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author | Norbert Preining <norbert@preining.info> | 2021-12-13 03:02:58 +0000 |
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committer | Norbert Preining <norbert@preining.info> | 2021-12-13 03:02:58 +0000 |
commit | 8164c2be3b4849a9c77e44b9659a3de8af010bd7 (patch) | |
tree | 4414d792306f426cccf2f370b64dee20cd239244 /macros/latex/contrib/numerica/numerica-basics.tex | |
parent | 9ed1a61f45b05f58016912a6bd6c13daab6db9eb (diff) |
CTAN sync 202112130302
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diff --git a/macros/latex/contrib/numerica/numerica-basics.tex b/macros/latex/contrib/numerica/numerica-basics.tex deleted file mode 100644 index 6e58e34822..0000000000 --- a/macros/latex/contrib/numerica/numerica-basics.tex +++ /dev/null @@ -1,4743 +0,0 @@ -%% 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@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@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@-1 } $` $\Longrightarrow$ $ \eval[^=@]{ 1.23@-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} |