:NN #1#2
+ {
+ \seq_pop:NN #1 \l_tmpb_fp
+ \__nmc_num_format:nNnN \l_tmpb_fp \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_table_bool
+ \bool_if:NT \l__nmc_table_QA_bool
+ { \__nmc_table_QA:N \l_tmpb_tl }
+ \seq_put_right:Nx #2 { \exp_not:o \l_tmpb_tl }
+ }
+\cs_new_protected:Npn \__nmc_table_form_cols_rii:NN #1#2
+ {
+ \__nmc_table_form_cols_rii_aux:NN #1#2
+ \__nmc_table_signs:
+ \__nmc_num_format:nNnN \l_tmpb_fp \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_table_num_format:N \l_tmpb_tl
+ \seq_put_right:Nx #2
+ { \exp_not:o \l_tmpb_tl \exp_not:o \l__nmc_table_diff_tl }
+ }
+\cs_new_protected:Npn \__nmc_table_form_cols_rii_aux:NN #1#2
+ {
+ \seq_pop:NN #1 \l_tmpb_fp
+ \int_if_zero:nF \l__nmc_table_diffs_int
+ { \__nmc_table_form_diffs_null:n { \l__nmc_table_diffs_int } }
+ \seq_pop:NN #1 \l_tmpa_fp
+ \int_set:Nn \l__nmc_table_c_int { 1 }
+ \int_set:Nn \l__nmc_table_b_int { \fp_sign:n { \l_tmpb_fp } }
+ \int_set:Nn \l__nmc_table_a_int { \fp_sign:n { \l_tmpa_fp } }
+ \__nmc_table_signs:
+ \__nmc_num_format:nNnN \l_tmpb_fp \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_table_num_format:N \l_tmpb_tl
+ \seq_put_right:Nx #2
+ { \exp_not:o \l_tmpb_tl \exp_not:o \l__nmc_table_diff_tl }
+ \int_if_zero:nF \l__nmc_table_diffs_int
+ { \__nmc_table_calc_diffs:nn
+ { \l_tmpb_fp } { \l_tmpa_fp } }
+ \fp_set_eq:NN \l_tmpb_fp \l_tmpa_fp
+ }
+\cs_new_protected:Npn \__nmc_table_form_cols_riii:NN #1#2
+ {
+ \__nmc_table_form_cols_rii_aux:NN #1#2
+ \int_set_eq:NN \l__nmc_table_c_int \l__nmc_table_b_int
+ \int_set_eq:NN \l__nmc_table_b_int \l__nmc_table_a_int
+ \seq_map_variable:NNn #1 \l_tmpa_fp
+ {
+ \int_set:Nn \l__nmc_table_a_int { \fp_sign:n { \l_tmpa_fp } }
+ \__nmc_table_signs:
+ \__nmc_num_format:nNnN \l_tmpb_fp \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_table_num_format:N \l_tmpb_tl
+ \seq_put_right:Nx #2
+ { \exp_not:o \l_tmpb_tl \exp_not:o \l__nmc_table_diff_tl }
+ \int_if_zero:nF \l__nmc_table_diffs_int
+ { \__nmc_table_calc_diffs:nn { \l_tmpb_fp } { \l_tmpa_fp } }
+ \fp_set_eq:NN \l_tmpb_fp \l_tmpa_fp
+ \int_set_eq:NN \l__nmc_table_c_int \l__nmc_table_b_int
+ \int_set_eq:NN \l__nmc_table_b_int \l__nmc_table_a_int
+ }
+ \int_set:Nn \l__nmc_table_a_int { 1 }
+ \__nmc_table_signs:
+ \__nmc_num_format:nNnN \l_tmpb_fp \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_table_num_format:N \l_tmpb_tl
+ \seq_put_right:Nx #2
+ { \exp_not:o \l_tmpb_tl \exp_not:o \l__nmc_table_diff_tl }
+ }
+\cs_new_protected:Npn \__nmc_table_num_format:N #1
+ {
+ \bool_if:NTF \l__nmc_sci_num_table_bool
+ {
+ \tl_if_head_eq_charcode:VNTF #1 (
+ { % for nums in [1,10)
+ \exp_last_unbraced:NV
+ \__nmc_table_t_format:wN #1\q_stop \l_tmpa_tl
+ \tl_set_eq:NN #1 \l_tmpa_tl
+ }
+ { \tl_set:Nx #1 { \__nmc_sci_num_table:n { #1 } } }
+ }
+ { \tl_set:Nx #1 { \__nmc_sci_num_table:n { #1 } } }
+ \bool_if:NT \l__nmc_table_QA_bool
+ { \__nmc_table_QA:N #1 }
+ }
+\cs_new_protected:Npn \__nmc_table_QA:N #1
+ {
+ \fp_compare:nNnT { \l__nmc_table_Q_tl } = { 1 }
+ {
+ \bool_lazy_and:nnT
+ { \l__nmc_num_only_bool } { !\l__nmc_table_num_only_bool }
+ {
+ \tl_set_eq:NN \l__nmc_table_num_only_tl \l_tmpb_tl
+ \bool_set_true:N \l__nmc_table_num_only_bool
+ }
+ \tl_set:Nx #1 { \l__nmc_table_A_tl }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_t_format:wN (#1)#2#3\q_stop #4
+ {
+ \tl_set:Nn \l__nmc_toss_tl { #1 }
+ \__nmc_table_prepad:Nnn \l__nmc_toss_tl
+ { \l__nmc_table_tsigns_int } { \exp_not:o \hphantom{-} }
+ \tl_set:Nx #4 { \__nmc_sci_num_table_plus:nn
+ { #3 } { \exp_not:o\l__nmc_toss_tl } }
+ }
+\cs_new_protected:Npn \__nmc_table_signs:
+ { % A prev, B current, C next
+ \int_compare:nNnTF \l__nmc_table_b_int > { -1 }
+ {
+ \bool_lazy_or:nnTF
+ { \int_compare_p:nNn \l__nmc_table_a_int = { -1 } }
+ { \int_compare_p:nNn \l__nmc_table_c_int = { -1 } }
+ { \bool_set_true:N \l__nmc_table_signs_bool }
+ { \bool_set_false:N \l__nmc_table_signs_bool }
+ }
+ { \bool_set_false:N \l__nmc_table_signs_bool }
+ }
+%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_table_calc_diffs:nn #1#2
+ {
+ \__nmc_num_format:nNnN
+ { \fp_abs:n {round( #1,\l__nmc_round_int )
+ - round( #2, \l__nmc_round_int) } } \l_tmpb_tl
+ { \l__nmc_round_int } \c_false_bool
+ \tl_set:Nx \l_tmpb_tl { \fp_eval:n { round( 10^{ \l__nmc_round_int }
+ * \l_tmpb_tl, \l__nmc_round_int + 1) } }
+ \__nmc_table_prepad:Nnn \l_tmpb_tl { \l__nmc_table_diffs_int } { 0 }
+ \tl_set:Nx \l__nmc_table_diff_tl { ^{\,^{ \l_tmpb_tl } } }
+ }
+\cs_new_protected:Npn \__nmc_table_form_diffs_null:n #1
+ {
+ \tl_set:Nn \l_tmpb_tl { 0 }
+ \__nmc_table_prepad:Nnn \l_tmpb_tl { #1 } { 0 }
+ \tl_set:Nx \l__nmc_table_diff_tl
+ { ^{\,^{ \exp_not:o \hphantom{ \l_tmpb_tl } } } }
+ }
+%%%%%%%%%%%%%%%%%%%
+\cs_new:Npn \__nmc_table_form_lower:
+ {
+ \tl_if_empty:NF \l__nmc_table_footer_tl
+ {
+ \__nmc_if_mod_zero:nnT { \l__nmc_table_rules_int } { 7 }
+ { \exp_not:N \midrule[\cmidrulewidth] }
+ \exp_not:o \l__nmc_table_footer_tl \tabularnewline
+ }
+ \__nmc_if_mod_zero:nnT { \l__nmc_table_rules_int } { 11 }
+ { \exp_not:N \bottomrule }
+ \exp_not:n { \end{tabular} }
+ }
+%%%%%%%%%%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_table_reuse:
+ {
+ \bool_if:NF \l__nmc_num_only_bool
+ {
+ \int_case:nnF { \int_sign:n { \l__nmc_table_reuse_int } }
+ {
+ { -1 } { \__nmc_table_reuse_row:n { \l__nmc_table_reuse_int } }
+
+ { 1 } { \__nmc_table_reuse_col:n { \l__nmc_table_reuse_int } }
+ }
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_reuse_row:n #1
+ {
+ \tl_set:Nx \l_tmpa_seq
+ { \seq_item:Nn \l__nmc_table_body_rows_seq { -#1 } }
+ \tl_gset:Nx \g__nmc_reuse_tl { \seq_use:Nn \l_tmpa_seq { , } }
+ }
+\cs_new_protected:Npn \__nmc_table_reuse_col:n #1
+ {
+ \int_zero:N \l_tmpa_int
+ \seq_map_variable:NNn \l__nmc_table_body_cols_seq \l_tmpa_seq
+ {
+ \int_incr:N \l_tmpa_int
+ \int_compare:nNnT { \l_tmpa_int } = { #1 }
+ {
+ \seq_mapthread_function:NNN \l__nmc_table_rvraw_seq
+ \l_tmpa_seq \__nmc_table_reuse:nn
+ \seq_map_break:
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_reuse:nn #1#2
+ {
+ \__nmc_num_format:nNnN { #1 } \l_tmpa_tl
+ { \l__nmc_table_rround_int } \c_false_bool
+ \__nmc_num_format:nNnN { #2 } \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \int_if_zero:nTF \l__nmc_table_rpos_int
+ { \seq_put_right:NV \l_nmc_table_reuse_seq \l_tmpb_tl }
+ {
+ \seq_put_right:Nx \l_nmc_table_reuse_seq
+ { { \l_tmpa_tl , \l_tmpb_tl } }
+ }
+ \tl_gset:Nx \g__nmc_reuse_tl { \seq_use:Nn \l_nmc_table_reuse_seq {,} }
+ }
+%------------------------------------------------
+\keys_define:nn { numerica }
+ {
+ table-rvar-rounding .int_set:N = \l__nmc_table_rround_int,
+ table-rvar-col-pos .int_set:N = \l__nmc_table_rpos_int,
+ table-rvar-align .tl_set:N = \l__nmc_table_rcalign_tl,
+ table-cvar-align .tl_set:N = \l__nmc_table_ccalign_tl,
+ table-block-sep .tl_set:N = \l__nmc_table_blocsep_tl,
+ table-rules .tl_set:N = \l__nmc_table_rules_tl,
+ table-form-diffs .int_set:N = \l__nmc_table_diffs_int,
+ table-reuse .int_set:N = \l__nmc_table_reuse_int,
+ table-rvar-rounding .default:n = 1,
+ table-rvar-align .default:n = r,
+ table-rvar-col-pos .default:n = 1,
+ table-cvar-align .default:n = r,
+ table-block-sep .default:n = 1ex,
+ table-rules .default:n = ThB,
+ table-form-diffs .default:n = 0,
+ table-reuse .default:n = 0
+ }
+\keys_set_known:nn { numerica }
+ {
+ table-rvar-rounding,table-rvar-align,table-rvar-col-pos,
+ table-cvar-align,table-block-sep,table-rules,table-fontsize,
+ table-form-diffs,table-reuse
+ }
+\keys_define:nn { numerica/table }
+ {
+ rvar .tl_set:N = \l__nmc_table_rvar_tl,
+ rstop .tl_set:N = \l__nmc_table_rstop_tl,
+ rstep .tl_set:N = \l__nmc_table_rstep_tl,
+ rows .int_set:N = \l__nmc_table_rnum_int,
+ rows .initial:n = -1,
+ rspec .clist_set:N = \l__nmc_table_rspec_clist,
+ rround .int_set:N = \l__nmc_table_rround_int,
+ rfont .code:n = \tl_set:Nn \l__nmc_table_rfont_tl
+ { \use:c { math#1 } },
+ ralign .tl_set:N = \l__nmc_table_rcalign_tl,
+ rhead .tl_set:N = \l__nmc_table_rhead_tl,
+ rhnudge .code:n = \tl_set:Nn \l__nmc_table_rhnudge_tl
+ { \mkern #1 mu },
+ rpos .int_set:N = \l__nmc_table_rpos_int,
+ rvar' .tl_set:N = \l__nmc_table_rvari_tl,
+ rhead' .tl_set:N = \l__nmc_table_rheadi_tl,
+ rhnudge' .code:n = \tl_set:Nn \l__nmc_table_rhnudgei_tl
+ { \mkern #1 mu },
+%
+ cvar .tl_set:N = \l__nmc_table_cvar_tl,
+ cstop .tl_set:N = \l__nmc_table_cstop_tl,
+ cstep .tl_set:N = \l__nmc_table_cstep_tl,
+ cols .int_set:N = \l__nmc_table_cnum_int,
+ cols .initial:n = -1,
+ cspec .clist_set:N = \l__nmc_table_cspec_clist,
+ calign .tl_set:N = \l__nmc_table_ccalign_tl,
+ chstyle .int_set:N = \l__nmc_table_chstyle_int,
+ chnudge .code:n = \tl_set:Nn \l__nmc_table_chnudge_tl
+ { \mkern #1 mu },
+ chead .tl_set:N = \l__nmc_table_chead_tl,
+ chround .int_set:N = \l__nmc_table_chround_int,
+%
+ ctitle .tl_set:N = \l__nmc_table_ctitle_tl,
+ rules .tl_set:N = \l__nmc_table_rules_tl,
+ foot .tl_set:N = \l__nmc_table_footer_tl,
+ rbloc .clist_set:N = \l__nmc_table_rbloc_clist,
+ rblocsep .tl_set:N = \l__nmc_table_blocsep_tl,
+ signs .int_set:N = \l__nmc_table_signs_int,
+ (pad) .int_set:N = \l__nmc_table_tsigns_int,
+ diffs .int_set:N = \l__nmc_table_diffs_int,
+%
+ Q? .tl_set:N = \l__nmc_table_Q_tl,
+ A! .tl_set:N = \l__nmc_table_A_tl,
+ reuse .int_set:N = \l__nmc_table_reuse_int,
+ valign .code:n = \tl_set:Nn \l__nmc_table_valign_tl { [#1] },
+ }
+% end of `numerica-tables.def'
\ No newline at end of file
Property changes on: trunk/Master/texmf-dist/tex/latex/numerica/numerica-tables.def
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Added: trunk/Master/texmf-dist/tex/latex/numerica/numerica.sty
===================================================================
--- trunk/Master/texmf-dist/tex/latex/numerica/numerica.sty (rev 0)
+++ trunk/Master/texmf-dist/tex/latex/numerica/numerica.sty 2021-02-15 22:22:05 UTC (rev 57759)
@@ -0,0 +1,2692 @@
+% This is file `numerica.sty',
+% the base of the `numerica' package.
+%
+% This work may be distributed and/or modified under the conditions
+% of the LaTeX Project Public License, either version 1.3c of this
+% license or any later version; see
+% http://www.latex-project.org/lppl.txt
+%
+% Andrew Parsloe (ajparsloe at gmail.com)
+%
+\RequirePackage{expl3}[2017/07/15]
+\RequirePackage{xparse,l3keys2e}
+\RequirePackage{amsmath,mathtools}
+
+\ProvidesExplPackage
+ {numerica}
+ {2021/02/15}
+ {1.0.0}
+ {Evaluate math expressions in the LaTeX form they are typeset}
+%------------------------------------------------
+\cs_if_free:NT \arccsc { \DeclareMathOperator{\arccsc}{arccsc} }
+\cs_if_free:NT \arcsec { \DeclareMathOperator{\arcsec}{arcsec} }
+\cs_if_free:NT \arccot { \DeclareMathOperator{\arccot}{arccot} }
+\cs_if_free:NT \csch { \DeclareMathOperator{\csch}{csch} }
+\cs_if_free:NT \sech { \DeclareMathOperator{\sech}{sech} }
+\cs_if_free:NT \asinh { \DeclareMathOperator{\asinh}{asinh} }
+\cs_if_free:NT \acosh { \DeclareMathOperator{\acosh}{acosh} }
+\cs_if_free:NT \atanh { \DeclareMathOperator{\atanh}{atanh} }
+\cs_if_free:NT \acsch { \DeclareMathOperator{\acsch}{acsch} }
+\cs_if_free:NT \asech { \DeclareMathOperator{\asech}{asech} }
+\cs_if_free:NT \acoth { \DeclareMathOperator{\acoth}{acoth} }
+\cs_if_free:NT \sgn { \DeclareMathOperator{\sgn}{sgn} }
+\cs_if_free:NT \lb { \DeclareMathOperator{\lb}{lb} }
+% mathtools
+\cs_if_free:NT \abs {\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} }
+\cs_if_free:NT \ceil { \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} }
+\cs_if_free:NT \floor { \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} }
+\RenewDocumentCommand \land {} { \,\wedge\, }
+\RenewDocumentCommand \lor {} { \,\vee\, }
+% Qleave apart; qleave to
+\NewDocumentCommand \Q {} { \prg_do_nothing: }
+\NewDocumentCommand \q {} { \prg_do_nothing: }
+\prg_new_conditional:Npnn \int_if_zero:n #1 { p,T,F,TF }
+ {
+ \int_compare:nNnTF { #1 } = { 0 }
+ { \prg_return_true: }
+ { \prg_return_false: }
+ }
+\prg_new_conditional:Npnn \__nmc_if_mod_zero:nn #1#2 { p,T,F,TF }
+ {
+ \bool_if:nTF
+ {
+ !\int_compare_p:nNn { #1 } = { 0 } &&
+ \int_compare_p:nNn { \int_mod:nn { #1 } { #2 } } = { 0 }
+ }
+ { \prg_return_true: }
+ { \prg_return_false: }
+ }
+% constants
+\tl_const:Nn \c__nmc_digits_tl { 123456789 }
+\tl_const:Nn \c__nmc_decimals_tl { 1234567890. }
+\tl_const:Nn \c__nmc_sdigits_tl { -1234567890 }
+\tl_const:Nn \c__nmc_comparisons_tl { <=> }
+\tl_const:Nn \c__nmc_modds_tl { |./ }
+\int_const:Nn \c__nmc_and_int { 0 }
+\int_const:Nn \c__nmc_cmp_int { 1 }
+\int_const:Nn \c__nmc_sum_int { 2 }
+\int_const:Nn \c__nmc_trg_int { 3 }
+\int_const:Nn \c__nmc_uny_int { 4 }
+\int_const:Nn \c__nmc_prn_int { 5 }
+\int_const:Nn \c__nmc_srd_int { 6 }
+\seq_const_from_clist:Nn \c__nmc_T_out_seq { 1,T,\texttt{T} }
+\seq_const_from_clist:Nn \c__nmc_F_out_seq { 0,F,\texttt{F} }
+\int_new:N \l__nmc_trg_int
+\int_new:N \l__nmc_num_sgn_int
+\tl_new:N \l__nmc_toss_tl
+\seq_new:N \l_tmpc_seq
+%-----------------------------------------------------------
+% processing and substitution property lists
+% #1 clist of keys; #2 tl of values
+\cs_new_protected:Npn \__nmc_fill_class:nn #1#2
+ {
+ \clist_map_inline:nn { #1 }
+ { \prop_gput:Nnn \g__nmc_class_prop { ##1 } { #2 } }
+ }
+% math token classes
+\prop_new:N \g__nmc_class_prop
+% decimals
+\__nmc_fill_class:nn { 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, . }
+ { \__nmc_fpify_dec: \__nmc_delim_dec: }
+% constants
+\__nmc_fill_class:nn { e, \pi, \gamma, \phi, \deg, \infty }
+ { \__nmc_fpify_const: \__nmc_delim_const: }
+% plus minus
+\__nmc_fill_class:nn { +, - }
+ { \__nmc_fpify_arith: \__nmc_delim_pm: }
+% arith.
+\prop_put:Nnn \g__nmc_class_prop { * }
+ { \__nmc_fpify_arith: \__nmc_delim_arith: }
+% slash
+\prop_put:Nnn \g__nmc_class_prop { / }
+ { \__nmc_fpify_slash: \__nmc_delim_arith: }
+% arith. alternatives
+\__nmc_fill_class:nn { \times, \cdot, \div }
+ { \__nmc_fpify_arith_alt: \__nmc_delim_arith: }
+% comparisons
+\__nmc_fill_class:nn
+ {
+ =, <, >,
+ \ne, \neq, \nless, \ngtr,
+ \ge, \geq, \geqq, \geqslant,
+ \le, \leq, \leqq, \leqslant,
+ \ngeq, \ngeqq, \ngeqslant,
+ \nleq, \nleqq, \nleqslant,
+ }
+ { \__nmc_fpify_comparison: \__nmc_delim_comparison: }
+% and or
+\__nmc_fill_class:nn { \wedge, \land, \vee, \lor }
+ { \__nmc_fpify_andor: \__nmc_delim_andor: }
+% lparen
+\__nmc_fill_class:nn { (, [, \{, \lparen, \lbrack, \lbrace }
+ { \__nmc_fpify_lparen: \__nmc_delim_lparen: }
+% lvert etc.
+\__nmc_fill_class:nn { |, \lvert, \lfloor, \lceil }
+ { \__nmc_fpify_lvert: \__nmc_delim_lvert: }
+% lmodifiers
+\__nmc_fill_class:nn { \left, \bigl, \Bigl, \biggl, \Biggl }
+ { \__nmc_fpify_lmod: \__nmc_delim_lmod: }
+% rmodifiers
+\__nmc_fill_class:nn { \right, \bigr, \Bigr, \biggr, \Biggr }
+ { \prg_do_nothing: \__nmc_delim_rparen: }
+% trig, hyper, other unary fns
+\__nmc_fill_class:nn { \sin, \cos, \tan, \csc, \sec, \cot }
+ {
+ { \__nmc_fpify_unary:n { \l__nmc_trg_int } }
+ { \__nmc_delim_unary:n { \l__nmc_trg_int } }
+ }
+\__nmc_fill_class:nn
+ {
+ \arcsin, \arccos, \arctan, \arccsc, \arcsec, \arccot,
+ \sinh, \cosh, \tanh, \csch, \sech, \coth,
+ \asinh, \acosh, \atanh, \acsch, \asech, \acoth,
+ \exp, \ln, \lg, \lb, \sgn
+ }
+ {
+ { \__nmc_fpify_unary:n { \c__nmc_uny_int } }
+ { \__nmc_delim_unary:n { \c__nmc_uny_int } }
+ }
+% power
+\prop_put:Nnn \g__nmc_class_prop { ^ }
+ { \__nmc_fpify_power: \__nmc_delim_power: }
+% (d)frac
+\__nmc_fill_class:nn { \frac, \dfrac }
+ { \__nmc_fpify_frac: \__nmc_delim_frac: }
+% tfrac
+\prop_put:Nnn \g__nmc_class_prop { \tfrac }
+ { \__nmc_fpify_frac: \__nmc_delim_tfrac: }
+% binom
+\__nmc_fill_class:nn { \tbinom, \binom, \dbinom }
+ { \__nmc_fpify_binom: \__nmc_delim_frac: }
+% unary with braced arg
+\__nmc_fill_class:nn { \sqrt, \abs, \floor, \ceil }
+ { \__nmc_fpify_unarybrace: \__nmc_delim_unarybrace: }
+% unary subscripted
+\prop_put:Nnn \g__nmc_class_prop { \log }
+ { \__nmc_fpify_unarysub: \__nmc_delim_log: }
+% surd, not
+\__nmc_fill_class:nn { \surd, \neg, \lnot }
+ { \__nmc_fpify_surd: \__nmc_delim_surd: }
+% factorial
+\prop_put:Nnn \g__nmc_class_prop { ! }
+ { \__nmc_fpify_fact: \__nmc_delim_fact: }
+% cleave
+\__nmc_fill_class:nn { \q, \Q }
+ { \prg_do_nothing: \__nmc_delim_qleave: }
+% n-ary
+\__nmc_fill_class:nn { \min, \max, \gcd }
+ { \__nmc_fpify_nary: \__nmc_delim_nary: }
+% comma (n-ary fns)
+\prop_put:Nnn \g__nmc_class_prop { , }
+ { \__nmc_fpify_comma: \__nmc_delim_comma: }
+% absorb
+\__nmc_fill_class:nn
+ {
+ {{}}, \\, &, \to, \q_nil,
+ \dots, \ldots, \cdots,
+ \ , \,, \;, \:, \!, \>,
+ \thinspace, \quad, \qquad , \hfill, \hfil,
+ \mathstrut, \displaystyle, \textstyle,
+ \scriptstyle, \scriptscriptstyle
+ }
+ { \prg_do_nothing: \__nmc_delim_absorb: }
+% absorbm
+\__nmc_fill_class:nn { \vphantom, \hphantom, \phantom, \label }
+ { \__nmc_fpify_absorbm: \__nmc_delim_absorbm: }
+% absorbom
+\__nmc_fill_class:nn { \xmathstrut }
+ { \__nmc_fpify_absorbom: \__nmc_delim_absorbom: }
+% sum, prod
+\__nmc_fill_class:nn { \sum, \prod }
+ { \__nmc_fpify_sum: \__nmc_delim_sum: }
+% begin end
+\__nmc_fill_class:nn { \begin, \end }
+ { \__nmc_fpify_BE: \__nmc_delim_BE: }
+% font
+\__nmc_fill_class:nn
+ {
+ \mathrm, \mathit, \mathcal, \mathtt, \mathbf, \mathbb,
+ \mathsf, \mathfrak, \mathscr, \mathnormal, \boldsymbol
+ }
+ { \__nmc_fpify_font: \__nmc_delim_font: }
+% meta
+\__nmc_fill_class:nn { \ensuremath, \text, \mbox }
+ { \__nmc_fpify_meta: \__nmc_delim_stop: }
+% meta2
+\__nmc_fill_class:nn { \splitfrac, \splitdfrac }
+ { \__nmc_fpify_metamm: \__nmc_delim_metamm: }
+% nmcFn
+\__nmc_fill_class:nn
+ {
+ \eval, \iter, \solve, \recur, \tabulate,
+ \nmcEvaluate, \nmcIterate, \nmcSolve,
+ \nmcRecur, \nmcTabulate
+ }
+ { \__nmc_fpify_cmd: \__nmc_delim_cmd: }
+
+\__nmc_fill_class:nn { \nmcInfo } { \__nmc_fpify_info: \__nmc_delim_info: }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% math token replacements for l3fp evaluation
+% #1 prop; #2 seq (keys); #3 seq (values)
+\cs_new_protected:Npn \__nmc_subst:NNN #1#2#3
+ {
+ \cs_set_protected:Npn \__nmc_substitutions:nn ##1##2
+ { \prop_gput:Nnn #1 { ##1 } { ##2 } }
+ \seq_mapthread_function:NNN #2 #3 \__nmc_substitutions:nn
+ }
+\prop_new:N \g__nmc_subst_fpfn_prop
+\seq_set_from_clist:Nn \l_tmpa_seq
+ {
+ e, \pi, \gamma, \phi, \deg, \infty,
+ \sin, \cos, \tan, \csc, \sec, \cot,
+ \arcsin, \arccos, \arctan, \arccsc, \arcsec, \arccot,
+ \sinh, \cosh, \tanh, \csch, \sech, \coth,
+ \asinh, \acosh, \atanh, \acsch, \asech, \acoth,
+ \exp, \ln, \lg, \lb, \sgn,
+ \max, \min, \surd, \sqrt,
+ \abs, \floor, \ceil,
+ \lvert, \lfloor, \lceil,
+ \times, \cdot, \div,
+ \neg, \lnot, \wedge, \land, \vee, \lor,
+ \le, \leq, \leqq, \leqslant,
+ \ge, \geq, \geqq, \geqslant,
+ \ne, \neq, \nless, \ngtr,
+ \nleq, \nleqq,\nleqslant, \ngeq, \ngeqq, \ngeqslant
+ }
+\seq_set_from_clist:Nn \l_tmpb_seq
+ {
+ exp(1), (pi), (0.5772156649015329),
+ (1.618033988749895), (0.0174532925199433), inf,
+ sin\__nmc_deg:, cos\__nmc_deg:, tan\__nmc_deg:,
+ csc\__nmc_deg:, sec\__nmc_deg:, cot\__nmc_deg:,
+ asin\__nmc_deg:, acos\__nmc_deg:, atan\__nmc_deg:,
+ acsc\__nmc_deg:, asec\__nmc_deg:, acot\__nmc_deg:,
+ sinh, cosh, tanh, csch, sech, coth,
+ asinh, acosh, atanh, acsch, asech, acoth,
+ exp, ln, (0.4342944819032518)ln, (1.442695040888963)ln, sign,
+ max, min, sqrt, sqrt,
+ abs, floor, ceil,
+ abs, floor, ceil,
+ *, *, /,
+ !, !, &&, &&, ||, ||,
+ <=, <=, <=, <=,
+ >=, >=, >=, >=,
+ !=, !=, !<, !>,
+ !<=, !<=, !<=, !>=, !>=, !>=
+ }
+\__nmc_subst:NNN \g__nmc_subst_fpfn_prop \l_tmpa_seq \l_tmpb_seq
+
+% property list for some miscellaneous substitutions
+\prop_new:N \g__nmc_subst_misc_prop
+\seq_set_from_clist:Nn \l_tmpa_seq
+ {
+ \cosh, \sinh, \tanh, \csch, \sech, \coth,
+ (, [, \{, \lparen, \lbrack, \lbrace,
+ \left, \bigl, \Bigl, \biggl, \Biggl,
+ \lvert, \lceil, \lfloor,
+ sum, prod, iter, solve
+ }
+\seq_set_from_clist:Nn \l_tmpb_seq
+ {
+ acosh, asinh, atanh, acsch, asech, acoth,
+ ), ], \}, \rparen, \rbrack, \rbrace,
+ \right, \bigr, \Bigr, \biggr, \Biggr,
+ \rvert, \rceil, \rfloor,
+ term, factor, iteration, step
+ }
+\__nmc_subst:NNN \g__nmc_subst_misc_prop \l_tmpa_seq \l_tmpb_seq
+
+% for variables & \reuse \cs
+\prop_new:N \g__nmc_subst_var_prop
+%---------------------------------------------------------------------------
+% base function variants
+\cs_generate_variant:Nn \tl_if_head_eq_meaning:nNTF { V }
+\cs_generate_variant:Nn \tl_if_head_eq_meaning:nNF { V }
+\cs_generate_variant:Nn \tl_if_head_eq_meaning_p:nN { V }
+\cs_generate_variant:Nn \tl_if_head_eq_charcode:nNTF { V }
+\cs_generate_variant:Nn \tl_if_head_eq_charcode:nNT { V }
+\cs_generate_variant:Nn \tl_if_head_eq_charcode:nNF { V }
+\cs_generate_variant:Nn \tl_if_head_is_group:nTF { V }
+\cs_generate_variant:Nn \tl_if_in:NnTF { NV }
+\cs_generate_variant:Nn \tl_if_in:NnT { NV }
+\cs_generate_variant:Nn \tl_if_in:nnTF { nV }
+\cs_generate_variant:Nn \tl_if_eq:nnTF { V }
+\cs_generate_variant:Nn \tl_if_eq:nnT { V }
+\cs_generate_variant:Nn \tl_if_eq:nnF { V }
+\cs_generate_variant:Nn \tl_replace_once:Nnn { Nnx }
+\cs_generate_variant:Nn \tl_count_tokens:n { V }
+\cs_generate_variant:Nn \int_compare:nNnTF { v }
+\cs_generate_variant:Nn \int_compare:nNnT { v }
+\cs_generate_variant:Nn \int_case:nnTF { v }
+\cs_generate_variant:Nn \clist_pop:NN { cc }
+\cs_generate_variant:Nn \prop_put_if_new:Nnn { No }
+\cs_generate_variant:Nn \iow_open:Nn { NV }
+\cs_generate_variant:Nn \iow_now:Nn { NV }
+\cs_generate_variant:Nn \keys_set_known:nn { x }
+\cs_generate_variant:Nn \file_get:nnN { V }
+\cs_generate_variant:Nn \file_get:nnNT { V }
+\cs_generate_variant:Nn \file_get:nnNTF { V }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% general purpose routines
+\tl_new:N \l__nmc_A_tl
+\tl_new:N \l__nmc_B_tl
+\tl_new:N \l__nmc_C_tl
+\tl_new:N \l__nmc_L_tl
+\tl_new:N \l__nmc_R_tl
+% prune head from A, place in B; record new head of A in C
+\cs_new_protected:Npn \__nmc_next:
+ { \exp_last_unbraced:NV\__nmc_next:w \l__nmc_A_tl\q_stop }
+\cs_new_protected:Npn \__nmc_next:w #1#2#3\q_stop
+ { \__nmc_assign_vals:nnnN {#1} {#2} {#3} \l__nmc_B_tl }
+% get |arg|
+\cs_new_protected:Npn \__nmc_absval_arg:N #1
+ { \exp_last_unbraced:NV \__nmc_absval_arg:wN \l__nmc_A_tl\q_stop #1 }
+\cs_new_protected:Npn \__nmc_absval_arg:wN #1|#2#3\q_stop #4
+ { \__nmc_assign_vals:nnnN {#1} {#2} {#3} #4 }
+\cs_new_protected:Npn \__nmc_assign_vals:nnnN #1#2#3#4
+ {
+ \tl_if_single:nTF { #2 }
+ { \tl_set:Nn \l__nmc_A_tl { #2#3 } }
+ { \tl_set:Nn \l__nmc_A_tl { {#2}#3 } }
+ \tl_set:Nn #4 { #1 }
+ \tl_set:Nn \l__nmc_C_tl { #2 }
+ }
+\cs_new_protected:Npn \__nmc_parenth:N #1
+ { \tl_set:Nx #1 { ( \exp_not:o #1 ) } }
+\cs_new_protected:Npn \__nmc_accum_fn_parenth:NNn #1#2#3
+ { \tl_put_right:Nx #1 { #3( \exp_not:o #2 ) } }
+
+\cs_new_protected:Npn \__nmc_accum_fpify_parenth:NN #1#2
+ { % fpify #2, wrap in ( ), append to #1
+ \group_begin:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN #2 \l_tmpa_tl
+ \exp_args:NNNV
+ \group_end:
+ \__nmc_accum_fn_parenth:NNn #1\l_tmpa_tl{}
+ }
+\cs_new_protected:Npn \__nmc_fpify_next_braced:N #1
+ { % fp-ify next braced argument, parenthesize, append to #1
+ \__nmc_next:
+ \__nmc_accum_fpify_parenth:NN #1 \l__nmc_B_tl
+ }
+% wrap #3 in #2 & #4, & append to #1
+\cs_new_protected:Npn \__nmc_put_right_wrap:NnNn #1#2#3#4
+ { \tl_put_right:Nx #1 { #2 \exp_not:o #3 #4 } }
+
+\cs_new_protected:Npn \__nmc_get_arg_L:nN #1#2
+ {
+ \tl_set:Nn \l__nmc_L_tl { #1 }
+ \prop_get:NnN \g__nmc_subst_misc_prop { #1 } \l__nmc_R_tl
+ \__nmc_get_arg_L_aux:NNV #1#2 \l__nmc_R_tl
+ }
+\cs_generate_variant:Nn \__nmc_get_arg_L:nN { V }
+
+\cs_new_protected:Npn \__nmc_get_arg_L_aux:NNn #1#2#3
+ { \__nmc_get_arg_LR:NNN #1#2#3 }
+\cs_generate_variant:Nn \__nmc_get_arg_L_aux:NNn { NNV }
+
+% #1 left delim; #2 <-- arg (delims *not* included)
+% #3 right delim (*must* differ from #1)
+\cs_new_protected:Npn \__nmc_get_arg_LR:NNN #1#2#3
+ {
+ \int_set:Nn \l_tmpa_int { 1 }
+ \tl_map_inline:Nn \l__nmc_A_tl
+ {
+ \str_case:nn { ##1 }
+ {
+ { #1 } { \int_incr:N \l_tmpa_int }
+ { #3 } { \__nmc_get_arg_LR_rdelim: }
+ { \q_nil } { \__nmc_get_arg_LR_qnil:N #1 }
+ }
+ \__nmc_next:
+ \tl_if_single:nTF { ##1 }
+ { \tl_put_right:Nn #2 { ##1 } }
+ { \tl_put_right:Nn #2 { {##1} } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_get_arg_LR_rdelim:
+ {
+ \int_decr:N \l_tmpa_int
+ \int_if_zero:nT { \l_tmpa_int }
+ { \__nmc_next: \tl_map_break: }
+ }
+\cs_new_protected:Npn \__nmc_get_arg_LR_qnil:N #1
+ {
+ \__nmc_error_what:n { Unmatched~\__nmc_verb:n { #1 }{}~in }
+ \tl_map_break:
+ }
+% #1 <== multi-char num poss. in sci notation
+% #2 delim only bool
+\cs_new_protected:Npn \__nmc_get_dec:NN #1#2
+ {
+ \tl_set:NV #1 \l__nmc_B_tl
+ \__nmc_get_dec_digits:NN #1 \c__nmc_decimals_tl
+ \bool_lazy_and:nnT { \l__nmc_sci_num_in_bool }
+ { \tl_if_eq_p:NN \l__nmc_C_tl \l__nmc_sci_num_in_tl }
+ { \__nmc_get_dec_aux:NN #1#2 }
+ }
+\cs_new_protected:Npn \__nmc_get_dec_aux:NN #1#2
+ { % l3fp uses "e"
+ \__nmc_next:
+ \tl_if_eq:VnTF \l__nmc_C_tl { - }
+ {
+ \__nmc_next:
+ \__nmc_get_dec_aux:NNn #1#2 { e- }
+ }
+ { \__nmc_get_dec_aux:NNn #1#2 { e } }
+ }
+ \cs_new_protected:Npn \__nmc_get_dec_aux:NNn #1#2#3
+ {
+ \tl_if_in:NVTF \c__nmc_digits_tl \l__nmc_C_tl
+ {
+ \bool_if:NTF #2
+ { \tl_put_right:NV #1 \l__nmc_sci_num_in_tl }
+ { \tl_put_right:Nn #1 { #3 } }
+ \__nmc_get_dec_digits:NN #1 \c__nmc_sdigits_tl
+ }
+ {
+ \tl_set_eq:NN \l__nmc_C_tl \l__nmc_sci_num_in_tl
+ \str_if_in:nnT { #3 } { - }
+ { \tl_put_left:Nn \l__nmc_A_tl { - } }
+ \tl_put_left:NV \l__nmc_A_tl \l__nmc_sci_num_in_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_get_dec_digits:NN #1#2
+ {
+ \tl_map_inline:Nn \l__nmc_A_tl
+ {
+ \tl_if_in:NnTF #2 { ##1 }
+ {
+ \tl_put_right:Nn #1 { ##1 }
+ \__nmc_next:
+ }
+ { \tl_map_break: }
+ }
+ }
+% for factorial, binom, n-th root, sum/prod
+\cs_new_protected:Npn \__nmc_intify:Nn #1#2
+ { % #1 value; #2 lower bound (e.g. 0,1,-1)
+ \int_set:Nn \l_tmpa_int { \l__nmc_intify_round_tl }
+ \fp_compare:nTF { { #2 } <= round(#1+0,\l_tmpa_int) = round(#1+0) }
+ { \tl_set:Nx #1 { \fp_eval:n { round(#1+0) } } }
+ {
+ \fp_compare:nNnTF { #2 } = { -inf }
+ { \__nmc_error_what:n { Integer~required~in } }
+ { \__nmc_error_what:n { Integer~$\ge#2$~required~in } }
+ }
+ }
+%------------------------------------------------
+% explicit/implicit calc. of fn vals (sum/prod, tables, etc.)
+% #1 var; #2 fn; #3 var val (fp); #4 <= fn val at #3
+\cs_new_protected:Npn \__nmc_calc_fn_val:nNnN #1#2#3#4
+ {
+ \prop_put:Nnx \g__nmc_subst_var_prop { #1 } { \fp_eval:n { #3 } }
+ \__nmc_fpify_set:NN #4 #2
+ }
+\cs_new_protected:Npn \__nmc_calc_mode:n #1
+ {
+ \int_set:Nn \l__nmc_mode_int { #1 }
+ \int_case:nn { #1 }
+ {
+ { 1 } { \cs_set_eq:NN \__nmc_calc_fn_val:nNnN
+ \__nmc_calc_fn_vali:nNnN }
+ { 2 } { \cs_set_eq:NN \__nmc_calc_fn_val:nNnN
+ \__nmc_calc_fn_valii:nNnN }
+ }
+ }
+\cs_generate_variant:Nn \__nmc_calc_fn_val:nNnN { V }
+
+\cs_new_protected:Npn \__nmc_calc_fn_vali:nNnN #1#2#3#4
+ { % mode 1 (all vals in vv-list may change)
+ \prop_put:Nnx \l__nmc_vv_change_prop { #1 } { \fp_eval:n { #3 } }
+ \__nmc_vv_get_vars_vals:NN \l__nmc_calc_fn_seq \l__nmc_vv_change_prop
+ \__nmc_fpify_set:NN #4 #2
+ }
+\cs_new_protected:Npn \__nmc_calc_fn_valii:nNnN #1#2#3#4
+ { % mode 2 (some vals in vv-list held const.)
+ \prop_put:Nnx \l__nmc_vv_change_prop { #1 } { \fp_eval:n { #3 } }
+ \clist_map_inline:Nn \g__nmc_unchanged_clist
+ {
+ \prop_get:NnN \g__nmc_subst_var_prop { ##1 } \l__nmc_subst_tl
+ \prop_put:NnV \l__nmc_vv_change_prop { ##1 } \l__nmc_subst_tl
+ }
+ \__nmc_vv_get_vars_vals:NN \l__nmc_calc_fn_seq \l__nmc_vv_change_prop
+ \__nmc_fpify_set:NN #4 #2
+ }
+\cs_new_protected:Npn \__nmc_fpify_set:NN #1#2
+ { % fp-set #1 to fpified #2
+ \group_begin:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN #2 \l_tmpa_tl
+ \exp_args:NNNV
+ \group_end:
+ \tl_set:Nn \l__nmc_fp_exprn_tl { \l_tmpa_tl }
+ \fp_set:Nn #1 { \l__nmc_fp_exprn_tl }
+ \__nmc_error_fpflag:
+ }
+\cs_generate_variant:Nn \__nmc_fpify_set:NN { cc }
+\cs_new_protected:Npn \__nmc_verb:n #1
+ { \texttt{ \tl_trim_spaces:n { \tl_to_str:n { #1 } } } }
+\cs_generate_variant:Nn \__nmc_verb:n { V }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Error handling
+\seq_new:N \g__nmc_error_where_seq
+\bool_new:N \g__nmc_error_bool
+%--------------------------------------
+\cs_new_protected:Npn \__nmc_error_where:n #1
+ { \seq_gpush:Nn \g__nmc_error_where_seq { #1 } }
+
+\cs_new_protected:Npn \__nmc_error_what:n #1
+ {
+ \bool_gset_true:N \g__nmc_error_bool
+ \tl_gclear:N \g__nmc_reuse_tl
+ \__nmc_error_msg:n { #1 }
+ }
+\cs_new_protected:Npn \__nmc_error_msg:n #1
+ {
+ \seq_if_empty:NF \g__nmc_error_where_seq
+ {
+ \seq_gpop:NN \g__nmc_error_where_seq \l_tmpa_tl
+ \hbox:n { !!!~ #1 :~\l_tmpa_tl.~!!! }
+ }
+ }
+% l3fp exceptions; invalid op triggers a LaTeX *error*, hence:
+\fp_trap:nn { invalid_operation } { flag }
+\cs_new_protected:Npn \__nmc_error_fpflag:
+ { % ln(1),cos(90),sin(360) => underflow. Ignored!
+ \flag_if_raised:nTF { fp_overflow } { \__nmc_flag:n { 1 } }
+ { \flag_if_raised:nTF { fp_division_by_zero } { \__nmc_flag:n { 2 } }
+ { \flag_if_raised:nT { fp_invalid_operation }
+ { \__nmc_flag:n { 3 } }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_flag:n #1
+ {
+ \__nmc_error_what:n
+ {
+ \__nmc_verb:n{l3fp}~error~
+ \int_case:nnT { #1 }
+ {
+ { 1 } { `Overflow' }
+ { 2 } { `Division~by~zero' }
+ { 3 } { `Invalid~operation' }
+ }
+ { ~in }
+ }
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% multi-tok vars (primed, subscripted, ...) to single tok
+% assumes math delims absent
+\int_new:N \g__nmc_prep_multitok_int
+\tl_new:N \l_nmc_multitoka_tl
+\tl_new:N \l_nmc_multitokb_tl
+\cs_new_protected:Npn \__nmc_prep_multitok:NN #1#2
+ { % #1 vv-list (seq); #2 formula (tl)
+ \seq_clear:N \l_tmpa_seq
+ \seq_map_inline:Nn #1
+ {
+ \seq_set_split:Nnn \l_tmpb_seq { = } { ##1 }
+ \seq_pop:NN \l_tmpb_seq \l_tmpb_tl
+ \int_compare:nNnT { \tl_count_tokens:V \l_tmpb_tl } > { 1 }
+ { \seq_push:NV \l_tmpa_seq \l_tmpb_tl }
+ }
+ \seq_sort:Nn \l_tmpa_seq
+ { % big --> small
+ \int_compare:nNnTF { \tl_count:n {##2} } > { \tl_count:n { ##1 } }
+ { \sort_return_swapped: }
+ { \sort_return_same: }
+ }
+ \seq_map_inline:Nn \l_tmpa_seq
+ { % multitoks => \_nmca, \_nmcb, etc
+ \int_gincr:N \g__nmc_prep_multitok_int
+ \tl_set:Nn \l_nmc_multitoka_tl { ##1 }
+ \tl_set:Nx \l_nmc_multitokb_tl { \cs:w _nmc\int_to_alph:n
+ { \g__nmc_prep_multitok_int } \cs_end: }
+ \tl_set:cn { _nmc\int_to_alph:n { \g__nmc_prep_multitok_int } }
+ { ##1 }
+ \regex_replace_all:nnN
+ { \u{l_nmc_multitoka_tl} } { \u{l_nmc_multitokb_tl} } #1
+ \regex_replace_all:nnN
+ { \u{l_nmc_multitoka_tl} } { \u{l_nmc_multitokb_tl} } #2
+ }
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \nmcInfo (no. of terms/loops/steps in "infinite" processes)
+\tl_new:N \g__nmc_sum_info_tl
+\tl_new:N \g__nmc_prod_info_tl
+\NewDocumentCommand \nmcInfo { s m }
+ {
+ \prop_if_in:NnTF \g__nmc_subst_misc_prop { #2 }
+ {
+ \tl_if_empty:cT { g__nmc_#2_info_tl }
+ { \tl_set:cn { g__nmc_#2_info_tl } { 0 } }
+ \tl_use:c { g__nmc_#2_info_tl }
+ \IfBooleanF { #1 }
+ {
+ \prop_get:NnN \g__nmc_subst_misc_prop { #2 } \l_tmpb_tl
+ \ \l_tmpb_tl % pluralise?
+ \int_if_zero:nF { \tl_use:c { g__nmc_#2_info_tl } - 1 } { s }
+ }
+ }
+ {
+ \__nmc_error_where:n { info~command }
+ \__nmc_error_what:n { Unknown~argument~\__nmc_verb:n { #2 }~in }
+ }
+ }
+\ProvideDocumentCommand \info {} { \nmcInfo }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\int_new:N \l__nmc_mode_int
+\tl_new:N \l__nmc_formula_tl
+\tl_new:N \l__nmc_formula_dup_tl
+\tl_new:N \l__nmc_fp_expr_tl
+\tl_new:N \l__nmc_fp_exprn_tl
+\tl_new:N \l_nmc_result_tl
+\bool_new:N \l__nmc_vv_val_only_bool
+\clist_new:N \l__nmc_vv_clist
+\seq_new:N \l__nmc_vv_visible_seq
+\seq_new:N \l__nmc_vv_all_seq
+\seq_new:N \l__nmc_calc_fn_seq
+\prop_new:N \l__nmc_vv_change_prop
+\tl_new:N \l__nmc_eq_var_tl
+\tl_new:N \l__nmc_eq_val_tl
+\tl_new:N \l__nmc_vv_fp_expr_tl
+\int_new:N \l__nmc_round_int
+\int_new:N \l__nmc_sci_num_int
+\int_new:N \l__nmc_prime_int
+\tl_new:N \l__nmc_bool_F_tl
+\tl_new:N \l__nmc_bool_T_tl
+\bool_new:N \l__nmc_TF_out_bool
+\bool_new:N \l__nmc_allow_TF_out_bool
+\bool_new:N \l__nmc_sci_num_x_bool
+\bool_new:N \l__nmc_sci_num_table_bool
+\tl_new:N \l__nmc_punc_tl
+\tl_new:N \l__nmc_math_delimi_tl
+\tl_new:N \l__nmc_math_delimii_tl
+\bool_new:N \l__nmc_wraps_math_bool
+% ##1 = number-only switch, ##2 = keyval settings,
+% ##3 = formula, ##4 = vv-list, ##5 = number format,
+% #1 = \nmcCommand, #2 = id, #3 =short-name cmd
+\cs_new_protected:Npn \nmc_define:nnN #1#2#3
+ {
+ \NewDocumentCommand { #1 } { s O{} m O{} O{} }
+ {
+ \bool_gset_false:N \g__nmc_error_bool
+ \group_begin:
+ \bool_set_eq:NN \l__nmc_num_only_bool ##1
+ \__nmc_get_inputs:nnnnn { #2 } { ##2 } { ##3 } {##4 } {##5 }
+ \bool_if:NF \g__nmc_error_bool
+ { \use:c { __nmc_#2_process: } }
+ \int_if_zero:nTF { \l__nmc_debug_int }
+ {
+ \bool_if:NF \g__nmc_error_bool
+ { \use:c { __nmc_#2_display: } }
+ }
+ { \__nmc_debug_display:nn { \l__nmc_debug_int } { #2 } }
+ \group_end:
+ }
+ \ProvideDocumentCommand { #3 } { } { #1 }
+ }
+\nmc_define:nnN { \nmcEvaluate } { eval } \eval
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_get_inputs:nnnnn #1#2#3#4#5
+ {
+ \__nmc_settings_get:nn { #1 }{ #2 }
+ \use:c { __nmc_#1_settings_digest: }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \mode_if_math:TF
+ { \tl_set:Nn \l__nmc_formula_tl { #3 } }
+ { \__nmc_math_delims:n { #3 } }
+ \tl_set_eq:NN \l__nmc_formula_dup_tl \l__nmc_formula_tl
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \tl_if_in:nnTF { #4 } { = }
+ { % vv-list
+ \__nmc_vv_extract_visible:nN { #4 } \l__nmc_vv_clist
+ \use:c { __nmc_#1_vv_digest:N } \l__nmc_vv_clist
+ \bool_if:NF \g__nmc_error_bool
+ { % number format
+ \tl_if_empty:nF { #5 }
+ { \__nmc_get_num_format:nn { #5 } { #1 } }
+ }
+ }
+ { % if empty vv-list
+ \tl_clear:N \l__nmc_vv_inline_tl
+ \tl_clear:N \l__nmc_vv_display_tl
+ \__nmc_get_num_format:nn { #4#5 } { #1 }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_settings_get:nn #1#2
+ { % #1 = iter,solve,recur,table; #2 = settings
+ \__nmc_error_where:n { settings }
+ \tl_if_empty:nF { #2 }
+ {
+ \keys_set_known:nn { numerica/generic } { #2 }
+ \keys_set_known:xn { numerica/#1 }{ #2 }
+ }
+ }
+\cs_new_protected:Npn \__nmc_debug_display:nn #1#2
+ { \__nmc_debug_display:xnn { \mode_if_math:TF { ed }{ * } } { #1 } { #2 } }
+\cs_new_protected:Npn \__nmc_debug_display:nnn #1#2#3
+ { % #2 dbg int #3 fn id
+ \__nmc_if_mod_zero:nnT { #2 } { 5 }
+ { \__nmc_debug_stored:N \l_tmpb_tl }
+ \raggedright
+ \begin{ align#1 }
+ \__nmc_debug_display:xnnn { \seq_use:Nn \l__nmc_vv_all_seq { ,~ } }
+ { #2 } { 2 } { vv-list }
+ \__nmc_debug_display:Vnnn \l__nmc_formula_tl { #2 } { 3 } { formula }
+ \__nmc_debug_display:Vnnn \l_tmpb_tl { #2 } { 5 } { stored }
+ \__nmc_debug_display:Vnnn \l__nmc_fp_expr_tl { #2 } { 7 } { fp-form }
+ \bool_if:nT
+ {
+ !\g__nmc_error_bool &&
+ \__nmc_if_mod_zero_p:nn { #2 } { 7 } &&
+ !\str_if_eq_p:nn { #3 } { table }
+ }
+ { \text{result:} & \quad \l_nmc_result_tl }
+ \__nmc_debug_display:Vnnn \l__nmc_reuse_retrieved_tl
+ { #2 } { 99991 } { Saved }
+ \end{ align#1 }
+ }
+\cs_generate_variant:Nn \__nmc_debug_display:nnn { x }
+
+\cs_new_protected:Npn \__nmc_debug_stored:N #1
+ {
+ \tl_clear:N \l_tmpa_tl
+ \tl_if_empty:NF \l__nmc_reuse_retrieved_tl
+ {
+ \clist_map_inline:Nn \l__nmc_reuse_retrieved_tl
+ { \tl_put_left:Nx \l_tmpa_tl { \exp_not:o { \use_i:nn ##1 }; } }
+ \tl_replace_all:Nnn \l_tmpa_tl { ~; } { ; }
+ }
+ \tl_clear:N #1
+ \prop_map_inline:Nn \g__nmc_subst_var_prop
+ {
+ \str_if_in:NnF \l_tmpa_tl { ##1; }
+ { \tl_put_right:Nn #1 { ##1=##2,~ } }
+ }
+ \tl_set:Nx #1 { \tl_range:Nnn #1 { 1 } { -2 } }
+ }
+\cs_new:Npn \__nmc_debug_display:nnnn #1#2#3#4
+ {
+ \__nmc_if_mod_zero:nnT { #2 } { #3 }
+ { \hbox:n { #4:} & \quad
+ \int_compare:nNnTF { #2 } > { 0 }
+ { \vbox_top:n } { \hbox:n } { \tl_to_str:n { #1 } } \\ }
+ }
+\cs_generate_variant:Nn \__nmc_debug_display:nnnn { V, x }
+%%%%%%%%% formula, delimiters %%%%%%%%%%%
+% delimiters
+% #1(tl) = formula
+\cs_new_protected:Npn \__nmc_math_delims:n #1
+ {
+ \tl_set:Nn \l__nmc_formula_tl { #1 }
+ \tl_trim_spaces:N \l__nmc_formula_tl
+ \bool_set_true:N \l__nmc_wraps_math_bool
+ % $ ?
+ \tl_if_head_eq_meaning:VNTF \l__nmc_formula_tl $
+ { \__nmc_math_delims_aux:n { 1 } }
+ { % \[ ?
+ \tl_if_head_eq_meaning:VNTF \l__nmc_formula_tl \[
+ { \__nmc_math_delims_aux:n { 1 } }
+ { %\begin{environ} ?
+ \tl_if_head_eq_meaning:VNTF \l__nmc_formula_tl \begin
+ { \__nmc_math_delims_aux:n { 2 } }
+ {
+ \__nmc_math_delims_aux:nn { \[ } { \] }
+ \bool_set_false:N \l__nmc_wraps_math_bool
+ }
+ }
+ }
+ \tl_trim_spaces:N \l__nmc_formula_tl
+ \bool_if:NTF \l__nmc_num_only_bool
+ { \__nmc_math_delims_aux:nn {}{} }
+ {
+ \bool_if:NT \l__nmc_vv_multline_bool
+ { \__nmc_math_delims_mliner:V \l__nmc_eq_num_tl }
+ }
+ }
+\cs_new:Npn \__nmc_math_delims_mliner:n #1
+ {
+ \bool_if:nT { \l__nmc_wraps_math_bool || !\mode_if_math_p: }
+ {
+ \tl_set:Nn \l__nmc_math_delimi_tl { \begin{multline#1} }
+ \tl_set:Nn \l__nmc_math_delimii_tl { \end{multline#1} }
+ }
+ }
+\cs_generate_variant:Nn \__nmc_math_delims_mliner:n { V }
+\cs_new_protected:Npn \__nmc_math_delims_aux:n #1
+ {
+ \__nmc_math_delims_aux:xx
+ { \tl_range:Nnn \l__nmc_formula_tl { 1 } { #1 } }
+ { \tl_range:Nnn \l__nmc_formula_tl { -#1 } { -1 } }
+ \int_set:Nn \l_tmpa_int { #1 + 1 }
+ \tl_set:Nx \l__nmc_formula_tl
+ { \tl_range:Nnn \l__nmc_formula_tl { \l_tmpa_int }
+ { -\l_tmpa_int } }
+ }
+\cs_new_protected:Npn \__nmc_math_delims_aux:nn #1#2
+ {
+ \tl_set:Nn \l__nmc_math_delimi_tl { #1 }
+ \tl_set:Nn \l__nmc_math_delimii_tl { #2 }
+ }
+\cs_generate_variant:Nn \__nmc_math_delims_aux:nn { xx }
+%%%%%%%%%% vv-list routines %%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_vv_extract_visible:nN #1#2
+ { % #2 reverses #1
+ \clist_clear:N #2
+ \seq_clear:N \l__nmc_vv_visible_seq
+ \clist_set:Nn \l_tmpa_clist { #1 }
+ \mode_if_math:F
+ {
+ \tl_remove_all:Nn \l_tmpa_clist { $ }
+ \tl_if_head_eq_charcode:VNT \l_tmpa_clist \[
+ {
+ \tl_remove_once:Nn \l_tmpa_clist { \[ }
+ \tl_remove_once:Nn \l_tmpa_clist { \] }
+ }
+ \tl_trim_spaces:N \l_tmpa_clist
+ }
+ \clist_map_inline:Nn \l_tmpa_clist
+ {
+ \tl_if_head_is_group:nTF { ##1 }
+ { % to remove { }
+ \clist_put_left:Nx #2
+ { \tl_head:n { ##1 } \tl_tail:n { ##1 } }
+ }
+ {
+ \seq_put_right:Nn \l__nmc_vv_visible_seq { ##1 }
+ \tl_if_head_eq_meaning:nNF { ##1 } \\
+ { \clist_put_left:Nn #2 { ##1 } }
+ }
+ }
+ \tl_replace_once:Nnx \l__nmc_vv_inline_tl { vv }
+ { \seq_use:Nn \l__nmc_vv_visible_seq { , } }
+ \tl_replace_once:Nnx \l__nmc_vv_display_tl { vv }
+ { \seq_use:Nn \l__nmc_vv_visible_seq { , } }
+ }
+\cs_new_protected:Npn \__nmc_vv_digest:N #1
+ {
+ \seq_set_from_clist:NN \l__nmc_vv_all_seq #1
+ \__nmc_error_where:n { variable\,=\,value~list }
+ \bool_if:NT \l__nmc_multitok_bool
+ { \__nmc_prep_multitok:NN \l__nmc_vv_all_seq \l__nmc_formula_tl }
+ \__nmc_vv_get_vars_vals:NN \l__nmc_vv_all_seq \c_empty_prop
+ }
+\cs_new_protected:Npn \__nmc_vv_get_vars_vals:NN #1#2
+ { % #1(seq) vv-list;#2(prop) changed vars (implicit-mode)
+ \seq_set_eq:NN \l_tmpa_seq #1
+ \seq_clear:N \l__nmc_calc_fn_seq
+ \seq_map_inline:Nn \l_tmpa_seq
+ {
+ \__nmc_vv_split_item:n { ##1 }
+ \bool_if:NT \g__nmc_error_bool { \seq_map_break: }
+ \__nmc_vv_record:NVN \l__nmc_eq_var_tl \l__nmc_eq_val_tl #2
+ \bool_if:NTF \g__nmc_error_bool { \seq_map_break: }
+ { \seq_put_right:Nn \l__nmc_calc_fn_seq { ##1 } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_vv_get_vars_vals_lims:NN #1#2
+ { % for summation, integration limits
+ \__nmc_vv_split_item:V #1
+ \bool_if:NF \g__nmc_error_bool
+ { \__nmc_vv_record:NVN \l__nmc_eq_var_tl \l__nmc_eq_val_tl #2 }
+ }
+\cs_new_protected:Npn \__nmc_vv_record:NnN #1#2#3
+ {
+ \bool_if:NF \l__nmc_vv_val_only_bool
+ { % record the var
+ \prop_put_if_new:Non \g__nmc_class_prop #1
+ { \__nmc_fpify_var: \__nmc_delim_var: }
+ }
+ \prop_get:NVNF #3 #1 \l__nmc_vv_fp_expr_tl
+ { % fp-ify & calc. (changed `var=val's in prop #3)
+ \tl_clear:N \l__nmc_vv_fp_expr_tl
+ \__nmc_fpify:nN { #2 } \l__nmc_vv_fp_expr_tl
+ }
+ \bool_if:nF
+ {
+ \l__nmc_vv_val_only_bool ||
+ \g__nmc_error_bool
+ }
+ { \__nmc_vv_write:VN #1 \l__nmc_vv_fp_expr_tl }
+ }
+\cs_generate_variant:Nn \__nmc_vv_record:NnN { NV }
+
+\cs_new_protected:Npn \__nmc_vv_write:nN #1#2
+ { % #1 = var; #2 = val as fp expr.
+ \tl_if_empty:NF #2
+ {
+ \tl_if_eq:VnTF #2 { inf }
+ { \prop_put:Nnn \g__nmc_subst_var_prop { #1 } { inf } }
+ {
+ \prop_put:Nnx \g__nmc_subst_var_prop
+ { #1 } { \fp_eval:n { #2 } }
+ \__nmc_error_fpflag:
+ }
+ }
+ }
+\cs_generate_variant:Nn \__nmc_vv_write:nN { V }
+% split var=val
+\cs_new_protected:Npn \__nmc_vv_split_item:n #1
+ {
+ \bool_set_false:N \l__nmc_vv_val_only_bool
+ \tl_if_in:nnTF { #1 } { = }
+ {
+ \__nmc_split_eq:w #1\q_stop
+ \tl_if_empty:NT \l__nmc_eq_val_tl
+ { \__nmc_error_what:n { No~value~for~$\l__nmc_eq_var_tl$~in } }
+ }
+ {
+ \bool_set_true:N \l__nmc_vv_val_only_bool
+ \tl_set:Nn \l__nmc_eq_val_tl { #1 }
+ }
+ }
+\cs_generate_variant:Nn \__nmc_vv_split_item:n { V }
+\cs_new_protected:Npn \__nmc_split_eq:w #1=#2\q_stop
+ {
+ \tl_set:Nn \l__nmc_eq_var_tl { #1 }
+ \tl_trim_spaces:N \l__nmc_eq_var_tl
+ \tl_set:Nn \l__nmc_eq_val_tl { #2 }
+ }
+%%%%%%%%%% number-formatting routines %%%%%%%%%%
+\cs_new_protected:Npn \__nmc_get_num_format:nn #1#2
+ {
+ \__nmc_error_where:n { result~format~spec. }
+ \tl_clear:N \l_tmpa_tl
+ \str_map_inline:nn { #1 }
+ {
+ \str_if_in:NnTF \c__nmc_sdigits_tl { ##1 }
+ { \tl_put_right:Nn \l_tmpa_tl { ##1 } }
+ {
+ \str_if_eq:nnTF { ##1 } { * }
+ { \bool_set_true:N \l__nmc_pad_zeros_bool }
+ {
+ \bool_set_true:N \l__nmc_sci_num_out_bool
+ \str_case:nnF { ##1 }
+ {
+ { x } { \bool_set_true:N \l__nmc_sci_num_x_bool
+ \tl_set:Nn \l__nmc_sci_num_out_tl { ##1 } }
+ { t } { \bool_set_true:N \l__nmc_sci_num_table_bool }
+ { ' } { \int_incr:N \l__nmc_prime_int }
+ }
+ { \tl_set:Nn \l__nmc_sci_num_out_tl { ##1 } }
+ }
+ }
+ }
+ \str_if_in:nnT { #1 } { ? }
+ {
+ \bool_set_eq:NN \l__nmc_TF_out_bool \l__nmc_allow_TF_out_bool
+ \bool_set_false:N \l__nmc_sci_num_out_bool
+ }
+ \tl_if_empty:NF \l_tmpa_tl
+ { \int_set:Nn \l__nmc_round_int \l_tmpa_tl }
+ \bool_if:NT \l__nmc_TF_out_bool
+ {
+ \bool_if:NTF \l__nmc_num_only_bool
+ { \__nmc_num_TF_format:n { 1 } }
+ { \__nmc_num_TF_format:n { 1 + \l__nmc_prime_int } }
+ }
+ \bool_if:NF \g__nmc_error_bool
+ { \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl }
+ }
+%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_num_TF_format:n #1
+ {
+ \tl_set:Nx \l__nmc_bool_F_tl { \seq_item:Nn \c__nmc_F_out_seq { #1 } }
+ \tl_set:Nx \l__nmc_bool_T_tl { \seq_item:Nn \c__nmc_T_out_seq { #1 } }
+ }
+\cs_new_protected:Npn \__nmc_num_format:nNnN #1#2#3#4
+ { % #1=fpf'ied in; #2=tl fmt'ed out; #3=int rnd; #4 sci bool
+ \bool_if:NTF \l__nmc_TF_out_bool
+ {
+ \tl_set:Nx #2 { \fp_compare:nNnTF { 0 } = { #1 }
+ { \exp_not:o \l__nmc_bool_F_tl }
+ { \exp_not:o \l__nmc_bool_T_tl } }
+ }
+ {
+ \bool_if:NTF #4
+ {
+ \exp_last_unbraced:Nx \__nmc_sci_output_aux:wNn
+ { \fp_to_scientific:n { #1 } } \q_stop #2 { #3 }
+ }
+ { \__nmc_num_format_decimal:nNn { #1 } #2 { #3 } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_num_format_decimal:nNn #1#2#3
+ {
+ \tl_set:Nx #2 { \fp_eval:n { 0 + round( #1, #3 ) } }
+ \__nmc_error_fpflag:
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_set:Nn \l__nmc_num_sgn_int { \fp_sign:n { #2 } }
+ \bool_lazy_and:nnT { \l__nmc_pad_zeros_bool }
+ { \int_compare_p:nNn { #3 } > { 0 } }
+ {
+ \tl_if_in:NnTF #2 { . }
+ { \__nmc_num_pad:Nnn #2 {} { #3 } }
+ { \__nmc_num_pad:Nnn { #2 } {.} { #3 } }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_num_pad:Nnn #1#2#3
+ { % pad with 0s: #1 number; #2 point #3 rounding
+ \tl_set:Nx #1 { \exp_last_unbraced:NV
+ \__nmc_num_pad:wnN #1#2\q_stop{ #3 } }
+ }
+\cs_new:Npn \__nmc_num_pad:wnN #1.#2\q_stop#3
+ {
+ \int_if_zero:nTF { #3 } { #1 }
+ { #1.#2\prg_replicate:nn { #3 - \tl_count:n { #2 } } { 0 } }
+ }
+\cs_new_protected:Npn \__nmc_sci_output_aux:wNn #1e#2\q_stop#3#4
+ { % #1e#2=fp input; #3=formatted tl out; #4 = rounding
+ \tl_set:Nx #3
+ { \fp_eval:n { 0 + round( #1, #4 ) } }
+ \__nmc_error_fpflag:
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_set:Nn \l__nmc_sci_num_int { #2 }
+ \tl_if_eq:VnT #3 { 10 }
+ {
+ \int_incr:N \l__nmc_sci_num_int
+ \tl_set:Nn #3 { 1 }
+ }
+ \bool_if:NT \l__nmc_pad_zeros_bool
+ {
+ \tl_if_in:NnTF #3 {.}
+ { \__nmc_num_pad:Nnn #3 {} { #4 } }
+ { \__nmc_num_pad:Nnn #3 {.} { #4 } }
+ }
+ \bool_lazy_or:nnT { !\int_if_zero_p:n { \l__nmc_sci_num_int } }
+ { \int_compare_p:nNn { \l__nmc_prime_int } > { 0 } }
+ { \__nmc_sci_write:Nx #3 { \int_use:N \l__nmc_sci_num_int } }
+ }
+ \int_set:Nn \l__nmc_num_sgn_int { \fp_sign:n { #1 } }
+ }
+\cs_new_protected:Npn \__nmc_sci_write:Nn #1#2
+ { % #1 significand #2 exponent
+ \bool_if:NTF \l__nmc_sci_num_x_bool
+ { \tl_put_right:Nn #1 { \times 10^{#2} } }
+ {
+ \bool_if:NTF \l__nmc_sci_num_table_bool
+ { \tl_set:Nx #1 { \__nmc_sci_num_table:nn { #1 } { #2 } } }
+ { \tl_put_right:Nx #1 { \hbox:n { \l__nmc_sci_num_out_tl {#2} } } }
+ }
+ }
+\cs_generate_variant:Nn \__nmc_sci_write:Nn { Nx }
+\cs_new:Npn \__nmc_sci_num_table:nn #1#2
+ { (#2)\exp_not:n {\,}{#1} }
+%%%%%%%%%%%%%% eval-specific routines %%%%%%%%%%%%%%%
+\cs_new:Npn \__nmc_eval_settings_digest:
+ { \bool_set_true:N \l__nmc_allow_TF_out_bool }
+\cs_set_eq:NN \__nmc_eval_vv_digest:N \__nmc_vv_digest:N
+\cs_new_protected:Npn \__nmc_eval_process:
+ {
+ \__nmc_error_where:n { formula }
+ \tl_clear:N \l__nmc_fp_expr_tl
+ \__nmc_fpify:VN \l__nmc_formula_tl \l__nmc_fp_expr_tl
+ \bool_if:NF \g__nmc_error_bool
+ { \__nmc_num_format:nNnN { \l__nmc_fp_expr_tl } \l_nmc_result_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool }
+ }
+\cs_new_protected:Npn \__nmc_eval_display:
+ {
+ \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl
+ \bool_if:NF \l__nmc_num_only_bool
+ {
+ \tl_set:Nx \l_nmc_result_tl
+ {
+ \exp_not:o \l__nmc_math_delimi_tl
+ \bool_if:NT \l__nmc_wraps_math_bool
+ {
+ \exp_not:o \l__nmc_formula_dup_tl
+ \bool_if:NTF \l__nmc_TF_out_bool
+ { \rightarrow } { = }
+ }
+ \exp_not:o \l_nmc_result_tl
+ \bool_if:NTF \l__nmc_vv_multline_bool
+ { \exp_not:o \l__nmc_vv_display_tl }
+ {
+ \mathchoice{ \exp_not:o \l__nmc_vv_display_tl }
+ { \exp_not:o \l__nmc_vv_inline_tl }{}{}
+ }
+ \l__nmc_punc_tl
+ \exp_not:o \l__nmc_math_delimii_tl
+ }
+ \int_if_zero:nT { \l__nmc_eval_reuse_int }
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ }
+ \l_nmc_result_tl
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \__nmc_fpify:nN
+\bool_new:N \l__nmc_insert_aster_bool
+\tl_new:N \l__nmc_subst_tl
+\tl_new:N \l__nmc_accum_tl
+\tl_new:N \l__nmc_arg_tl
+% unary functions
+\tl_new:N \l__nmc_fn_tl
+\tl_new:N \l__nmc_fn_arg_tl
+\tl_new:N \l__nmc_power_arg_tl
+\bool_new:N \l__nmc_hyperbolic_bool
+% factorial
+\fp_new:N \l__nmc_fact_fp
+\tl_new:N \l__nmc_fact_arg_tl
+% n-ary
+\int_new:N \l_tmpc_int
+\int_new:N \l_tmpd_int
+\bool_new:N \l__nmc_comma_nary_bool
+% sqrt
+\tl_new:N \l__nmc_ubrace_n_tl
+% binom
+\tl_new:N \l__nmc_binom_top_tl
+\tl_new:N \l__nmc_binom_bot_tl
+% sum/prod
+\tl_new:N \l__nmc_sum_var_tl
+\tl_new:N \l__nmc_summand_tl
+\tl_new:N \l__nmc_sum_op_tl
+\tl_new:N \l__nmc_sum_type_tl
+\tl_new:N \g__nmc_sum_A_tl
+\bool_new:N \l__nmc_sum_bool
+\int_new:N \l__nmc_sum_index_int
+\int_new:N \l__nmc_sum_end_int
+\int_new:N \l__nmc_sum_round_int
+\int_new:N \l__nmc_suma_int
+\int_new:N \l__nmc_sumb_int
+\fp_new:N \l__nmc_sum_total_fp
+\fp_new:N \l__nmc_sum_prev_fp
+\fp_new:N \l__nmc_sum_rounded_fp
+% #1 = latex expr #2 = fp-ified result
+\cs_new_protected:Npn \__nmc_fpify:nN #1#2
+ {
+ \group_begin:
+ \bool_set_false:N \l__nmc_superscript_bool
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ \tl_clear:N \l__nmc_accum_tl
+ \tl_clear:N \l__nmc_C_tl
+ \tl_set:Nn \l__nmc_A_tl { #1\q_nil }
+ \tl_if_empty:nT { #1 }
+ { \__nmc_error_what:n { Empty~argument~to~fp-ify~in } }
+ \bool_until_do:nn
+ {
+ \quark_if_nil_p:N \l__nmc_C_tl
+ || \g__nmc_error_bool
+ }
+ {
+ \tl_clear:N \l__nmc_arg_tl
+ \__nmc_next:
+ \prop_get:NVNTF \g__nmc_class_prop \l__nmc_B_tl \l_tmpb_tl
+ { \exp_last_unbraced:NV \use_i:nn \l_tmpb_tl }
+ { \__nmc_fpify_unknown: }
+ }
+ \exp_args:NNNV
+ \group_end:
+ \tl_put_right:Nn #2 \l__nmc_accum_tl
+ }
+\cs_generate_variant:Nn \__nmc_fpify:nN { V }
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_fpify_unknown:
+ {
+ \tl_if_in:nVTF { )]\}\rbrace } \l__nmc_B_tl
+ { \__nmc_error_what:n { Unmatched~\__nmc_verb:V \l__nmc_B_tl{}~in } }
+ { \__nmc_error_what:n { Unknown~token~\__nmc_verb:V \l__nmc_B_tl{}~in } }
+ }
+\cs_new_protected:Npn \__nmc_insert_aster:
+ {
+ \bool_if:NT \l__nmc_insert_aster_bool
+ { \tl_put_right:Nn \l__nmc_accum_tl { * } }
+ }
+\cs_new_protected:Npn \__nmc_fpify_dec:
+ {
+ \__nmc_get_dec:NN \l__nmc_fact_arg_tl \c_false_bool
+ \tl_if_eq:VnF \l__nmc_C_tl { ! }
+ {
+ \__nmc_insert_aster:
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_fact_arg_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_var:
+ { % ( ) in case var < 0 raised to power
+ \prop_get:NVNTF \g__nmc_subst_var_prop \l__nmc_B_tl \l__nmc_subst_tl
+ {
+ \tl_set_eq:NN \l__nmc_fact_arg_tl \l__nmc_subst_tl
+ \tl_if_in:noTF { !' } \l__nmc_C_tl
+ {
+ % \tl_if_eq:VnT \l__nmc_C_tl { ' }
+ % { \__nmc_deriv_dot_from_prime: }
+ }
+ {
+ \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl \l__nmc_subst_tl {}
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+ }
+ { \__nmc_fpify_unknown: }
+ }
+\cs_new_protected:Npn \__nmc_fpify_const:
+ { % if const used as var
+ \prop_get:NVNTF \g__nmc_subst_var_prop \l__nmc_B_tl \l__nmc_subst_tl
+ { \__nmc_fpify_var: }
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l__nmc_subst_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ { \tl_clear:N \l__nmc_fact_arg_tl }
+ { \tl_put_right:NV \l__nmc_accum_tl \l__nmc_subst_tl }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_arith:
+ {
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_B_tl
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_slash:
+ {
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_B_tl
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c__nmc_uny_int }
+ \__nmc_accum_fpify_parenth:NN \l__nmc_accum_tl \l__nmc_arg_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_arith_alt:
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l__nmc_B_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_B_tl
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_comparison:
+ {
+ \tl_if_in:NVTF \c__nmc_comparisons_tl \l__nmc_C_tl
+ { \__nmc_error_what:n { Multi-token~comparison~in } }
+ {
+ \prop_get:NVNF \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l_tmpa_tl
+ { \tl_set_eq:NN \l_tmpa_tl \l__nmc_B_tl } % < = >
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c__nmc_cmp_int }
+ \tl_clear:N \l_tmpb_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpb_tl
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl \l_tmpb_tl { - }
+ \tl_set:Nx \l__nmc_accum_tl
+ { round( \l__nmc_accum_tl, \int_use:N \l__nmc_round_int ) }
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+ \tl_put_right:Nn \l__nmc_accum_tl { 0 }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_andor:
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l_tmpa_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+% \tl_show:N \l__nmc_A_tl
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c__nmc_and_int }
+% \tl_show:N \l__nmc_arg_tl
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+ \bool_set_false:N \l__nmc_insert_aster_bool
+
+ }
+\cs_new_protected:Npn \__nmc_fpify_lparen:
+ {
+ \__nmc_get_arg_L:VN \l__nmc_B_tl \l__nmc_arg_tl
+ \tl_set_eq:NN \l__nmc_fact_arg_tl \l__nmc_arg_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ { \bool_set_false:N \l__nmc_insert_aster_bool }
+ {
+ \__nmc_accum_fpify_parenth:NN \l__nmc_accum_tl \l__nmc_arg_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_lvert:
+ {
+ \tl_if_eq:VnTF \l__nmc_B_tl { | }
+ {
+ \tl_set:Nn \l__nmc_L_tl { \lvert }
+ \__nmc_absval_arg:N \l__nmc_arg_tl
+ \tl_set:Nn \l__nmc_R_tl { \rvert }
+ }
+ {
+ \tl_set_eq:NN \l__nmc_L_tl \l__nmc_B_tl
+ \__nmc_get_arg_L:VN \l__nmc_B_tl \l__nmc_arg_tl
+ }
+ \__nmc_fpify_lvert_aux:
+ }
+\cs_new_protected:Npn \__nmc_fpify_lvert_aux:
+ {
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ {
+ \tl_set:NV \l__nmc_fact_arg_tl \l__nmc_L_tl
+ \tl_put_right:NV \l__nmc_fact_arg_tl \l__nmc_arg_tl
+ \tl_put_right:NV \l__nmc_fact_arg_tl \l__nmc_R_tl
+ }
+ {
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ \str_case:onF \l__nmc_L_tl
+ {
+ { \lvert } { \tl_put_right:Nn \l__nmc_accum_tl { abs( } }
+ { \lceil } { \tl_put_right:Nn \l__nmc_accum_tl { ceil( } }
+ { \lfloor } { \tl_put_right:Nn \l__nmc_accum_tl { floor( } }
+ { | } { \tl_put_right:Nn \l__nmc_accum_tl { abs( } }
+ { / } { \tl_put_right:Nn \l__nmc_accum_tl { / } }
+ }
+ { \tl_put_right:Nn \l__nmc_accum_tl { ( } }
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+ \tl_if_eq:VnF \l__nmc_R_tl { . }
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { ) }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_lmod:
+ {
+ \tl_set_eq:NN \l__nmc_L_tl \l__nmc_B_tl
+ \prop_get:NVN \g__nmc_subst_misc_prop \l__nmc_L_tl \l__nmc_R_tl
+ \str_case:on \l__nmc_C_tl
+ {
+ { | } { \__nmc_fpify_lmod_abs:V \l__nmc_L_tl }
+ { . } { \__nmc_fpify_lmod_aux:V \l__nmc_L_tl }
+ { / }
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { / }
+ \__nmc_fpify_lmod_aux:V \l__nmc_L_tl
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_lmod_abs:n #1
+ {
+ \__nmc_next:
+ \tl_put_right:Nn \l__nmc_accum_tl { abs( }
+ \exp_args:NNNV\__nmc_get_arg_LR:NNN { #1 } \l__nmc_arg_tl \l__nmc_R_tl
+ \__nmc_next:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+ \tl_put_right:Nn \l__nmc_accum_tl { ) }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_generate_variant:Nn \__nmc_fpify_lmod_abs:n { V }
+
+\cs_new_protected:Npn \__nmc_fpify_lmod_aux:n #1
+ {
+ \__nmc_next:
+ \exp_args:NNNV\__nmc_get_arg_LR:NNN #1 \l__nmc_arg_tl \l__nmc_R_tl
+ \tl_if_eq:VnT \l__nmc_C_tl { . }
+ { \__nmc_next: }
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_generate_variant:Nn \__nmc_fpify_lmod_aux:n { V }
+
+\cs_new_protected:Npn \__nmc_fpify_rmod:
+ {
+ \tl_if_eq:VnT \l__nmc_C_tl { . }
+ { \__nmc_next: }
+ }
+\cs_new_protected:Npn \__nmc_fpify_unary:n #1
+ {
+ \tl_set:NV \l__nmc_fn_tl \l__nmc_B_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { ^ }
+ { \__nmc_fpify_unary_superscript: }
+ { \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_fn_tl \l__nmc_fn_tl }
+ \str_if_in:NnTF \l__nmc_fn_tl { h }
+ { \bool_set_true:N \l__nmc_hyperbolic_bool }
+ { \bool_set_false:N \l__nmc_hyperbolic_bool }
+ % get fn arg
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { #1 }
+ \tl_clear:N \l__nmc_fn_arg_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l__nmc_fn_arg_tl
+ \bool_if:NT \l__nmc_hyperbolic_bool
+ {
+ \tl_set:Nx \l__nmc_fn_tl
+ { \__nmc_fpify_unary_hyperbolic:N \l__nmc_fn_arg_tl }
+ }
+ % append fn
+ \bool_if:NTF \l__nmc_hyperbolic_bool
+ { \tl_put_right:NV \l__nmc_accum_tl \l__nmc_fn_tl }
+ {
+ \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl
+ \l__nmc_fn_arg_tl \l__nmc_fn_tl
+ }
+ \bool_if:NT \l__nmc_superscript_bool
+ { % e.g. \sin^{2}
+ \tl_put_right:Nn \l__nmc_accum_tl { ^ }
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_power_arg_tl
+ \bool_set_false:N \l__nmc_superscript_bool
+ }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_unary_superscript:
+ {
+ \__nmc_next:
+ \tl_clear:N \l__nmc_power_arg_tl
+ \__nmc_fpify_next_braced:N \l__nmc_power_arg_tl
+ \tl_if_eq:VnTF \l__nmc_power_arg_tl { (-1) }
+ { % inverse trig/hyp. function?
+ \prop_get:NVNF \g__nmc_subst_misc_prop \l__nmc_fn_tl \l__nmc_fn_tl
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop
+ \l__nmc_fn_tl \l__nmc_subst_tl
+ \tl_concat:NNN \l__nmc_fn_tl a \l__nmc_subst_tl
+ }
+ }
+ { % not inverse fn
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_fn_tl \l__nmc_fn_tl
+ \bool_set_true:N \l__nmc_superscript_bool
+ }
+ }
+\cs_new:Npn \__nmc_fpify_unary_hyperbolic:N #1
+ {
+ \str_case:on \l__nmc_fn_tl
+ {
+ { sinh } { ( 0.5( exp(#1)- exp(-(#1)) ) ) }
+ { cosh } { ( 0.5( exp(#1) + exp(-(#1)) ) ) }
+ { tanh } { ( (exp(2(#1))-1)/(exp(2(#1))+1) ) }
+ { asinh } { ln( #1 + sqrt((#1)^2+1) ) }
+ { acosh } { ln( #1 + sqrt((#1)^2-1) ) }
+ { atanh } { ( 0.5 ln( (1+(#1))/(1-(#1)) ) ) }
+ { csch } { 1/( 0.5( exp(#1)- exp(-(#1)) ) ) }
+ { sech } { 1/( 0.5( exp(#1) + exp(-(#1)) ) ) }
+ { coth } { ( (exp(2(#1))+1)/(exp(2(#1))-1) ) }
+ { acsch } { ( ln(1/(#1)+sqrt(1/(#1)^2+1)) ) }
+ { asech } { ( ln(1/(#1)+sqrt(1/(#1)^2-1)) ) }
+ { acoth } { ( 0.5 ln( ((#1)+1)/((#1)-1) ) ) }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_power:
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { ^ }
+ \__nmc_fpify_next_braced:N \l__nmc_accum_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_frac:
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { ( }
+ \__nmc_fpify_next_braced:N \l__nmc_accum_tl
+ \tl_put_right:Nn \l__nmc_accum_tl { / }
+ \__nmc_fpify_next_braced:N \l__nmc_accum_tl
+ \tl_put_right:Nn \l__nmc_accum_tl { ) }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_unarybrace:
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l__nmc_subst_tl
+ \tl_if_eq:VnT \l__nmc_C_tl { * }
+ { \__nmc_next: }
+ \tl_if_eq:VnTF \l__nmc_C_tl { [ }
+ { % n-th root
+ \tl_if_eq:VnTF \l__nmc_subst_tl { sqrt }
+ {
+ \__nmc_next:
+ \__nmc_fpify_ubrace_n:
+ }
+ {
+ \__nmc_next: \__nmc_next: \__nmc_next:
+ \__nmc_fpify_ubrace_aux:
+ }
+ }
+ { \__nmc_fpify_ubrace_aux: }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_ubrace_aux:
+ {
+ \tl_set_eq:NN \l_tmpa_tl \l__nmc_subst_tl
+ \__nmc_fpify_next_braced:N \l_tmpa_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ { \__nmc_fpify_ubrace_fact: }
+ { \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl }
+ }
+\cs_new_protected:Npn \__nmc_fpify_ubrace_n:
+ {
+ \__nmc_error_where:n { [~]~argument~of~\__nmc_verb:n { \sqrt } }
+ \tl_clear:N \l__nmc_arg_tl
+ \tl_clear:N \l__nmc_ubrace_n_tl
+ \__nmc_get_arg_LR:NNN [ \l__nmc_arg_tl ]
+ \__nmc_fpify:VN \l__nmc_arg_tl \l__nmc_ubrace_n_tl
+ \__nmc_intify:Nn \l__nmc_ubrace_n_tl { 1 }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ \__nmc_next:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_B_tl \l_tmpa_tl
+ \bool_if:nTF
+ {
+ \fp_compare_p:nNn { \l_tmpa_tl } < { 0 }
+ && \int_if_even_p:n { \l__nmc_ubrace_n_tl }
+ }
+ { \flag_raise:n { fp_invalid_operation } \__nmc_error_fpflag: }
+ {
+ \tl_put_right:Nx \l__nmc_accum_tl
+ { \__nmc_fpify_ubrace_n_aux:VV {\l_tmpa_tl}{ \l__nmc_ubrace_n_tl }}
+ }
+ }
+ }
+\cs_new:Npn \__nmc_fpify_ubrace_n_aux:nn #1#2
+ { sign( #1 )( abs( #1 ) )^( 1/( #2 ) ) }
+\cs_generate_variant:Nn \__nmc_fpify_ubrace_n_aux:nn { VV }
+
+\cs_new_protected:Npn \__nmc_fpify_ubrace_fact:
+ {
+ \tl_if_eq:VnF \l__nmc_subst_tl { sqrt }
+ {
+ \str_case:onT \l__nmc_subst_tl
+ {
+ { abs } { \tl_set:Nn \l__nmc_fact_arg_tl { \abs } }
+ { ceil } { \tl_set:Nn \l__nmc_fact_arg_tl { \ceil } }
+ { floor} { \tl_set:Nn \l__nmc_fact_arg_tl { \floor } }
+ }
+ {
+ \tl_put_right:Nx \l__nmc_fact_arg_tl
+ { { \exp_not:o \l__nmc_B_tl } }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_unarysub:
+ { % e.g. \log_{10}
+ \str_if_eq:VnTF \l__nmc_C_tl { _ }
+ {
+ \__nmc_next: \__nmc_next:
+ \__nmc_fpify_unarysub:N \l__nmc_B_tl
+ }
+ { \__nmc_fpify_unarysub:N \l__nmc_log_base_tl }
+ }
+\cs_new_protected:Npn \__nmc_fpify_unarysub:N #1
+ {
+ \__nmc_fpify:VN #1 \l__nmc_arg_tl
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \fp_compare:nF { 0 < \l__nmc_arg_tl != 1 }
+ {
+ \__nmc_error_what:n
+ { Valid~base~required~for~\__nmc_verb:n { \log } in }
+ }
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \tl_set:Nx \l_tmpa_tl
+ { \fp_eval:n { 1 / ( ln ( \l__nmc_arg_tl ) ) } }
+ \__nmc_error_fpflag:
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl \l_tmpa_tl {}
+ \tl_put_left:Nn \l__nmc_A_tl { \ln }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_surd:
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l__nmc_subst_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_subst_tl
+ \tl_clear:N \l__nmc_arg_tl
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c__nmc_srd_int }
+ \__nmc_parenth:N \l__nmc_arg_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l__nmc_accum_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_fact:
+ {
+ \__nmc_error_where:n { factorial }
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_fact_arg_tl \l_tmpa_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ { \__nmc_fpify_fact_repeated: }
+ {
+ \__nmc_intify:Nn \l_tmpa_tl { 0 }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl \l_tmpa_tl { fact }
+ }
+ }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_fact_repeated:
+ {
+ \__nmc_intify:Nn \l_tmpa_tl { -1 }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ \__nmc_fpify_double_fact:N \l_tmpa_tl
+ \tl_set:Nx \l_tmpa_tl { \fp_use:N \l__nmc_fact_fp }
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ { \tl_set_eq:NN \l__nmc_fact_arg_tl \l_tmpa_tl }
+ { \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl \l_tmpa_tl {} }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_double_fact:N #1
+ {
+ \__nmc_next:
+ \int_compare:nNnTF { #1 } > { 0 }
+ {
+ \int_if_even:nTF { #1 }
+ { \__nmc_fpify_fact_do:nNn { 2 } #1 { 2 } }
+ { \__nmc_fpify_fact_do:nNn { 1 } #1 { 2 } }
+ }
+ { \fp_set:Nn \l__nmc_fact_fp { 1 } }
+ }
+% multiply nos from #1 to #2, put in \l__nmc_fact_fp
+% #3 = 1 (!), = 2 (!!)
+\cs_new_protected:Npn \__nmc_fpify_fact_do:nNn #1#2#3
+ {
+ \fp_set:Nn \l_tmpa_fp { #1 }
+ \fp_set:Nn \l_tmpb_fp { #2 }
+ \int_set:Nn \l_tmpb_int { #3 }
+ \fp_set_eq:NN \l__nmc_fact_fp \l_tmpa_fp
+ \fp_while_do:nNnn { \l_tmpa_fp } < { \l_tmpb_fp }
+ {
+ \fp_add:Nn \l_tmpa_fp { \l_tmpb_int }
+ \fp_set:Nn \l__nmc_fact_fp { \l__nmc_fact_fp * \l_tmpa_fp }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_nary:
+ {
+ \bool_set_true:N \l__nmc_comma_nary_bool
+ \str_case:on \l__nmc_B_tl
+ {
+ { \gcd } { \__nmc_fpify_nary_gcd: }
+ { \max } { \__nmc_fpify_nary_maxmin: }
+ { \min } { \__nmc_fpify_nary_maxmin: }
+ }
+ \bool_set_false:N \l__nmc_comma_nary_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_nary_gcd:
+ {
+ \__nmc_insert_aster:
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c_max_int }
+ \tl_set:Nx \l__nmc_arg_tl { \tl_range:Nnn \l__nmc_arg_tl { 2 } { -2 } }
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ \seq_set_split:NnV \l_tmpa_seq {,} \l_tmpa_tl
+ % eval. 1st expr.
+ \seq_pop:NN \l_tmpa_seq \l_tmpb_tl
+ \int_set:Nn \l_tmpb_int { \fp_to_int:n \l_tmpb_tl }
+ \seq_map_inline:Nn \l_tmpa_seq
+ {
+ \int_set:Nn \l_tmpd_int { \fp_to_int:n { ##1 } }
+ \int_set:Nn \l_tmpa_int { \int_max:nn { \l_tmpd_int } { \l_tmpb_int } }
+ \int_set:Nn \l_tmpb_int { \int_min:nn { \l_tmpd_int } { \l_tmpb_int } }
+ \int_set:Nn \l_tmpc_int { \int_mod:nn { \l_tmpa_int } { \l_tmpb_int } }
+
+ \int_until_do:nNnn { \l_tmpc_int } = { 0 }
+ {
+ \int_set_eq:NN \l_tmpa_int \l_tmpb_int
+ \int_set_eq:NN \l_tmpb_int \l_tmpc_int
+ \int_set:Nn \l_tmpc_int { \int_mod:nn
+ { \l_tmpa_int } { \l_tmpb_int } }
+ }
+ }
+ \tl_put_right:Nx \l__nmc_accum_tl { \int_use:N \l_tmpb_int }
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_nary_maxmin:
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l__nmc_subst_tl
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c_max_int }
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ % em-brace in case nested
+ \tl_put_right:Nx \l__nmc_accum_tl
+ { { \l__nmc_subst_tl \exp_not:o \l_tmpa_tl } }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_comma:
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { , }
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_binom:
+ {
+ \__nmc_fpify_next_braced:N \l__nmc_binom_top_tl
+ \__nmc_fpify_next_braced:N \l__nmc_binom_bot_tl
+ \__nmc_error_where:n { 2nd~argument~of~\__nmc_verb:n { \binom } }
+ \__nmc_intify:Nn \l__nmc_binom_bot_tl { 0 }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ \__nmc_fpify_binom_calc:NN
+ \l__nmc_binom_top_tl \l__nmc_binom_bot_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_binom_calc:NN #1 #2
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { ( }
+ \int_if_zero:nTF { #2 }
+ { \tl_put_right:Nn \l__nmc_accum_tl { 1 } }
+ { % calc. top
+ \__nmc_fpify_binom_calc_do:nN { #1 - #2 + 1 } #1
+ \tl_put_right:Nn \l__nmc_accum_tl { / }
+ \__nmc_fpify_binom_calc_do:nN { 1 } #2 % calc. bot.
+ }
+ \tl_put_right:Nn \l__nmc_accum_tl { ) }
+ }
+\cs_new_protected:Npn \__nmc_fpify_binom_calc_do:nN #1 #2
+ {
+ \__nmc_fpify_fact_do:nNn { #1 } #2 { 1 }
+ \tl_put_right:Nx \l__nmc_accum_tl { \fp_use:N \l__nmc_fact_fp }
+ }
+\cs_new_protected:Npn \__nmc_fpify_BE:
+ {
+ \__nmc_next:
+ \tl_if_in:VnT \l__nmc_B_tl { alignat }
+ { \__nmc_next: }
+ }
+% \cs { m }
+\cs_new_protected:Npn \__nmc_fpify_absorbm:
+ { \__nmc_next: }
+% \cs { o m }
+\cs_new_protected:Npn \__nmc_fpify_absorbom:
+ {
+ \str_if_eq:VnT \l__nmc_C_tl { [ }
+ {
+ \__nmc_next:
+ \__nmc_get_arg_LR:NNN [ \l__nmc_toss_tl ]
+ }
+ \__nmc_next:
+ }
+\cs_new_protected:Npn \__nmc_fpify_font:
+ { \__nmc_fpify_meta: }
+
+\cs_new_protected:Npn \__nmc_fpify_meta:
+ {
+ \__nmc_next:
+ \__nmc_fpify:VN \l__nmc_B_tl \l__nmc_accum_tl
+ }
+\cs_new_protected:Npn \__nmc_fpify_metamm:
+ { \prg_replicate:nn { 2 } { \__nmc_fpify_meta: } }
+%%%%%%%%%%%%%%%%%%%
+% sum/prod
+\cs_new_protected:Npn \__nmc_fpify_sum:
+ {
+ \__nmc_insert_aster:
+ \group_begin:
+ \tl_if_eq:VnTF \l__nmc_B_tl { \sum }
+ { % sum
+ \tl_set:Nn \l__nmc_sum_op_tl { + }
+ \fp_zero:N \l__nmc_sum_total_fp
+ \fp_set:Nn \l__nmc_sum_prev_fp { 1 }
+ \tl_set:Nn \l__nmc_sum_type_tl { sum }
+ \bool_set_true:N \l__nmc_sum_bool
+ }
+ { % product
+ \tl_set:Nn \l__nmc_sum_op_tl { * }
+ \fp_set:Nn \l__nmc_sum_total_fp { 1 }
+ \fp_set:Nn \l__nmc_sum_prev_fp { 2 }
+ \tl_set:Nn \l__nmc_sum_type_tl { product }
+ \bool_set_false:N \l__nmc_sum_bool
+ }
+ \__nmc_error_where:n { \l__nmc_sum_type_tl }
+
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_error_where:n { \l__nmc_sum_type_tl~\ limits }
+ \__nmc_sum_get_limits:NNN \l__nmc_sum_var_tl
+ \l__nmc_sum_index_int \l__nmc_sum_end_int
+ }
+ \bool_if:NTF \g__nmc_error_bool
+ { \group_end: }
+ { % get the summand/multiplicand
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c__nmc_sum_int }
+ \tl_if_eq:VnF \l__nmc_C_tl { ( }
+ { \__nmc_parenth:N \l__nmc_arg_tl }
+ \tl_set_eq:NN \l__nmc_summand_tl \l__nmc_arg_tl
+ % do the sum/prod; result in \l__nmc_sum_total_fp
+ \__nmc_error_where:n { \l__nmc_sum_type_tl }
+ \__nmc_sum_do:nnN { \l__nmc_sum_index_int }
+ { \l__nmc_sum_end_int } \l__nmc_sum_total_fp
+ \fp_gset_eq:NN \g_tmpa_fp \l__nmc_sum_total_fp
+ \tl_gset_eq:NN \g_tmpa_tl \l__nmc_A_tl
+ \exp_args:NNNV
+ \group_end:
+ \tl_set:Nn \l__nmc_C_tl \l__nmc_C_tl
+ \tl_put_right:Nx \l__nmc_accum_tl { \fp_use:N \g_tmpa_fp }
+ \tl_set_eq:NN \l__nmc_A_tl \g_tmpa_tl
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ }
+ }
+\cs_new_protected:Npn \__nmc_sum_get_limits:NNN #1#2#3
+ { % #1 (tl) sum var #2, #3 (ints) lower, upper limits
+ \str_if_eq:VnTF \l__nmc_C_tl { _ }
+ {
+ \__nmc_sum_limit_lower:NN #1#2
+ \bool_if:NF \g__nmc_error_bool
+ { \__nmc_sum_limit_upper:N #3 }
+ }
+ {
+ \__nmc_sum_limit_upper:N #3
+ \__nmc_sum_limit_lower:NN #1#2
+ }
+ }
+\cs_new_protected:Npn \__nmc_sum_limit_lower:NN #1#2
+ {
+ \__nmc_next: \__nmc_next:
+ \bool_if:NT \l__nmc_multitok_bool
+ { % may not be in vv-list
+ \seq_clear:N \l_tmpc_seq
+ \seq_push:NV \l_tmpc_seq \l__nmc_B_tl
+ \__nmc_prep_multitok:NN \l_tmpc_seq \l__nmc_A_tl
+ \seq_pop:NN \l_tmpc_seq \l__nmc_B_tl
+ }
+ \str_if_in:NnTF \l__nmc_B_tl { = }
+ {
+ \__nmc_vv_get_vars_vals_lims:NN \l__nmc_B_tl \c_empty_prop
+ \tl_set_eq:NN #1 \l__nmc_eq_var_tl
+ \__nmc_sum_limit_assign:NN #2 \l__nmc_vv_fp_expr_tl
+ \seq_if_in:NVF \l__nmc_calc_fn_seq \l__nmc_B_tl
+ { \seq_put_left:NV \l__nmc_calc_fn_seq \l__nmc_B_tl }
+ }
+ {
+ \__nmc_error_what:n
+ { No~\__nmc_verb:V \l__nmc_sum_type_tl\ variable~in }
+ }
+ }
+\cs_new_protected:Npn \__nmc_sum_limit_upper:N #1
+ {
+ \__nmc_next: \__nmc_next:
+ \__nmc_vv_get_vars_vals_lims:NN \l__nmc_B_tl \c_empty_prop
+ \__nmc_sum_limit_assign:NN #1 \l__nmc_vv_fp_expr_tl
+ }
+\cs_new_protected:Npn \__nmc_sum_limit_assign:NN #1#2
+ {
+ \tl_if_in:VnTF #2 { inf }
+ { \int_set:Nn #1 { \fp_sign:n { #2 } * \c_max_int } }
+ {
+ \fp_compare:nNnTF { abs(#2) } > { \c_max_int }
+ { \int_set:Nn #1 { \fp_sign:n { #2 } * \c_max_int } }
+ {
+ \__nmc_intify:Nn #2 { -inf }
+ \bool_if:NF \g__nmc_error_bool
+ { \int_set:Nn #1 { #2 } }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_sum_do:nnN #1#2#3
+ { % #1,#2 lower, upper limits (int), #3 <-- total (fp)
+ \int_compare:nNnTF { \c_max_int } =
+ { \int_max:nn { \int_abs:n { #1 } } { \int_abs:n { #2 } } }
+ {
+ \bool_if:NTF \l__nmc_sum_bool
+ { \__nmc_sum_do_infinite:Nn #1 { sum } }
+ { \__nmc_sum_do_infinite:Nn #1 { prod } }
+ }
+ { \int_step_function:nnnN { #1 } { 1 } { #2 } \__nmc_sum_eval:n }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_compare:nTF { #1 <= #2 }
+ { \fp_set:Nn #3 { \l__nmc_sum_total_fp } }
+ {
+ \tl_if_eq:VnTF \l__nmc_sum_op_tl { * }
+ { \fp_set:Nn #3 { 1 } }
+ { \fp_set:Nn #3 { 0 } }
+ }
+ }
+ }
+\cs_set:Npn \__nmc_sum_do_infinite:Nn #1#2
+ {
+ \int_set:Nn \l__nmc_sum_round_int
+ { \l__nmc_round_int + \use:c { l__nmc_#2_extra_int } }
+ \int_set:Nn \l__nmc_suma_int { #1 - 1 }
+ % check loop
+ \bool_do_while:nn { \l__nmc_sum_cont_bool }
+ { % main loop
+ \bool_set_false:N \l__nmc_sum_cont_bool
+ \fp_until_do:nNnn { \l__nmc_sum_prev_fp }
+ = { \l__nmc_sum_rounded_fp }
+ {
+ \int_incr:N \l__nmc_suma_int
+ \fp_set_eq:NN \l__nmc_sum_prev_fp \l__nmc_sum_rounded_fp
+ \__nmc_sum_eval:n { \l__nmc_suma_int }
+ \fp_set:Nn \l__nmc_sum_rounded_fp
+ { round( \l__nmc_sum_total_fp, \l__nmc_sum_round_int ) }
+ }
+ % query terms
+ \int_set:Nn \l__nmc_sumb_int
+ { \l__nmc_suma_int + \use:c { l__nmc_#2_query_int } }
+ \int_while_do:nNnn { \l__nmc_suma_int } < { \l__nmc_sumb_int }
+ {
+ \int_incr:N \l__nmc_suma_int
+ \__nmc_sum_query:n { \l__nmc_suma_int }
+ }
+ }
+ \tl_gset:cx { g__nmc_ \tl_range:Nnn \l__nmc_sum_type_tl{1}{4} _info_tl }
+ { \int_eval:n { \l__nmc_suma_int - #1
+ - \use:c { l__nmc_#2_query_int } } }
+ }
+\cs_new_protected:Npn \__nmc_sum_eval:n #1
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_sum_var_tl \l__nmc_summand_tl
+ { #1 } \l_tmpb_fp
+ \fp_set:Nn \l__nmc_sum_total_fp
+ { \l__nmc_sum_total_fp \l__nmc_sum_op_tl \l_tmpb_fp }
+ }
+\cs_new_protected:Npn \__nmc_sum_query:n #1
+ {
+ \__nmc_sum_eval:n { #1 }
+ \fp_set:Nn \l__nmc_sum_rounded_fp
+ { round( \l__nmc_sum_total_fp, \l__nmc_sum_round_int ) }
+ \fp_compare:nNnF { \l__nmc_sum_prev_fp } = { \l__nmc_sum_rounded_fp }
+ {
+ \bool_set_true:N \l__nmc_sum_cont_bool
+ \int_set_eq:NN \l__nmc_sumb_int \l__nmc_suma_int
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_cmd:
+ {
+ \__nmc_insert_aster:
+ \str_case:onT \l__nmc_B_tl
+ {
+ { \eval } { \__nmc_cmd:n { eval } }
+ { \iter } { \__nmc_cmd:n { iter } }
+ { \solve } { \__nmc_cmd:n { solve } }
+ { \recur } { \__nmc_cmd:n { recur } }
+ { \tabulate } { \__nmc_cmd:n { table } }
+ { \nmcInfo } { \nmcInfo }
+ { \nmcEvaluate } { \__nmc_cmd:n { eval } }
+ { \nmcIterate } { \__nmc_cmd:n { iter } }
+ { \nmcSolve } { \__nmc_cmd:n { solve } }
+ { \nmcRecur } { \__nmc_cmd:n { recur } }
+ { \nmcTabulate } { \__nmc_cmd:n { table } }
+ }
+ {
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ \bool_set_false:N \l__nmc_wraps_math_bool
+ }
+ }
+\cs_new_protected:Npn \__nmc_cmd:n #1
+ { % \cm { s O m O O }
+ \tl_set:Nn \l__nmc_delim_arg_tl { { #1 } } % cmd
+ \tl_if_eq:VnT \l__nmc_C_tl { * }
+ { \__nmc_next: }
+ \__nmc_append_sqbrarg: % settings
+ \__nmc_next:
+ \tl_put_right:Nx \l__nmc_delim_arg_tl { { \exp_not:o \l__nmc_B_tl } }
+ \__nmc_append_sqbrarg: % vv-list
+ \__nmc_append_sqbrarg: % num format
+ \tl_put_right:Nn \l__nmc_accum_tl { ( }
+ \group_begin:
+ \bool_set_true:N \l__nmc_num_only_bool
+ \exp_last_unbraced:No \__nmc_get_inputs:nnnnn \l__nmc_delim_arg_tl
+ \bool_set_false:N \l__nmc_sci_num_out_bool
+ \bool_if:NF \g__nmc_error_bool
+ { \use:c { __nmc_#1_process: } }
+ \exp_args:NNNV
+ \group_end:
+ \tl_put_right:Nn \l__nmc_accum_tl { \l_nmc_result_tl ) }
+ }
+\cs_new_protected:Npn \__nmc_append_sqbrarg:
+ {
+ \tl_if_eq:VnTF \l__nmc_C_tl { [ }
+ { \__nmc_append_sqbrarg_aux: }
+ { \tl_put_right:Nx \l__nmc_delim_arg_tl { {} } }
+ }
+\cs_new_protected:Npn \__nmc_append_sqbrarg_aux:
+ {
+ \__nmc_next:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_get_arg_LR:NNN [ \l_tmpa_tl ]
+ \tl_put_right:Nx \l__nmc_delim_arg_tl { { \exp_not:o \l_tmpa_tl } }
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \nmc_delim_arg:Nnn
+\tl_new:N \l__nmc_delim_arg_tl
+\tl_new:N \l__nmc_delim_argi_tl
+\int_new:N \l__nmc_delim_pas_int
+\int_new:N \l__nmc_delim_pos_int
+\bool_new:N \l__nmc_delim_stop_bool
+\bool_new:N \l__nmc_delim_lmod_bool
+
+\prg_new_conditional:Npnn \__nmc_if_pospas_gtr:nn #1#2 { p, TF }
+ {
+ \bool_if:nTF
+ {
+ \int_compare_p:nNn { \l__nmc_delim_pos_int } > { #1 }
+ &&
+ \int_compare_p:nNn { \l__nmc_delim_pas_int } > { #2 }
+ }
+ { \prg_return_true: }
+ { \prg_return_false: }
+ }
+%--------------------------------------------------------------------
+% #1(tlvar) receives arg (\l__nmc_delim_arg_tl)
+% #2 (int) initial pos
+% #3 (int) parsing state
+\cs_new_protected:Npn \__nmc_delim_arg:Nnn #1#2#3
+ {
+ \group_begin:
+ \bool_set_false:N \__nmc_delim_stop_bool
+ \bool_set_false:N \l__nmc_delim_lmod_bool
+ \int_set:Nn \l__nmc_delim_pos_int { #2 }
+ \int_set:Nn \l__nmc_delim_pas_int { #3 }
+ \tl_clear:N \l__nmc_delim_arg_tl
+ \bool_until_do:nn
+ {
+ \l__nmc_delim_stop_bool ||
+ \quark_if_nil_p:N \l__nmc_C_tl
+ }
+ {
+ \prop_get:NVNTF \g__nmc_class_prop \l__nmc_C_tl \l__nmc_class_tl
+ { \exp_last_unbraced:NV \use_ii:nn \l__nmc_class_tl }
+ { \__nmc_delim_var: } % unknown; treat as var
+ \int_incr:N \l__nmc_delim_pos_int
+ }
+ \tl_gset_eq:NN \g_tmpa_tl \l__nmc_A_tl
+ \tl_gset_eq:NN \g_tmpb_tl \l__nmc_delim_arg_tl
+ \exp_args:NNNV
+ \group_end:
+ \tl_set:Nn \l__nmc_C_tl \l__nmc_C_tl
+ \tl_set_eq:NN \l__nmc_A_tl \g_tmpa_tl
+ \tl_set_eq:NN #1 \g_tmpb_tl
+ }
+%%%%%%%%% delimit arg routines %%%%%%%%
+\cs_new_protected:Npn \__nmc_delim_stop:
+ {
+ \tl_if_eq:VnTF \l__nmc_C_tl { \q }
+ { \bool_set_false:N \l__nmc_delim_stop_bool }
+ { \bool_set_true:N \l__nmc_delim_stop_bool }
+ }
+\cs_new_protected:Npn \__nmc_delim_stop_if_gtr:n #1
+ {
+ \int_compare:nNnT { \l__nmc_delim_pas_int } > { #1 }
+ { \__nmc_delim_stop: }
+ }
+\cs_new_protected:Npn \__nmc_delim_append:
+ {
+ \__nmc_next:
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_B_tl
+ }
+\cs_new_protected:Npn \__nmc_delim_append_braced:
+ {
+ \__nmc_next:
+ \tl_put_right:Nx \l__nmc_delim_arg_tl { { \exp_not:o\l__nmc_B_tl } }
+ }
+\cs_new_protected:Npn \__nmc_delim_append_twobraced:
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_append_braced:
+ }
+\cs_new_protected:Npn \__nmc_delim_append_sqbra:
+ {
+ \__nmc_next:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_get_arg_LR:NNN [ \l_tmpa_tl ]
+ \__nmc_put_right_wrap:NnNn \l__nmc_delim_arg_tl
+ { [ } \l_tmpa_tl { ] }
+ }
+\cs_new_protected:Npn \__nmc_delim_dec:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_uny_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next:
+ \__nmc_get_dec:NN \l_tmpa_tl \c_true_bool
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l_tmpa_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_var:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_uny_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append: }
+ }
+\cs_new_protected:Npn \__nmc_delim_const:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_uny_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append: }
+ }
+\cs_new_protected:Npn \__nmc_delim_arith:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append: }
+ }
+\cs_new_protected:Npn \__nmc_delim_pm:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_cmp_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append: }
+ }
+\cs_new_protected:Npn \__nmc_delim_comparison:
+ {
+ \int_compare:nNnT { \l__nmc_delim_pas_int } = { \c__nmc_cmp_int }
+ {
+ \tl_put_left:NV \l__nmc_A_tl \l__nmc_delim_arg_tl
+ \tl_put_left:Nn \l__nmc_A_tl { \land }
+ \tl_put_left:Nn \l__nmc_C_tl { \land }
+ }
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_and_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 1 } { \c__nmc_cmp_int }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_andor:
+ { \__nmc_delim_stop: }
+
+\cs_new_protected:Npn \__nmc_delim_lparen:
+ {
+ \bool_lazy_or:nnTF { \__nmc_if_pospas_gtr_p:nn { 1 } { \c__nmc_trg_int } }
+ { \__nmc_if_pospas_gtr_p:nn { \l__nmc_posi_int } { \c__nmc_sum_int } }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next:
+ \str_case:on \l__nmc_B_tl
+ {
+ { ( } { \__nmc_delim_lparen:nn { ( } { ) } }
+ { [ } { \__nmc_delim_lparen:nn { [ } { ] } }
+ { \{ } { \__nmc_delim_lparen:nn {\{ } {\} } }
+ { \lparen } { \__nmc_delim_lparen:nn { \lparen } { \rparen } }
+ { \lbrack } { \__nmc_delim_lparen:nn { \lbrack } { \rbrack } }
+ { \lbrace } { \__nmc_delim_lparen:nn { \lbrace } { \rbrace } }
+ }
+ \__nmc_delim_lparen_auxi:n { \l__nmc_posi_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_lparen:nn #1#2
+ {
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { ( }
+ \__nmc_get_arg_LR:NNN #1 \l__nmc_delim_arg_tl #2
+ \bool_if:NT \l__nmc_delim_lmod_bool
+ {
+ \tl_set:Nx \l__nmc_delim_arg_tl
+ { \tl_range:Nnn \l__nmc_delim_arg_tl { 1 } { -2 } }
+ }
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { ) }
+ \int_incr:N \l__nmc_delim_pos_int
+ }
+\cs_generate_variant:Nn \__nmc_delim_lparen:nn { VV }
+
+\cs_new_protected:Npn \__nmc_delim_lparen_auxi:n #1
+ { % #1 is first pos.
+ \int_compare:nNnTF { \l__nmc_delim_pas_int } = { \c__nmc_trg_int }
+ { % +1 to prevent ( )( )
+ \__nmc_delim_arg:Nnn
+ \l__nmc_delim_argi_tl { #1 + 1 } { \c__nmc_trg_int }
+ }
+ {
+ \int_compare:nNnT { \l__nmc_delim_pas_int } = { \c__nmc_uny_int }
+ {
+ \__nmc_delim_arg:Nnn
+ \l__nmc_delim_argi_tl { 2 } { \c__nmc_prn_int }
+ }
+ }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+\cs_new_protected:Npn \__nmc_delim_rparen:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_cmp_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next:
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_B_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_lvert:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next:
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_B_tl
+ \str_case:on \l__nmc_B_tl
+ {
+ { \lvert } { \__nmc_delim_lvert:nn { \lvert } { \rvert } }
+ { \lceil } { \__nmc_delim_lvert:nn { \lceil } { \rceil } }
+ { \lfloor } { \__nmc_delim_lvert:nn { \lfloor } { \rfloor } }
+ { | }
+ {
+ \__nmc_absval_arg:N \l__nmc_delim_arg_tl
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { | }
+ }
+ }
+ \__nmc_delim_lparen_auxi:n { 0 }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_lvert:nn #1#2
+ {
+ \__nmc_delim_lparen:nn { #1 } { #2 }
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { #2 }
+ }
+\cs_new_protected:Npn \__nmc_delim_lmod:
+ {
+ \bool_lazy_and:nnT { \int_compare_p:nNn { \l__nmc_posi_int } < { 3 } }
+ { \int_compare_p:nNn { \l__nmc_delim_pas_int } = { \l__nmc_trg_int } }
+ { \int_incr:N \l__nmc_delim_pas_int }
+ \__nmc_if_pospas_gtr:nnTF { \l__nmc_posi_int } { \l__nmc_trg_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next:
+ \tl_if_in:NVTF \c__nmc_modds_tl \l__nmc_C_tl
+ {
+ \prop_get:NVN \g__nmc_subst_misc_prop \l__nmc_B_tl \l_tmpb_tl
+ \__nmc_delim_lparen:VV \l__nmc_B_tl \l_tmpb_tl
+ \__nmc_next:
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_B_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { | }
+ { \__nmc_delim_lparen_auxi:n { 0 } }
+ {
+ \__nmc_next:
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_B_tl
+ \__nmc_delim_stop:
+ }
+ }
+ {
+ \int_decr:N \l__nmc_delim_pos_int
+ \bool_set_true:N \l__nmc_delim_lmod_bool
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_unary:n #1
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \tl_if_eq:VnT \l__nmc_C_tl { ^ }
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ }
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 1 } { #1 }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_power:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_prn_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_tfrac:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_uny_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append_twobraced: }
+ }
+\cs_new_protected:Npn \__nmc_delim_frac: % also \(d)binom
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append_twobraced:
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_unarybrace:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \str_case:on \l__nmc_C_tl
+ {
+ { * } { \__nmc_delim_append: }
+ { [ } { \__nmc_delim_append_sqbra: }
+ }
+ \__nmc_delim_append_braced:
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 2 }
+ { \c__nmc_prn_int }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_log:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 1 } { \c__nmc_uny_int }
+ \__nmc_accum_fn_parenth:NNn \l__nmc_delim_arg_tl
+ \l__nmc_delim_argi_tl {}
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_surd:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_uny_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \tl_clear:N \l__nmc_delim_argi_tl
+ \tl_if_eq:VnT \l__nmc_B_tl { \prod }
+ {
+ \prg_replicate:nn { 2 }
+ { \__nmc_delim_append: \__nmc_delim_append_braced: }
+ }
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 1 } { \c__nmc_srd_int }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_fact:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_prn_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \tl_if_eq:VnT \l__nmc_C_tl { ! }
+ { \__nmc_delim_append: }
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_qleave:
+ {
+ \__nmc_next:
+ \tl_if_eq:VnTF \l__nmc_B_tl { \q }
+ { \int_zero:N \l__nmc_delim_pos_int } % adhere
+ { \bool_set_true:N \l__nmc_delim_stop_bool } % sever
+ \tl_if_eq:VnT \l__nmc_C_tl { ' }
+ { \__nmc_next: }
+ }
+\cs_new_protected:Npn \__nmc_delim_nary:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \__nmc_get_arg_LR:NNN ( \l__nmc_B_tl )
+ \__nmc_put_right_wrap:NnNn \l__nmc_delim_arg_tl
+ { ( } \l__nmc_B_tl { ) }
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_comma:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_cmp_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append: }
+ }
+\cs_new_protected:Npn \__nmc_delim_sum:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 1 } { \c__nmc_sum_int }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_cmp_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_absorb:
+ { % \cs
+ \__nmc_next:
+ \int_decr:N \l__nmc_delim_pos_int
+ }
+\cs_new_protected:Npn \__nmc_delim_absorbm:
+ { % \cs { m }
+ \__nmc_next: \__nmc_next:
+ \int_decr:N \l__nmc_delim_pos_int
+ }
+\cs_new_protected:Npn \__nmc_delim_absorbom:
+ { % \cs { o m }
+ \__nmc_next:
+ \str_if_eq:VnT \l__nmc_C_tl { [ }
+ {
+ \__nmc_next:
+ \__nmc_get_arg_LR:NNN [ \l__nmc_toss_tl ]
+ }
+ \__nmc_next:
+ \int_decr:N \l__nmc_delim_pos_int
+ }
+\cs_new_protected:Npn \__nmc_delim_BE:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_cmp_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next: \__nmc_next:
+ \str_if_in:VnTF \l__nmc_B_tl { alignat }
+ { \__nmc_next: }
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { \c_group_begin_token }
+ \__nmc_get_arg_LR:NNN \begin \l__nmc_delim_arg_tl \end
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { \c_group_end_token }
+ \__nmc_next:
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_font:
+ {
+ \__nmc_next:
+ \tl_put_left:NV \l__nmc_A_tl \l__nmc_B_tl
+ \int_decr:N \l__nmc_delim_pos_int
+ }
+\cs_new_protected:Npn \__nmc_delim_metamm:
+ {
+ \__nmc_next:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_append_braced:
+ \int_decr:N \l__nmc_delim_pos_int
+ }
+\cs_new_protected:Npn \__nmc_delim_cmd:
+ { % \cmd { s O m O O }
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_C_tl
+ \__nmc_next:
+ \tl_if_eq:VnT \l__nmc_C_tl { * }
+ { \__nmc_next: }
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { * }
+ \__nmc_delim_nmcfn_aux:NN [ ]
+ \__nmc_next:
+ \tl_put_right:Nx
+ \l__nmc_delim_arg_tl { { \exp_not:o \l__nmc_B_tl } }
+ \__nmc_delim_nmcfn_aux:NN [ ]
+ \__nmc_delim_nmcfn_aux:NN [ ]
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_nmcfn_aux:NN #1#2
+ {
+ \tl_if_eq:VnT \l__nmc_C_tl { [ }
+ {
+ \__nmc_next:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_get_arg_LR:NNN #1 \l_tmpa_tl ]
+ \__nmc_put_right_wrap:NnNn \l__nmc_delim_arg_tl
+ { #1 } \l_tmpa_tl { #2 }
+ }
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% save & retrieve
+\tl_new:N \g__nmc_reuse_tl
+\tl_new:N \g__nmc_reuse_filename_tl
+\tl_new:N \l__nmc_reuse_retrieved_tl
+\bool_new:N \l__nmc_reuse_retrieved_bool
+\bool_new:N \g__nmc_lyx_bool
+\ior_new:N \g__nmc_ior
+\iow_new:N \g__nmc_iow
+\tl_gset:NV \g__nmc_reuse_filename_tl \jobname
+\tl_gput_right:Nn \g__nmc_reuse_filename_tl { .nmc }
+%
+\NewDocumentCommand \nmcReuse { s >{ \TrimSpaces } O{} }
+ { % save \eval result to csname #2
+ \tl_if_empty:nTF { #2 }
+ { \__nmc_reuse_retrieve: }
+ {
+ \tl_if_empty:NTF \g__nmc_reuse_tl
+ {
+ \__nmc_error_where:n { \__nmc_verb:n { \nmcReuse }~command }
+ \__nmc_error_what:n { No~value~to~save~with }
+ }
+ { \__nmc_reuse_result:c { #2 } }
+ }
+ \IfBooleanT #1 { \__nmc_debug_display:nn { 99991 } { reuse } }
+ }
+\ProvideDocumentCommand\reuse {} { \nmcReuse }
+\cs_new_protected:Npn \__nmc_reuse_retrieve:
+ {
+ \file_get:VnNT \g__nmc_reuse_filename_tl {} \l__nmc_reuse_retrieved_tl
+ {
+ \bool_lazy_or:nnF
+ { \tl_if_blank_p:V \l__nmc_reuse_retrieved_tl }
+ { \tl_if_head_eq_meaning_p:VN \l__nmc_reuse_retrieved_tl \par }
+ {
+ \seq_set_split:NnV \l_tmpa_seq {,} \l__nmc_reuse_retrieved_tl
+ \seq_map_inline:Nn \l_tmpa_seq
+ { \__nmc_reuse_defcmds:Nnn ##1 { _props } }
+ }
+ }
+ \bool_set_true:N \l__nmc_reuse_retrieved_bool
+ }
+\cs_new_protected:Npn \__nmc_reuse_result:N #1
+ {
+ \tl_set:Nn \l_tmpb_tl { #1 }
+ \tl_put_right:Nx \l_tmpb_tl { { \tl_to_str:N \g__nmc_reuse_tl } }
+ \bool_if:NF \l__nmc_reuse_retrieved_bool
+ { \__nmc_reuse_retrieve: }
+ \clist_put_right:NV \l__nmc_reuse_retrieved_tl \l_tmpb_tl
+ \exp_last_unbraced:NV\__nmc_reuse_defcmds:Nnn \l_tmpb_tl {}
+ \tl_gclear:N \g__nmc_reuse_tl
+ }
+\cs_generate_variant:Nn \__nmc_reuse_result:N { c }
+
+\cs_new_protected:Npn \__nmc_reuse_defcmds:Nnn #1#2#3
+ {
+ \prop_get:NnNTF \g__nmc_subst_var_prop { #1 } \l__nmc_toss_tl
+ { \RenewDocumentCommand #1{}{#2} }
+ { \NewDocumentCommand #1{}{#2} }
+ \tl_set_rescan:Nnn #1 { \ExplSyntaxOn } { #2 }
+ \use:c { __nmc_reuse_write#3:Nn } #1{#2}
+ }
+\cs_new_protected:Npn \__nmc_reuse_write:Nn #1#2
+ {
+ \__nmc_reuse_write_props:Nn #1{#2}
+ \file_get:VnNTF \g__nmc_reuse_filename_tl {} \l_tmpa_tl
+ {
+ \tl_if_head_eq_meaning:VNTF \l_tmpa_tl \par
+ { \tl_clear:N \l_tmpa_tl }
+ {
+ \__nmc_reuse_delete:NN #1 \l_tmpa_tl
+ \tl_if_blank:VF \l_tmpa_tl
+ {
+ \tl_trim_spaces:N \l_tmpa_tl
+ \tl_put_right:Nn \l_tmpa_tl {,}
+ }
+ }
+ }
+ { \tl_clear:N \l_tmpa_tl }
+ \tl_put_right:Nn \l_tmpa_tl { #1 {#2} }
+ \iow_open:NV \g__nmc_iow \g__nmc_reuse_filename_tl
+ \iow_now:NV \g__nmc_iow \l_tmpa_tl
+ \iow_close:N \g__nmc_iow
+ }
+\cs_new_protected:Npn \__nmc_reuse_delete:NN #1#2
+ {
+ \seq_clear:N \l_tmpb_seq
+ \seq_set_split:NnV \l_tmpa_seq {,} #2
+ \seq_map_inline:Nn \l_tmpa_seq
+ {
+ \tl_if_head_eq_meaning:nNF { ##1 } #1
+ { \seq_put_right:Nn \l_tmpb_seq { ##1 } }
+ }
+ \tl_set:Nx #2 { \seq_use:Nn \l_tmpb_seq {,} }
+ }
+\cs_new_protected:Npn \__nmc_reuse_write_props:Nn #1#2
+ {
+ \prop_put_if_new:Nnn \g__nmc_class_prop { #1 }
+ { \__nmc_fpify_var: \__nmc_delim_var: }
+ \prop_put:Nnn \g__nmc_subst_var_prop { #1 } { #2 }
+ }
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% settings
+\bool_new:N \l__nmc_multitok_bool
+\bool_new:N \l__nmc_vv_multline_bool
+\bool_new:N \l__nmc_deg_bool
+\tl_new:N \l__nmc_eq_num_tl
+\int_new:N \l__nmc_posi_int
+\int_new:N \l__nmc_eval_reuse_int
+\int_new:N \l__nmc_debug_int
+\tl_new:N \l__nmc_reuse_sep_tl % numerica-plus
+\int_set_eq:NN \l__nmc_posi_int \c_one_int
+
+\cs_new_protected:Npn \__nmc_set_trig_parsing:n #1
+ { % #1=1,2,3
+ \int_set:Nn\l__nmc_posi_int { #1 }
+ \int_compare:nNnTF { #1 } > { 1 }
+ { % #1=2,3
+ \int_set_eq:NN \l__nmc_trg_int \c__nmc_trg_int
+ \int_compare:nNnT { #1 } > { 2 }
+ { % #1=3
+ \__nmc_fill_class:nn { \frac, \dfrac }
+ { \__nmc_fpify_frac: \__nmc_delim_tfrac: }
+ }
+ }
+ { \int_set_eq:NN \l__nmc_trg_int \c__nmc_uny_int } % #1=1
+ }
+\cs_new:Npn \__nmc_deg:
+ { \bool_if:NT { \l__nmc_deg_bool } { d } }
+\cs_new:Npn \__nmc_int_to_bool:Nn #1#2
+ { \bool_set:Nn #1 { !\int_if_zero_p:n { #2 } } }
+\keys_define:nn { numerica }
+ {
+ rounding .int_set:N = \l__nmc_round_int,
+ pad .code:n = \__nmc_int_to_bool:Nn
+ \l__nmc_pad_zeros_bool { #1 },
+ output-sci-notation .code:n = \__nmc_int_to_bool:Nn
+ \l__nmc_sci_num_out_bool { #1 },
+ output-exponent-char .tl_set:N = \l__nmc_sci_num_out_tl,
+ input-sci-notation .code:n = \__nmc_int_to_bool:Nn
+ \l__nmc_sci_num_in_bool { #1 },
+ input-exponent-char .tl_set:N = \l__nmc_sci_num_in_tl,
+ multitoken-variables .code:n = \__nmc_int_to_bool:Nn
+ \l__nmc_multitok_bool { #1 },
+ logarithm-base .tl_set:N = \l__nmc_log_base_tl,
+ intify-rounding .tl_set:N = \l__nmc_intify_round_tl,
+ vv-display .tl_set:N = \l__nmc_vv_display_tl,
+ vv-inline .tl_set:N = \l__nmc_vv_inline_tl,
+ sum-extra-rounding .int_set:N = \l__nmc_sum_extra_int,
+ sum-query-terms .int_set:N = \l__nmc_sum_query_int,
+ prod-extra-rounding .int_set:N = \l__nmc_prod_extra_int,
+ prod-query-terms .int_set:N = \l__nmc_prod_query_int,
+ eval-reuse .int_set:N = \l__nmc_eval_reuse_int,
+%
+ rounding .default:n = 6,
+ pad .default:n = 0,
+ output-sci-notation .default:n = 0,
+ output-exponent-char .default:n = e,
+ input-sci-notation .default:n = 0,
+ input-exponent-char .default:n = e,
+ multitoken-variables .default:n = 1,
+ logarithm-base .default:n = 10,
+ intify-rounding .default:n = 14,
+ vv-display .default:n = {,}\mskip 36mu minus 24mu(vv),
+ vv-inline .default:n = {,}\mskip 12mu plus 6mu minus 9mu(vv),
+ sum-extra-rounding .default:n = 2,
+ sum-query-terms .default:n = 0,
+ prod-extra-rounding .default:n = 2,
+ prod-query-terms .default:n = 0,
+ eval-reuse .default:n = 0,
+ }
+\keys_set_known:nn { numerica }
+ {
+ rounding,pad,output-sci-notation,output-exponent-char,
+ input-sci-notation,input-exponent-char,
+ multitoken-variables,logarithm-base,intify-rounding,
+ vv-display,vv-inline,sum-extra-rounding,sum-query-terms,
+ prod-extra-rounding,prod-query-terms,eval-reuse
+ }
+\keys_define:nn { numerica/generic }
+ {
+ dbg .code:n = \int_set:Nn \l__nmc_debug_int
+ { \int_compare:nNnTF { \int_abs:n { #1 } } = { 1 }
+ { \int_sign:n { #1 }*2*3*5*7 } { #1 } },
+ ^ .code:n = \bool_set_true:N \l__nmc_sci_num_in_bool
+ \tl_set:Nn \l__nmc_sci_num_in_tl { #1 },
+ xx .code:n = \__nmc_int_to_bool:Nn \l__nmc_multitok_bool { #1 },
+ () .code:n = \__nmc_set_trig_parsing:n { 2+\int_sign:n { #1-1 } },
+ () .initial:n = 0,
+ o .code:n = \bool_set_true:N \l__nmc_deg_bool,
+ log .tl_set:N = \l__nmc_log_base_tl,
+ vvmode .code:n = \__nmc_calc_mode:n { \int_sign:n { #1*#1 } },
+ vvd .code:n = \tl_set:Nn \l__nmc_vv_display_tl { #1 }
+ \tl_if_in:NnT \l__nmc_vv_display_tl { \\ }
+ { \bool_set_true:N \l__nmc_vv_multline_bool },
+ vvi .tl_set:N = \l__nmc_vv_inline_tl,
+ * .code:n = \tl_set:Nn \l__nmc_eq_num_tl { * },
+ p .tl_set:N = \l__nmc_punc_tl,
+ p .default:n = {,},
+ S+ .int_set:N = \l__nmc_sum_extra_int,
+ S? .int_set:N = \l__nmc_sum_query_int,
+ P+ .int_set:N = \l__nmc_prod_extra_int,
+ P? .int_set:N = \l__nmc_prod_query_int,
+ }
+\keys_define:nn { numerica/eval }
+ { reuse .int_set:N = \l__nmc_eval_reuse_int }
+\keys_define:nn { numerica/package }
+ {
+ plus .code:n = \file_input:n { numerica-plus.def },
+ tables .code:n = \file_input:n { numerica-tables.def },
+ % calculus .code:n = \file_input:n { numerica-calculus.def },
+ lyx .code:n = \file_input:n { numerica-lyx.def }
+ }
+\ProcessKeysPackageOptions { numerica/package }
+
+\file_if_exist:nT { numerica.cfg }
+ {
+ \file_get:nnN { numerica.cfg } {} \l_tmpa_tl
+ \exp_args:NnV \keys_set_known:nn { numerica } \l_tmpa_tl
+ }
+% end of `numerica.sty'
\ No newline at end of file
Property changes on: trunk/Master/texmf-dist/tex/latex/numerica/numerica.sty
___________________________________________________________________
Added: svn:eol-style
## -0,0 +1 ##
+native
\ No newline at end of property
Modified: trunk/Master/tlpkg/bin/tlpkg-ctan-check
===================================================================
--- trunk/Master/tlpkg/bin/tlpkg-ctan-check 2021-02-15 22:21:25 UTC (rev 57758)
+++ trunk/Master/tlpkg/bin/tlpkg-ctan-check 2021-02-15 22:22:05 UTC (rev 57759)
@@ -546,7 +546,7 @@
novel nowidow nox npp-for-context
nrc ntgclass nth ntheorem ntheorem-vn nuc nucleardata
numberedblock numberpt
- numericplots numname numnameru numprint numspell nunito nwejm
+ numerica numericplots numname numnameru numprint numspell nunito nwejm
oberdiek objectz obnov
ocg-p ocgx ocgx2 ocherokee ocr-b ocr-b-outline ocr-latex octave octavo
odsfile ofs
Modified: trunk/Master/tlpkg/tlpsrc/collection-mathscience.tlpsrc
===================================================================
--- trunk/Master/tlpkg/tlpsrc/collection-mathscience.tlpsrc 2021-02-15 22:21:25 UTC (rev 57758)
+++ trunk/Master/tlpkg/tlpsrc/collection-mathscience.tlpsrc 2021-02-15 22:22:05 UTC (rev 57759)
@@ -144,6 +144,7 @@
depend nicematrix
depend nuc
depend nucleardata
+depend numerica
depend objectz
depend oplotsymbl
depend ot-tableau
Added: trunk/Master/tlpkg/tlpsrc/numerica.tlpsrc
===================================================================
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