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diff --git a/macros/latex/contrib/numerica-tables/numerica-tables.tex b/macros/latex/contrib/numerica-tables/numerica-tables.tex index 29f0e48129..72a814632a 100644 --- a/macros/latex/contrib/numerica-tables/numerica-tables.tex +++ b/macros/latex/contrib/numerica-tables/numerica-tables.tex @@ -1,4 +1,4 @@ -%% LyX 2.4.0-beta3-devel created this file. For more info, see https://www.lyx.org/. +%% LyX 2.4.2.1 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} @@ -16,7 +16,7 @@ \usepackage{booktabs} \usepackage{framed} \usepackage{url} -\usepackage{multirow} +\usepackage{varwidth} \usepackage{amsmath} \usepackage{makeidx} \makeindex @@ -30,6 +30,10 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. %% Because html converters don't know tabularnewline \providecommand{\tabularnewline}{\\} +%% Variable width box for table cells +\newenvironment{cellvarwidth}[1][t] + {\begin{varwidth}[#1]{\linewidth}} + {\@finalstrut\@arstrutbox\end{varwidth}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands. \newenvironment{centred}% @@ -48,73 +52,67 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. \usepackage{numerica} \usepackage{numerica-tables} -\usepackage{numerica-plus} -\newcommand\rel{\,\varrho\;} -\DeclareMathOperator{\erf}{erf} -\DeclareMathOperator{\gd}{gd} \usepackage{upquote} \usepackage{xfrac} \usepackage[indentunit=1em,totoc,columnsep=1.5em, initsep=10.25pt plus 5pt minus 3pt]{idxlayout} +\usepackage{diagbox} \reuse{} \makeatother \begin{document} \title{\texttt{numerica-tables}~\\ -{\large version 3.1.0}} +{\large version 3.2.0}} \author{Andrew Parsloe\\ (\url{ajparsloe@gmail.com})} \maketitle \begin{abstract} -The \texttt{numerica-tables} package defines a command \verb`\nmcTabulate` -which enables the creation of multi-column tables of mathematical -function values. Key\textendash value assignments allow presentation -in a wide variety of table styles within the `formal table' framework -of the \texttt{booktabs} package. \texttt{numerica-tables} requires -the \texttt{numerica} package to be loaded.\medskip{} +The \texttt{numerica-tables} package enables the creation of multi-row, +multi-column tables of values of mathematical functions. Key\textendash value +assignments allow presentation in a wide variety of table styles, +both vertically by column or horizontally by row, within the `formal +table' framework of the \texttt{booktabs} package. \texttt{numerica-tables} +requires the prior loading of the \texttt{numerica} package.\medskip{} \end{abstract} \noindent{}% \noindent\begin{minipage}[t]{1\columnwidth}% \begin{shaded}% \begin{itemize} -\item This document applies to version 3.1.0 of \texttt{numerica-tables}. +\item This document applies to version 3.2.0 of \texttt{numerica-tables} +and requires the \texttt{booktabs} package. \item Version 3 of \texttt{numerica} needs to be loaded before \texttt{numerica-tables}; -(\texttt{numerica} requires \texttt{amsmath}, \texttt{mathtools} and -a reasonably recent \LaTeX{} system). -\item The \texttt{booktabs} package is required. -\item I refer many times in this document to \emph{Handbook of Mathematical -Functions}, edited by Milton Abramowitz and Irene A. Stegun, Dover, -1965, abbreviated to \emph{HMF}, and often followed by a reference, -like Table 1.2, to a specific table. +(\texttt{numerica} requires \texttt{amsmath} and \texttt{mathtools}). +\item I refer many times in this document to tables included in \emph{Handbook +of Mathematical Functions}, edited by Milton Abramowitz and Irene +A. Stegun, Dover, 1965, and referenced as \emph{HMF}. +\item Version 3.2.0 of \texttt{numerica-tables} +\begin{itemize} +\item adds a setting \verb`transpose` to convert a table with function +values in columns into a table with function values in rows; +\item adds the package options \verb`rules` and \verb`norules` to specify +the default pattern of horizontal rules; +\item adds new settings and enhances existing settings to adjust the appearance, +size and positioning of items in both the row variable column and +the header row; +\item enhances and adjusts elements of fraction form output (e.g. with a +`semi-verbatim' setting); +\item fixes a bug when tabulating factorials; +\item updates documentation. +\end{itemize} \item Version 3.1.0 of \texttt{numerica-tables} \begin{itemize} \item adds an index to the documentation; -\item adds the ability to round table entries to different values depending -on row or column (or both) in the table; -\item fixes two minor presentation bugs. +\item adds the ability to round table entries to different values. \end{itemize} \item Version 3.0.0 of \texttt{numerica-tables} \begin{itemize} -\item adds the ability to use numbers or expressions from -\begin{itemize} -\item a comma list for row variable values, either as values or verbatim; -\item a macro containing a comma list for row variable values, either as -values or verbatim; -\item a file for row variable values, either as values or verbatim; -\end{itemize} -\item adds the ability to use functions of a stepped variable to generate -varyingly stepped row variable values; +\item adds the ability to use as row variable values numbers or expressions +from a comma list, a macro, a file, or a step function, to be displayed +either as values or verbatim; \item adds the ability to suppress the header row; \item is compatible with the additional features of \texttt{numerica} version -3.0.0, -\begin{itemize} -\item including the decimal comma if the \verb`comma` package option is -used with \texttt{numerica}; and -\item fraction-form row variables, or fraction-form output if the \verb`/` -or \verb`//` `triggers' are used in the number-format option; -\end{itemize} -\item amends and adds to documentation. +3.0.0, including the decimal comma and fraction-form output. \end{itemize} \end{itemize} \end{shaded}% @@ -124,14 +122,14 @@ or \verb`//` `triggers' are used in the number-format option; \chapter{Introduction} -Entering +\label{chap:Introduction}Entering \begin{verbatim} \usepackage[<options>]{numerica} - \usepackage{numerica-tables} + \usepackage[<options>]{numerica-tables} \end{verbatim} \noindent in the preamble of a document gives access to a command for creating tables of function values in a wide variety of styles. -Contrary to previous practice, for version 3.0.0 of \texttt{numerica-tables}, +Contrary to previous practice, from version 3.0.0 of \texttt{numerica-tables}, the \texttt{numerica} package is not loaded automatically but must be loaded explicitly (as above), with options if desired, \emph{before} \texttt{numerica-tables}. It is \emph{essential} that the version @@ -139,32 +137,64 @@ of \texttt{numerica} loaded is version 3. All tables are `formal tables' in the sense of the \texttt{booktabs}\index{booktabs@\texttt{booktabs}} package, which is loaded automatically. Such tables have no vertical -rules\index{rules} and few horizontal rules. +rules and few horizontal rules. Quoting from the documentation for +that package:\index{rules@\texttt{rules}!edicts} +\begin{enumerate} +\item Never, ever use vertical rules. +\item Never use double rules. +\end{enumerate} + +\section{Package options} + +\label{sec:Package-options}New \index{package options}\index{rules@\texttt{rules}!package option}\index{norules@\texttt{norules}!package option} +with version 3.2 of \texttt{numerica-tables} are the package options +\verb`rules` and \verb`norules`, and \verb`Q?*`, which is more +complicated and discussed later (\S\ref{sec:Star-option}). The \verb`rules` +option determines which (if any) horizontal rules are drawn in a table +if none are explicitly set for that table. Because most tables in +this document are set amongst text, they are delimited by top and +bottom rules, and one beneath the header row. For consistency with +earlier versions, `out of the box' the \verb`rules` package option +is initialized to \verb`rules=ThB`. + +For the user this may not be what is wanted. You may prefer by default +to insert a rule only beneath the header row of your tables in which +case the package option would be \verb`rules=h`. For other possible +values for the package option see the identical values of the \verb`rules` +\emph{setting} discussed in §\ref{subsec:Rules:-rules-setting}. If, +as a default, you want no rules at all in your tables, you can either +give \verb`rules` an empty value (\verb`rules= ` ) or use the \verb`norules` +package option: +\begin{verbatim} + \usepackage[norules]{numerica-tables} +\end{verbatim} \section{Table structure} -I\index{table structure} take as my source of models of mathematical +I\index{tables!structure} take as my source of models of mathematical tables those presented in \emph{Handbook of Mathematical Functions}, edited by Milton Abramowitz and Irene A. Stegun, Dover, 1965, not -because the typesetting is elegant (it often isn't) but because \emph{HMF} +because the typesetting is elegant (it often is not) but because \emph{HMF} displays a wide variety of table styles. The editors of that volume were faced with a host of different problems requiring a host of different solutions. The \verb`\nmcTabulate` command of \texttt{numerica-tables} -aims to reproduce most of those different solutions, within \texttt{booktabs} -\index{booktabs@\texttt{booktabs}} elegance. - -To create a table we need to specify a function or functions to tabulate. -The values a function takes will generally depend on a primary parameter -and, possibly, a number of secondary parameters (which is where much -of the complexity comes from). Mathematical tables are structured -in \emph{columns}. We (nearly always) read \emph{down} a column as -the primary parameter is incremented, generally in regular steps. -We need to decide on the range of values the primary parameter will -take and how fine-grained the tabulation will be \textendash{} what -the step size of its increments will be. Assigning different values -to a second parameter generates a second, third,\ldots{} column. Sometimes -rather than a second parameter, a second, third, \ldots{} function -of the first parameter is tabulated in the successive columns. +aims to reproduce most of those different solutions, within \texttt{booktabs}\index{booktabs@\texttt{booktabs}} +elegance. + +Creating a table presumes we have a function or functions we wish +to tabulate. The values a function takes will generally depend on +a primary parameter and, possibly, a number of secondary parameters +(which is where much of the complexity comes from). Books of mathematical +tables are structured (nearly always) in \emph{columns} (but see below), +and we read (nearly always) \emph{down} a column as the primary parameter +is incremented, generally in regular steps. We need to decide on the +range of values the primary parameter will take and how fine-grained +the tabulation will be \textendash{} what the step size of its increments +will be. Assigning different values to a second parameter generates +a second, third, \ldots{} column of function values. Sometimes rather +than a second parameter, a second, third, \ldots{} function of the +first parameter is tabulated in the successive columns \textendash{} +like adjacent columns of different trigonometric functions. In this document the first parameter is called the \emph{row variable}\index{row variable} \textendash{} its value determines which row we are in; the second @@ -181,28 +211,76 @@ rules are absent, horizontal rules used sparingly \textendash{} for example, at the top and bottom of the table, or under the header row, but not in the body of the table. +\subsection{Transposed tables} + +From\index{tables!`horizontal'/`vertical'} version 3.2 of \texttt{numerica-tables} +it is also possible to present tables `horizontally', in which function +values are read in rows (generally) left-to-right across the page. +\texttt{numerica-tables} handles this situation by constructing the +table in the usual column-based `vertical' manner and then treating +the columns of calculated function values as rows; the row variable +column and the header row swap roles. The placement of the new row +variable column and header row and the `decoration' of the values +contained therein (font, alignment, etc.) is all done \emph{after} +transposition. + +The process is effected through the setting \verb`transpose`\index{transpose@\texttt{transpose}} +and is discussed at §\ref{subsec:Transpose}, but examples are presented +throughout the document. + +\subsection{\texttt{tabular} environment} + +The tables created by \texttt{numerica-tables} use the \verb`tabular`\index{tabular environment@\texttt{tabular} environment} +environment of \LaTeX . This means that the vertical separation between +rows can be altered by means of the parameter \verb`\arraystretch`\index{arraystretch@\texttt{\textbackslash arraystretch}}. +Writing, for instance, +\begin{verbatim} + \renewcommand\arraystretch{1.2} +\end{verbatim} +increases the separation between rows by a scale factor of $1.2$. +An example of use occurs in §\ref{subsec:rmath}. + +The separation between columns is determined by the parameter \verb`\tabcolsep`\index{tabcolsep@\texttt{\textbackslash tabcolsep}} +which is half the width of the space between columns and defaults +to $6$ pt. To change \verb`\tabcolsep` you can use either of the +commands \verb`\addtolength` or \verb`\setlength`. Since the default +is $6$pt, the commands +\begin{verbatim} + \addtolength\tabcolsep{3pt} + \setlength\tabcolsep{9pt} +\end{verbatim} +have the same effect and increase the half-width between columns to +$9$ pt; see the final example in §\ref{subsec:Nudgingtheheaders:-chnudge}. + +After changing either of these parameters you will need to reset it +back to the default if you do not want all subsequent tables to be +stretched vertically or horizontally as the case may be. + \section{Shared syntax} The \verb`\nmcTabulate`\index{nmcTabulate@\texttt{\textbackslash nmcTabulate}} -command (short-name form \verb`\tabulate`\index{tabulate@\texttt{\textbackslash tabulate}|see{\texttt{\textbackslash nmcTabulate}}}) +command, short-name form \verb`\tabulate`\index{tabulate@\texttt{\textbackslash tabulate}|see{\texttt{\textbackslash nmcTabulate}}}, shares the syntax of \verb`\nmcEvaluate` (see \texttt{numerica.pdf}). When all options are used the command looks like \begin{verbatim} \nmcTabulate*[settings]{expr.}[vv-list][num. format] \end{verbatim} +\selectlanguage{english}% \begin{enumerate} \item \verb`*` optional switch; if present ensures a single number output with no formatting, or an appropriate error message if the single number cannot be produced; see §\ref{sec:Star-option}; -\item \verb`[settings]` comma-separated list of \emph{key=value }settings; -this option is at the heart of creating a table of function values; -see Chapter~\ref{chap:=00005CnmcTabulate-settings}; +\item \verb`[settings]` \emph{mandatory }comma-separated list of \emph{key=value +}settings; this option is at the heart of creating a table of function +values, and is discussed in the next chapter; \item \verb`{expr.}` mandatory argument specifying the mathematical expression or expressions in \LaTeX{} form to be tabulated; -\item \verb`[vv-list]` comma-separated list (or semicolon-separated list -if the \verb`comma` package option is used with \texttt{numerica}) -of \emph{variable=value }items, in particular containing the initial -value of the row variable, and column variable if one is used; +\item \verb`[vv-list]`\index{vv-list} \emph{mandatory} comma-separated +list (or semicolon-separated\index{decimal comma!semicolon use} list +if the \verb`comma` package option\index{comma package option@\texttt{comma} package option} +is used with \texttt{numerica}) of \emph{variable=value }items, in +particular containing the initial value of the row variable (essential) +and column variable (if one is used); \item \verb`[num. format]` optional format specification for presentation of the numerical results (rounding, padding with zeros, scientific notation, fraction-form output); see §\ref{subsec:Trailing-optional-argument}.\index{rounding value} @@ -216,47 +294,81 @@ contains items \emph{necessary} for the construction of any table of function values. Should \texttt{numerica} be loaded with the \verb`comma` package -option, numbers in tables will be displayed with a decimal comma\index{decimal comma}. -In this case items in the vv-list\index{vv-list!and decimal comma} -\emph{must} be separated by a semicolon\index{semicolon and decimal comma} -since the decimal comma is likely to feature there. Similarly, $n$-ary -functions \textendash{} \verb`\max`, \verb`\min` and \verb`\gcd`\index{n-ary@$n$-ary functions} -\textendash{} must use the semicolon as their argument separator. - -Math environments are significant for \verb`\nmcEvaluate`. They are -essentially irrelevant for \verb`\nmcTabulate`\index{nmcTabulate@\texttt{\textbackslash nmcTabulate}}\index{environments}, -although an external environment may determine how a table is positioned -on the page\index{table placement!on the page} (\verb`\[ <table> \]` -for instance will centre the table between the margins). Environments -specified in the main argument (the formula) should be avoided. \verb`$ $`, -\verb`\( \)` or \verb`\[ \]` will not cause error, but a \verb`\begin`\textendash \verb`\end` -environment there will. - -Just as the syntax is shared, so the settings available to the command -\verb`\nmcEvaluate` in \texttt{numerica} are also available to \verb`\nmcTabulate`, -although not all will be relevant. See the associated document \texttt{numerica.pdf} -for a list of these and associated discussion. I will point out instances -of their use in the following examples. +option\index{comma package option@\texttt{comma} package option}, +numbers in tables will be displayed with a decimal comma\index{decimal comma!semicolon use}. +In this case, to avoid confusion, items in the vv-list\index{vv-list!and decimal comma} +\emph{must} be separated by a semicolon. Similarly, $n$-ary functions +\textendash{} \verb`\max`, \verb`\min` and \verb`\gcd`\index{n-ary@$n$-ary functions} +\textendash{} must use semicolons as their argument separator. From +version 3.2 of \texttt{numerica-tables} row variable data in the form +of lists or files for the \verb`rdata` and \verb`rfile` settings +do not need to be \emph{comma} separated. If the \verb`comma` package +option\index{comma package option@\texttt{comma} package option} +is used, the semicolon is assumed (see §\ref{subsec:rdata-rfile-rverb}) +or some other item separator can be set with the (new in version 3.2) +\verb`rdelim` setting. + +Although math environments are significant for \verb`\nmcEvaluate`, +they should be avoided with \verb`\nmcTabulate`\index{nmcTabulate@\texttt{\textbackslash nmcTabulate}}\index{environments}. +Placing a \verb`\tabulate` command within a math environment, or +vice versa, is likely to cause a \LaTeX{} error. From version 3.2 of +\texttt{numerica-tables}, it is possible to specify the math style +(\verb`\displaystyle`, \verb`\textstyle` or \verb`\scriptstyle`) +of entries in the row variable column and in the header row. Function +values (numbers) in the table cells are presented between inline math +delimiters (\verb`$` \verb`$`). \chapter{\texttt{\textbackslash nmcTabulate} settings} -\label{chap:=00005CnmcTabulate-settings}In addition to the shared -settings, \verb`\nmcTabulate` has many settings specific to it. They -are discussed in groups in subsequent sections, some in more than -one place. For the main discussion of row variable settings, see §\ref{sec:Row-variable-settings}; -for column variable settings see §\ref{sec:Column-variable-settings}; -for whole-of-table formatting see §\ref{sec:Whole-of-table-formatting}; -for formatting the function values in table cells see §\ref{sec:Function-value-formatting}. - -\section{Row-variable settings} +Just as \verb`\nmcTabulate` shares the syntax of \verb`\nmcEvaluate` +(of \texttt{numerica}), it also shares the \emph{settings} of the +latter command \textendash{} although not all will be relevant. See +the associated document \texttt{numerica.pdf} for a list of those +settings and associated discussion. I will point out instances of +their use in the following examples. + +In addition to the shared settings, there are many specific to \verb`\nmcTabulate`, +which is what this document is about. They are discussed in groups +in later sections, some in more than one place. For the main discussion +of row variable settings, see immediately below; for column variable +settings see §\ref{sec:Column-variable-settings}; for multi-function +tables see §\ref{sec:Multiple-function-tables}; for whole-of-table +formatting see §\ref{sec:Whole-of-table-formatting}; for formatting +the function values in table cells see §\ref{sec:Function-value-formatting}. + +\section{Row variable settings} + +\label{sec:Row-variable-settings}Okay, assume we have chosen the +function we want tabulated. Now we need to choose the row variable, +how fine-grained we want the tabulation to be and between which values. + +\subsection{Row variable specification: uniform case} + +\label{subsec:Row-var-spec-uniform}\index{row variable!start, step, stop values|(}The +\emph{row} variable is set in the settings option of the \verb`\tabulate` +command with the key \verb`rvar`\index{rvar@\texttt{rvar}} \textendash{} +say \verb`rvar=x`. What value to start tabulating from is specified +in the vv-list\index{vv-list!row/col.variables@row/col.\ variables}\index{row variable!initial value in vv-list} +\textendash{} \verb`x=0` perhaps \textendash{} and so does not need +a specific key, but what value to tabulate to, \verb`rstop`\index{rstop@\texttt{rstop}}, +and the step size, how fine-grained the tabulation is to be, \verb`rstep`\index{rstep@\texttt{rstep}}, +do need to be specified in the settings option. In the uniform case +(which makes up the overwhelming majority of cases in \emph{HMF}) +the step size is constant. It does not change as the value of the +row variable changes. (The non-uniform case, available from version +3.0.0 of \texttt{numerica-tables}, is discussed in §\ref{subsec:Row-var-spec-nonuniform} +below. Quite different keys are required.) -\label{sec:Row-variable-settings} - -\subsection{Row-variable specification: uniform case} +The two tables in the example below tabulate $\sin x$ and $\cos x$ +between $0$ and $1$ in increments of $0.2$. By placing the start +value of the tabulation variable in the vv-list, the possibility is +opened for other parameters in more complicated functions to depend +on the row variable. Although it will often be the first entry in +the vv-list, it does not need to be. -\label{subsec:Row-var-spec-uniform} -\begin{table}[b] -\centering{}\caption{Row-variable specification} +\begin{table}[t] +\centering +\centering{}\caption{Row variable specification} \begin{center} \begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{3cm}} \toprule @@ -264,462 +376,415 @@ for formatting the function values in table cells see §\ref{sec:Function-value-f \midrule {\small\texttt{rvar}} & {\small token(s)} & {\small row variable} & \tabularnewline {\small\texttt{rstep}} & {\small real num.} & {\small step size} & \tabularnewline -{\small\texttt{rstop}} & {\small real num.} & {\small stop value} & \multirow{2}{3cm}{{\small use only one of rstop or rows}}\tabularnewline -{\small\texttt{rows}} & {\small int} & {\small number of rows} & \tabularnewline -{\small\texttt{rspec}} & {\small comma list} & {\small\texttt{\{rvar}}{\small , }{\small\texttt{step}}{\small , }{\small\texttt{rows\}}} & {\small short form spec.}\tabularnewline +{\small\texttt{rstop}} & {\small real num.} & {\small stop value} & {\small excludes }{\small\texttt{rows}}\tabularnewline +{\small\texttt{rows}} & {\small int} & {\small number of rows} & {\small excludes }{\small\texttt{rstop}}\tabularnewline +{\small\texttt{rspec}} & {\small comma list} & {\small\texttt{\{rvar}}{\small , }{\small\texttt{rstep}}{\small , }{\small\texttt{rows\}}} & {\small short form spec.}\tabularnewline +{\small\texttt{rround}} & {\small int} & {\small rounding} & {\small default: }{\small\texttt{1}}\tabularnewline \bottomrule \end{tabular} \par\end{center} \end{table} -Deciding\index{row variable!start, step, stop values|(} on a function -to tabulate (entered in the main or mandatory argument of \verb`\nmcTabulate`) -will inevitably also mean deciding on the tabulation variable, the -\emph{row} variable, \verb`rvar`\index{rvar@\texttt{rvar}}, what -value to start tabulating from \textendash{} \emph{that} is specified -in the vv-list\index{vv-list!row/col. variable initial values}\index{row variable!initial value in vv-list} -and so does not need a specific key \textendash{} what value to tabulate -to, \verb`rstop`\index{rstop@\texttt{rstop}}, and how fine-grained -the tabulation is to be, the step size \verb`rstep`\index{rstep@\texttt{rstep}}. -In the uniform case (which makes up the overwhelming majority of cases -in \emph{HMF} for instance) the step size is constant. It does not -change as the value of the row variable changes. (The non-uniform -case, available from version 3.0.0 of \texttt{numerica-tables}, is -discussed in §\ref{subsec:Row-var-spec-nonuniform} below. Quite different -keys are required.) - -The two tables in the example below tabulate $\sin x$ and $\cos x$ -between $0$ and $1$ in increments of $0.2$. Note that the start -value of the tabulation variable is entered in the vv-list. The reason -for placing it there is that for more complicated functions other -parameters in the function and therefore in the vv-list may depend -on the row variable. Although it will often be the first entry in -the vv-list, it does not need to be. The initial value of the row -variable may depend on other quantities which must necessarily precede -it \textendash{} lie to the right of it \textendash{} in the list. - -In the vv-list, the start value of the row variable may be a \LaTeX{} +The initial value of the row variable may depend on other quantities +which must necessarily precede it \textendash{} lie to the right of +it \textendash{} in the vv-list. The start value may be a \LaTeX{} expression. Both \verb`rstep` and \verb`rstop` may also be \LaTeX{} expressions. They are evaluated \emph{after} the vv-list is evaluated and so may depend on the values of variables in the vv-list, including -the \emph{initial} value of the row variable. Thus setting \verb`rstep=1/x`, -where \verb`x` is the row variable, will give a \emph{constant} step -size equal to the reciprocal of the \emph{initial }value of the row -variable. - -The difference in appearance of the two tables results from padding -with zeros\index{function value formatting!padding with zeros}\index{padding with zeros} -in the second (the asterisk in the trailing optional argument has -the same effect in \verb`\nmcTabulate` as in \verb`\nmcEvaluate`). -As you can see, padding applies not only to the values of the function -but also to the values of the row variable, although that has been -padded to only $1$ decimal place rather than the $6$ of the function -values. Padding makes an obvious improvement to the appearance. (How -many digits to pad to in the row variable column is discussed in §\ref{subsec:Row-var-col-formatting}.) +the \emph{initial} value of the row variable. As of v.3.2 of \texttt{numerica-tables} +the \verb`vv@=1`\index{vv-list!vva@\texttt{vv"@} (mode) setting} +(or \verb`vvmode=1`) setting from \texttt{numerica} does not affect +\verb`rstep` and \verb`rstop`. For instance if \verb`rvar=x`, setting +\verb`rstep=1/x` when \verb`vv@=1` gives a \emph{constant} step +size equal to the reciprocal of the initial\emph{ }value of the row +variable. The following examples illustrate the use of \verb`rvar`, +\verb`rstep` and \verb`rstop` and the presence of the initial value +of the row variable in the vv-list. \begin{verbatim} \tabulate[rvar=x,rstep=0.2,rstop=1] { \sin x }[x=0]\qquad \tabulate[rvar=x,rstep=0.2,rstop=1] - { \cos x }[x=0][*] + { \sin x }[x=0][*] \end{verbatim} $\Longrightarrow$ \tabulate[rvar=x,rstep=0.2,rstop=1] { \sin x }[x=0] \qquad \tabulate[rvar=x,rstep=0.2,rstop=1] - { \cos x }[x=0][*]\medskip{} - -Sometimes (perhaps often) it may prove more convenient to specify -the number of rows, \verb`rows`\index{rows@\texttt{rows}}, to tabulate -rather than a stop value. Only one of \verb`rows` and \verb`rstop` -should be given, but if both (inadvertently) are present, it is the -value of \verb`rows` that prevails. The first of the following three -tables shows an example where \verb`rows` is specified. - -The second and third tables use an abbreviated form of the row variable -specification, \verb`rspec`\index{rspec@\texttt{rspec}}. This is -a three-element comma list, \verb`{rvar,rstep,rows}`. The second -table gives a straightforward example. In the third table a \LaTeX{} -expression has been inserted for \verb`rows` in the \verb`rspec` -comma list. Like \verb`rstep` and \verb`rstop`, \verb`rows` can -be a \LaTeX{} expression, but it is evaluated \emph{before} the vv-list -and therefore, unlike \verb`rstep` and \verb`rstop`, cannot depend -on quantites specified there like the initial row variable value. + { \sin x }[x=0][*]\medskip{} + +The difference in appearance of the tables results from padding with +zeros\index{function-value formatting!padding with zeros}\index{padding with zeros} +in the second. The asterisk in the trailing optional argument has +the same effect in \verb`\nmcTabulate` as in \verb`\nmcEvaluate`. +As you can see, padding applies not only to the values of the function +but also to the values of the row variable, although that has been +padded to only $1$ decimal place (the default) rather than the $6$ +of the function values. (How many digits to round to, and therefore +pad the row variable values to, is discussed in §\ref{subsec:Rounding:rround}.) +Padding makes an obvious improvement to the appearance, both in the +function-value column and the row variable column. + +You may feel a single column of function values makes poor use of +space on the page. To turn the second of these tables into a `horizontal' +table, add the key \verb`transpose` to the settings option. I have +also reduced the number of decimal places from the default $6$ to +$4$ to avoid any issues with page width. +\begin{verbatim} + \tabulate[rvar=x,rstep=0.2,rstop=1,transpose] + { \sin x }[x=0][4*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rvar=x,rstep=0.2,rstop=1,transpose] + { \sin x }[x=0][4*] + +\medskip{} + +\index{transpose@\texttt{transpose}!examples of use}To my eye, for +such a `shallow' (vertically compressed) table, the horizontal rules +here are intrusive. They can be turned off; see §\ref{subsec:Rules:-rules-setting} +or, anticipating the later discussion, add \verb`norules`\index{norules@\texttt{norules}} +to the settings option (prior to version 3.2 of \texttt{numerica-tables} +it was necessary to write \verb`rules=`\texttt{ }, an empty setting, +to achieve this), and just for variety use the cosine: +\begin{verbatim} + \tabulate[rvar=x,rstep=0.2,rstop=1,transpose,norules] + { \cos x }[x=0][4*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rvar=x,rstep=0.2,rstop=1,transpose,norules] + { \cos x }[x=0][4*] \index{transpose@\texttt{transpose}!examples of use} + +\medskip{} + +The remaining quibble is the placement of the column headers. Can +they be centred or otherwise adjusted? Yes; see §\ref{subsec:Row-var-col-formatting} +for ways of adjusting the row variable column and header, and §\ref{subsec:Column-header-formatting} +for ways of adjusting the function-value columns and headers. + +It may be more convenient at times to specify the number of rows, +\verb`rows`\index{rows@\texttt{rows}}, to tabulate rather than a +stop value. Only one of \verb`rows` and \verb`rstop` should be given, +but if both (inadvertently) are present, it is \verb`rows` that +prevails. The first of the following three tables shows an example +where \verb`rows` is specified. The second and third tables use an +abbreviated form of the row variable specification, \verb`rspec`\index{rspec@\texttt{rspec}}. +This is a three-element comma list, \verb`{rvar,rstep,rows}` (with +an optional fourth element, \verb`rround`; see §\ref{subsec:Rounding:rround}). +The second table gives a straightforward example. In the third table +a simple \LaTeX{} expression has been inserted for \verb`rows` in +the \verb`rspec` comma list. Like \verb`rstep` and \verb`rstop`, +\verb`rows` can be a \LaTeX{} expression but unlike \verb`rstep` +and \verb`rstop` it is evaluated \emph{before} the vv-list and therefore +cannot depend on quantites specified there like the initial row variable +value. On evaluation, the \LaTeX{} expression is rounded to an integer. \begin{verbatim} \tabulate[rvar=x,rstep=0.2,rows=6] { \sin x/\cos x }[x=0][*] \qquad \tabulate[rspec={x,0.2,6}] { \tan x }[x=0][*] \qquad - \tabulate[rspec={x,0.2,1+(1/0.2)}] - { \sqrt{\sec^2 x - 1} }[x=0][*] + \tabulate[rspec={x,0.2,1.2/0.2}] + { \sqrt{\sec^2 x - 1} }[x=0.2][*] \end{verbatim} $\Longrightarrow$ \tabulate[rvar=x,rstep=0.2,rows=6] - { \sin x/\cos x }[x=0][*] \qquad + { \sin x/\cos x }[x=0.2][*] \qquad \tabulate[rspec={x,0.2,6}] - { \tan x }[x=0][*] \qquad - \tabulate[rspec={x,0.2,1+(1/0.2)}] - { \sqrt{\sec^2 x - 1} }[x=0][*]\index{row variable!start, step, stop values|)}\medskip{} + { \tan x }[x=0.2][*] \qquad + \tabulate[rspec={x,0.2,1.2/0.2}] + { \sqrt{\sec^2 x - 1} }[x=0.2][*]\index{row variable!start, step, stop values|)} + +\subsubsection{Rounding: \texttt{rround}} +\label{subsec:Rounding:rround}After\index{rounding value!row variable|(}\index{rround@\texttt{rround}|(} +studying some of the previous tables, we might decide to adjust the +step size, say from $0.2$ to $0.25$. But changing \verb`rstep` +to the new value gives a disconcerting result, as you can see from +the first of the tables below. \texttt{numerica-tables} uses a default +rounding value of $1$ for the row variable and has rounded $0.25$ +down to $0.2$, then $0.2+0.25=0.45$ down to $0.4$, then $0.4+0.25=0.65$ +down to $0.6$, then $0.6+0.25=0.85$ down to $0.8$, at which point +it stops since $0.8+0.25>1$ which is the stopping value. The rounded-down +values are the values used \textendash{} always it is the \emph{displayed} +rounded value that is used for calculating function values. The second +table corrects matters by adjusting the row variable rounding with +\verb`rround=2`. From version 3.2 of \texttt{numerica-tables} the +\verb`rround` value can also be included as an optional fourth element +to \verb`rspec`\index{rspec@\texttt{rspec}}. The third table illustrates +this abbreviated form of specification: +\begin{verbatim} + \tabulate[rvar=x,rstep=0.25,rstop=1] + { \exp x }[x=0][*] \qquad + \tabulate[rvar=x,rstep=0.25,rstop=1,rround=2] + { \exp x }[x=0][*] \qquad + \tabulate[rspec={x,0.25,5,2}] + { \exp x }[x=0][*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rvar=x,rstep=0.25,rstop=1] + { \exp x }[x=0][*] \qquad + \tabulate[rvar=x,rstep=0.25,rstop=1,rround=2] + { \exp x }[x=0][*] \qquad + \tabulate[rspec={x,0.25,5,2}] + { \exp x }[x=0][*]\medskip{} + +The reason for allowing the inclusion of the \verb`rround` value +in \verb`rspec` is because the values included in \verb`rspec`, +together with the initial value assigned to the row variable in the +vv-list, determine the numbers in the row variable column used to +calculate function values. The further settings discussed below in +§\ref{subsec:Row-var-col-formatting} are \emph{formatting} elements +affecting appearance but irrelevant to the actual values calculated.\index{rround@\texttt{rround}|)}\index{rounding value!row variable|)} \subsection{Row variable specification: non-uniform case} -\label{subsec:Row-var-spec-nonuniform} -\begin{table}[b] +\label{subsec:Row-var-spec-nonuniform}Occasionally one wants to form +a table in which the row variable does not increase or decrease in +regular steps; for examples, see \emph{HMF} Tables 1.1 and 3.1. (Tables +9.7, 10.5, 10.10 use two step values and could also be handled with +these settings.) For instance, one might want a table of values of +simple functions of a list of constants, or a table of function values +at $\pi$, $\pi/2$, $\pi/3$, $\pi/4$, $\dots$, or at $1$, $10$, +$100$, $1000$, $\dots$, or a table of function values at thoroughly +irregular, perhaps experimentally determined, values. + +\texttt{numerica-tables} provides two means of specifying such row +variables, either by means of a row variable function (\verb`rfunc`\index{rfunc@\texttt{rfunc}}), +when the row variable values change in a non-uniform but formulaic +way, or by explicitly listing the row variable values in a comma list +(\verb`rdata`\index{rdata@\texttt{rdata}}, \verb`rfile`\index{rfile@\texttt{rfile}}), +or with some other separator. In the latter case, the row variable +can be displayed either as a sequence of \emph{values}, or verbatim +as a sequence of \emph{expressions} \textendash{} like fractions of +$\pi$ \textendash{} with the \verb`rverb`\index{rverb@\texttt{rverb}} +setting. + +\subsubsection{\texttt{rfunc}} + +\begin{table}[t] +\centering \centering{}\caption{Non-uniform row variable specification} \begin{center} -\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{3cm}} +\begin{tabular}{l>{\raggedright}p{2cm}>{\raggedright}p{4cm}>{\raggedright}p{3cm}} \toprule {\small key} & {\small type} & {\small meaning} & {\small comment}\tabularnewline \midrule -{\small\texttt{rdata}} & {\small comma list} & {\small list of row-var. values } & {\small comma list may be stored in a macro}\tabularnewline -{\small\texttt{rfile}} & {\small chars} & {\small filepath/name} & {\small file of comma separated values}\tabularnewline -{\small\texttt{rverb}} & {\small int (}{\small\texttt{0}}/\texttt{1}) & {\small display }{\small\texttt{rdata }}{\small or}{\small\texttt{ rfile}}{\small{} -values verbatim} & {\small initialized to }{\small\texttt{0}}\tabularnewline -{\small\texttt{rfunc}} & {\small token(s)} & {\small function of a stepped variable} & \tabularnewline +{\small\texttt{rfunc}} & {\small token(s)} & {\small formula for row var. values} & \tabularnewline +{\small\texttt{rdata}} & {\small comma list or macro} & {\small list or macro (containing list) of row var. values} & \tabularnewline +{\small\texttt{rfile}} & {\small chars} & {\small filepath/filename} & {\small file contains list of row var. values}\tabularnewline +{\small\texttt{rdelim}} & {\small char} & {\small item separator for }{\small\texttt{rdata}}{\small{} or }{\small\texttt{rfile}}{\small{} +lists} & {\small defaults to }{\small\texttt{,}}{\small{} or }{\small\texttt{;}}{\small{} +depending as }{\small\texttt{.}}{\small{} or }{\small\texttt{,}}{\small{} +is decimal mark}\tabularnewline +{\small\texttt{rverb}} & {\small fp (}{\small\texttt{0/0.5/1}}{\small )} & {\small display }{\small\texttt{rdata}}{\small , }{\small\texttt{rfile}}{\small{} +values verbatim (}{\small\texttt{1}}{\small ), or slash fractions formatted +(}{\small\texttt{0.5}}{\small )} & {\small initialized to }{\small\texttt{0}}\tabularnewline \bottomrule \end{tabular} \par\end{center} \end{table} -Occasionally one wants to form a table in which the row variable does -not increase or decrease in regular steps; for examples, see \emph{HMF} -Tables 1.1 and 3.1. (Tables 9.7, 10.5, 10.10 use two step values and -could also be handled with these settings.) For instance, one might -want a table of values of simple functions of a list of constants, -or a table of function values at $\pi$, $\pi/2$, $\pi/3$, $\pi/4$, -$\dots$, or at $1$, $10$, $100$, $1000$, $\dots$, or a table -of function values at thoroughly irregular, perhaps experimentally -determined, values. - -\texttt{numerica-tables} provides two means of specifying such row -variables, either by means of a row variable function (\verb`rfunc`\index{rfunc@\texttt{rfunc}}), -when the row variable values change in a non-uniform but formulaic -way, or by explicitly listing the row variable values in a comma list -(\verb`rdata`\index{rdata@\texttt{rdata}}, \verb`rfile`\index{rfile@\texttt{rfile}}). -In this latter case, a setting \verb`rverb`\index{rverb@\texttt{rverb}} -enables the row variable column to be displayed either as a sequence -of \emph{values} or verbatim as a sequence of \emph{expressions} \textendash{} -like fractions of $\pi$. - -\subsubsection{\texttt{rfunc}} - -\index{rfunc@\texttt{rfunc}|(}\index{row variable!variably stepped|(}Suppose +\index{rfunc@\texttt{rfunc}|(}\index{row variable!variably stepped|(}\label{subsec:rfunc}Suppose \textendash{} perhaps with an interest in the distribution of prime numbers \textendash{} that we want to create a small table of values of $n/\ln n$ for, say, $n=10$, $100$, 1000, $10000$, $\ldots$ -The prospective row variable $n$ is not increasing in constant steps, -although clearly in a regular manner. We handle such cases with the -\verb`rfunc` setting; in the present instance \verb`rfunc=10^n` -where now \verb`n` does increment by a constant amount: -\begin{verbatim} - \tabulate[rpos=1,rfunc=10^n,rvar=n,rstep=1, - rows=7,rround=0] - { n/\ln n }[n=1][0] -\end{verbatim} -$\Longrightarrow$ \tabulate[rpos=1,rfunc=10^n,rvar=n,rstep=1, - rows=7,rround=0] - { n/\ln n }[n=1][0]\medskip{} - -The variable \verb`n` has two different meanings here. Initially -it is the variable of a simple step function incrementing by $1$ -at each step. The function takes values \verb`10^n`. Once the table -is compiled however \verb`n` is used to denote these successive function -values, $10,100,\ldots,10000000$. To the \emph{reader} of the table, -only this latter meaning is evident. Yes, it would be possible to -add further keys to specify a distinct step function variable and -its start, step and stop values, but the list of keys to specify a -table is already large. The `double usage' implemented, perhaps -confusing initially, economizes on keys and is invisible to the reader -of the table. The initial value \verb`n=1`\index{vv-list!row/col. variable initial values} +The prospective row variable $n$ is not increasing uniformly, although +clearly in a formulaic way. The key \verb`rfunc` is for such cases; +in the present instance \verb`rfunc=10^n` where now \verb`n` does +increment by a constant amount: +\begin{verbatim} + \tabulate[rfunc=10^n,rvar=n,rstep=1,rows=7] + { n/\ln n }[n=1][0] \qquad + \tabulate[rfunc=10^n,rvar=n,rstep=1,rows=7,norules] + { n/\ln n }[n=1][0] +\end{verbatim} +$\Longrightarrow$ \tabulate[rpos=1,rfunc=10^n,rvar=n,rstep=1,rows=7] + { n/\ln n }[n=1][0]\qquad + \tabulate[rfunc=10^n,rvar=n,rstep=1,rows=7,norules] + { n/\ln n }[n=1][0]\medskip{} + +The benefit of the \verb`norules`\index{norules@\texttt{norules}} +setting in the second table is obvious. + +The variable \verb`n` has two meanings in these tables although only +one is publicly visible. Initially \verb`n` is the variable of a +step function, incremented by \verb`rstep=1` and taking values \verb`10^n`. +Once the table is compiled, \verb`n` denotes these successive function +values, $10,100,\ldots,10000000$. To the \emph{reader}, only this +latter meaning is evident, the potentially confusing `double usage' +not so. The initial value \verb`n=1`\index{vv-list!row/col.variables@row/col.\ variables}\index{row variable!initial value in vv-list} in the vv-list applies to the row variable function \verb`10^n`, -not to the function being tabulated (so the error-producing expression -\verb`1/\ln 1` does not arise).\index{rfunc@\texttt{rfunc}|)}\index{row variable!variably stepped|)} - -\subsubsection{\texttt{rdata}, \texttt{rfile}, \texttt{rverb} } - -\label{subsec:rdata-rfile-rverb}One\index{row variable!values from list, macro or file|(} -source of difficulty in reading the previous table is working out -just how many zeros there are in the larger numbers in the left column. -It would be more readable `at a glance' if we could write those -in scientific notation. To do that we use the \verb`rdata`\index{rdata@\texttt{rdata}|(} -and \verb`rverb`\index{rverb@\texttt{rverb}|(} keys. In the first -of the examples below, \verb`rverb` is absent (corresponding to the -default \verb`rverb=0`); in the second \verb`rverb=1`, the effect -of which is to use verbatim the row variable values provided in the -\verb`rdata` comma list:\index{row variable!verbatim values|(} +not to the function \verb`n/\ln n` being tabulated (so the error-producing +expression \verb`1/\ln 1` does not arise).\index{rfunc@\texttt{rfunc}|)}\index{row variable!variably stepped|)} + +\subsubsection{\texttt{rdata}, \texttt{rfile}, \texttt{rdelim}, \texttt{rverb} } + +\label{subsec:rdata-rfile-rverb}\index{row variable!values from list/macro/file|(}A +difficulty in reading the last two tables is working out just how +many zeros there are in the larger numbers in the left column. They +would be more readable `at a glance' if we could write them in scientific +notation. To do that, use the \verb`rdata`\index{rdata@\texttt{rdata}|(} +and \verb`rverb`\index{rverb@\texttt{rverb}|(} keys. In the tables +above, \verb`rverb` is absent (corresponding to \verb`rverb=0`); +in the tables below \verb`rverb=1`, the effect of which is to use +the row variable values in the \verb`rdata` list verbatim.\index{row variable!verbatim values|(} +In the first example below, the larger values of the row variable +are now expressed in scientific notation, which gives a table that +is easier to grasp \textendash{} but less pleasing to the eye. The +right alignment\index{ralign@\texttt{ralign}} of the column is the +culprit, as is seen when presented left-aligned in §\ref{subsec:RValignment:ralign}. \begin{verbatim} \tabulate[rdata={10,100,1000,10^4,10^5,10^6,10^7}, - rvar=n,rround=0,ralign=l] - { n/\ln n }[0]\qquad - \tabulate[rdata={10,100,1000,10^4,10^5,10^6,10^7}, - rverb=1,rvar=n,rround=0,ralign=l] - { n/\ln n }[0] + rverb=1,rvar=n]{ n/\ln n }[0] \qquad + \def\mydata{\sfrac14\,\pi,\sfrac13\,\pi,\sfrac12\,\pi, + \sfrac23\,\pi,\sfrac34\,\pi,\pi} + \tabulate[rdata=\mydata,rverb=1,rvar=k] + { k }[*] \end{verbatim} $\Longrightarrow$ \tabulate[rdata={10,100,1000,10^4,10^5,10^6,10^7}, - rvar=n,rround=0] - { n/\ln n }[0]\qquad - \tabulate[rdata={10,100,1000,10^4,10^5,10^6,10^7}, - rverb=1,rvar=n,rround=0] - { n/\ln n }[0]\medskip{} - -\verb`rverb` is particularly useful if you want to make a table of -simple functions of constants. In the following example, with its -initial setting (\verb`rverb=0`), the constants would be listed in -the first column as numbers and repeated in the second column (perhaps -to a different number of decimal places), which would all be rather -pointless. With \verb`rverb=1`, the constants are listed verbatim -in the first column against their numerical values in the second. -In the example, the data has been stored in macros, \verb`\mydatai` -and \verb`\mydataii`, prior to calling the \verb`\tabulate` command. -\verb`\mydataii` uses the \verb`\sfrac` command from the \texttt{xfrac} -package. By setting \verb`rdata` equal to these macros, the \verb`\tabulate` -command gains access to the values stored in them. -\begin{verbatim} - \def\mydatai{\tfrac12\pi,\tfrac13\pi,\tfrac23\pi, - \tfrac14\pi,\tfrac34\pi} - \renewcommand\arraystretch{1.2} - \tabulate[rdata=\mydatai,rverb=1,rvar=k,headless=1] - { k }[6*] - \renewcommand\arraystretch{1} \qquad - \def\mydataii{\sfrac\pi2,\sfrac\pi3,\sfrac{2\pi}3, - \sfrac\pi4,\sfrac{3\pi}4} - \tabulate[rdata=\mydataii,rverb=1,rvar=k,headless=1] - { k }[6*] -\end{verbatim} -$\Longrightarrow$ \renewcommand\arraystretch{1.2} - \def\mydatai{\tfrac12\pi,\tfrac13\pi, - \tfrac23\pi,\tfrac14\pi,\tfrac34\pi} - \tabulate[rdata=\mydatai,rverb=1,rvar=k,headless=1 ] - { k }[6*] - \renewcommand\arraystretch{1} \qquad - \def\mydataii{\sfrac\pi2,\sfrac\pi3, - \sfrac{2\pi}3,\sfrac\pi4,\sfrac{3\pi}4} - \tabulate[rdata=\mydataii,rverb=1,rvar=k,headless=1] - { k }[6*]\medskip{} - -The setting \verb`headless=1`\index{headless@\texttt{headless}} -(see §\ref{subsec:Header-row-suppress})\index{header row|see{column variable header, \texttt{headless}, row variable header}} -means the tables have no header row. To accommodate the tower of \verb`\tfrac`-s -in the first column of the first table, \verb`\arraystretch`\index{arraystretch@\texttt{\textbackslash arraystretch}} -has been used to add more space between the rows (and re-set afterwards). -Through the virtues of \verb`\sfrac` \index{sfrac@\texttt{\textbackslash sfrac}} -it is unnecessary in the second table which is neater and more readable -to my eye. But even better is the example at §\ref{subsec:rfracFraction-form-rvar}. - -Note that the row variable can be chosen arbitrarily \textendash{} -I earlier used \verb`n` and have used \verb`k` here but it could -be anything. Nor does the function tabulated need to be the identity -\textendash{} see the earlier $n/\ln n$ example or a later example -in a multi-column setting in §\ref{subsec:Headerstyle:-multi-column-case}. -The identity function was appropriate for tables showing the decimal -values of symbolic constants, as was the \verb`rverb=1` setting. + rverb=1,rvar=n,rround=0]{ n/\ln n }[0] \qquad + \def\mydata{\sfrac14\,\pi,\sfrac13\,\pi,\sfrac12\,\pi, + \sfrac23\,\pi,\sfrac34\,\pi,\pi} + \tabulate[rdata=\mydata,rverb=1,rvar=k] + { k }[*] \medskip{} + +In the second example \verb`rverb=1` is used to make a table of simple +fractions of $\pi$, listed verbatim in the row variable column against +their decimal values in the second column (the pointless header row +is discussed shortly). When \verb`rverb=1`, \verb`rround`\index{rround@\texttt{rround}} +does not apply. The full value is used in the calculation. For the +fractions I have used the \verb`\sfrac`\index{sfrac@\texttt{\textbackslash sfrac}} +command from the \texttt{xfrac}\index{xfrac package@\texttt{xfrac} package} +package (loaded in the preamble to the present document). \verb`\sfrac` +produces an elegant inline fraction with a smaller vertical footprint +than \verb`\tfrac` and so is better suited to use in a table column +that aligns fraction above fraction. The data has been stored in the +macro \verb`\mydata`. By setting \verb`rdata` equal to this macro, +the \verb`\tabulate` command gains access to the values stored in +it. + +In addition to a list or a macro, data for the row variable can also +be stored as a list in a file \textendash{} say \verb`mydata.txt`. +If \verb`mydata.txt` is placed in the directory of the current document +and \verb`rfile=mydata.txt`\index{rfile@\texttt{rfile}} entered +in the settings option of the \verb`\tabulate` command, the file +will be found and the values in the file used for the row variable. +Alternatively, the file could be placed in your \verb`texmf` tree +and your \TeX{} distro alerted to its presence (by refreshing the filename +database). Again \verb`rfile=mydata.txt` in the settings option will +ensure the file is found and the contents used for the row variable +values. Or, the file could be stored elsewhere and the \verb`rfile` +key equated to the full path and filename \textendash{} something +like \verb`rfile=e:/mydocs/mydatafiles/mydata.txt`. This ensures +the file will be found and the contents used for the row variable +values. Note that even in Windows systems (where file paths use backslashes) +the path requires that forward slashes only be used. \index{row variable!values from list/macro/file|)} + +The example above uses commas to separate the values accessed by \verb`rdata`. +In earlier versions of \texttt{numerica-tables} this was the case +even if the \verb`comma` package option was used with \texttt{numerica} +(meaning values using a comma as the decimal mark would need to be +enclosed in braces). From version 3.2 of \texttt{numerica-tables}, +if the \verb`comma` package option\index{comma package option@\texttt{comma} package option} +is used when calling \texttt{numerica}, then the list that \verb`rdata` +is equated to (or the file that \verb`rfile` references) should be, +by default, a \emph{semicolon} list.\index{decimal comma!semicolon use}\index{decimal comma!and rdata, rfile@and \texttt{rdata}, \texttt{rfile}} +It is also possible, from version 3.2, to specify your own delimiter +(item separator) by using the \verb`rdelim`\index{rdelim@\texttt{rdelim}} +setting. For instance, \verb`rdelim=|` would mean that the `pipe' +character separates items in the \verb`rdata` list (or the file referenced +by \verb`rfile`). + +An irritant with the second table above is the pointless header row. +To suppress it I could anticipate and enter \verb`headless`\index{headless@\texttt{headless}} +(see §\ref{subsec:Header-row-suppress}) in the settings option. But +perhaps a better solution is to present the table horizontally by +means of the \verb`transpose` setting. Now the pointless header in +the vertical table becomes a pointless first column in the horizontal +table. The way to suppress that is by giving a zero value to the setting +\verb`rpos`\index{rpos@\texttt{rpos}} (see §\ref{subsec:Row-var-col-pos} +for a discussion of \verb`rpos`). \verb`rpos` determines the placement +of the row variable column and a zero value suppresses its display +entirely. I have also suppressed all rules with the \verb`norules`\index{norules@\texttt{norules}} +setting, and reduced the rounding value to $4$. At which point the +remaining quibble is with the default right alignment of the entries +in the header row. I have anticipated again and centred each column +with the \verb`calign=c`\index{calign@\texttt{calign}} setting (§\ref{subsec:Alignment:-calign}). +\begin{verbatim} + \def\mydata{\sfrac14\,\pi, \sfrac13\,\pi, \sfrac12\,\pi, + \sfrac23\,\pi, \sfrac34\,\pi, \pi} + \tabulate[rdata=\mydata,rverb=1,rvar=k,rpos=0,calign=c, + norules,transpose]{ k }[4*] +\end{verbatim} +$\Longrightarrow$ \def\mydata{\sfrac14\,\pi, \sfrac13\,\pi, \sfrac12\,\pi, + \sfrac23\,\pi, \sfrac34\,\pi, \pi } + \tabulate[rdata=\mydata,rverb=1,rvar=k,rpos=0,calign=c, + norules,transpose]{ k }[4*] \index{transpose@\texttt{transpose}!examples of use}\index{norules@\texttt{norules}}\medskip{} + + +\subsubsection{Fraction-form values} + +\label{subsec:rfracFraction-form-rvar}\index{row variable!fraction form|(}\index{fraction form!row variable values|(} The \verb`rverb` setting applies \emph{only} to the \verb`rdata` -and \verb`rfile` keys. It has no effect otherwise.\index{row variable!verbatim values|)}\index{rverb@\texttt{rverb}|)}\index{rdata@\texttt{rdata}|)} - -In addition to a comma list or macro, data for the row variable can -also be stored in a file of comma-separated values \textendash{} say -\verb`mydata.txt`. If \verb`mydata.txt` is placed in the directory -of the current document and \verb`rfile=mydata.txt`\index{rfile@\texttt{rfile}} -entered in the settings option of the \verb`\tabulate` command, the -file will be found and the values used for the row variable. Alternatively, -the file could be placed in your \verb`texmf` tree and your \TeX{} -distro alerted to its presence (by refreshing the filename database). -Again \verb`rfile=mydata.txt` in the settings option will ensure -the file is found and the contents used for the row variable values. -Or, the file could be stored elsewhere and the \verb`rfile` key equated -to the full path and filename \textendash{} something like \verb`rfile=e:/mydocs/mydatafiles/mydata.txt`. -This ensures the file will be found and the contents used for the -row variable values. Note that even in Windows systems (where file -paths use backslashes) the path requires that forward slashes only -be used. \index{row variable!values from list, macro or file|)} - -\subsection{Formatting row variable values \& column} +and \verb`rfile` keys. It has no effect otherwise. Besides \verb`0` +and \verb`1`, it can also take a third value, \verb`rverb=1/2`, +which can be thought of as `semi-verbatim'\index{rverb@\texttt{rverb}!`semi-verbatim'}. +(\verb`rverb=0.5` or indeed any expression in the syntax of \texttt{l3fp} +\textendash{} like \verb`sin(pi/6)` \textendash{} that evaluates +to one half will serve as well.) This renders `naked' slash fractions +from \verb`rdata` or \verb`rfile` input in formatted form using +the \verb`\sfrac` command from the \texttt{xfrac} package if it is +loaded (e.g. $\sfrac 23$), or in scriptstyle (e.g. $\scriptstyle 2/3$) +if it isn't. A `naked' slash fraction here means the fraction isn't +buried within parentheses or other constructs. + +I repeat the vertical presentation of the last table but use the `semi-verbatim' +setting. By moving the repeated factor \verb`\pi` from the data file +to the function being tabulated, the data file becomes a list of `naked' +slash fractions (and one integer) and the header row becomes meaningful. +Note that the integer is displayed as is with the `semi-verbatim' +setting; only the slash fractions are reformatted. Again, I have centred +the second column with the \verb`calign=c` setting: +\begin{verbatim} + \def\mydata{1/4,1/3,1/2,2/3,3/4,1} + \tabulate[rdata=\mydata,rverb=1/2, + rvar=k,calign=c] + { k\pi }[k=0][6*] +\end{verbatim} +$\Longrightarrow$ \def\mydata{1/4,1/3,1/2,2/3,3/4,1} + \tabulate[rdata=\mydata,rverb=0.5, + rvar=k,rhnudge=6,chnudge=18] + { k\pi }[k=0][6*] + +\subsection{Formatting the row variable column \& header} \label{subsec:Row-var-col-formatting}The padding option ({*}) of the trailing optional argument is one way of formatting the row variable values, but to how many decimal places? Aligned left or right or centred? -Under what heading \textendash{} the example tables so far have simply -used the row variable for the header? And should the row variable -column be at the left of the table, or the right \textendash{} or -both? These and related questions are answered by assigning values -to the keys listed in Table~\ref{tab:Formatting-the-row-variable}. -\begin{table}[tb] -\centering{}\caption{\protect\label{tab:Formatting-the-row-variable}Formatting the row -variable column} +Under what heading \textendash{} the examples so far have simply used +the row variable for the header? And should the row variable column +be at the left of the table, or the right \textendash{} or both? These +and related questions are answered by assigning values to the keys +listed in Table~\ref{tab:Formatting-the-row-variable}. + +\begin{table}[t] +\centering +\centering{}\caption{\label{tab:Formatting-the-row-variable}Formatting the row variable +column \& header} \begin{center} -\begin{tabular}{ll>{\raggedright}p{4cm}l} +\begin{tabular}{l>{\raggedright}p{2cm}>{\raggedright}p{4cm}>{\raggedright}p{2cm}} \toprule {\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline \midrule -{\small\texttt{rround}} & {\small int} & {\small rounding} & {\small\texttt{1}}\tabularnewline -{\small\texttt{rfont}} & {\small chars} & {\small font (}{\small\verb`\math<chars>`}{\small )} & \tabularnewline -{\small\texttt{ralign}} & {\small char (}{\small\texttt{r/c/l}}{\small )} & {\small horizontal alignment} & {\small\texttt{r}}\tabularnewline -{\small\texttt{rhead}} & {\small tokens} & {\small header} & {\small\texttt{rvar}}\tabularnewline -{\small\texttt{rhnudge}} & int & {\small nudge header }{\small{\small\verb`<int>`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline {\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small column placement} & {\small\texttt{1}}\tabularnewline -{\small\texttt{rvar'}} & {\small tokens} & {\small 2nd row variable col. spec. } & {\small\texttt{rvar}}\tabularnewline -{\small\texttt{rhead'}} & {\small tokens} & {\small header of 2nd rv col. (if it exists)} & {\small\texttt{rvar'}}\tabularnewline -{\small\texttt{rhnudge'}} & int & {\small nudge 2nd rv col. header }{\small{\small\verb`<int>`}}{\small{} +{\small\texttt{ralign}} & {\small char (}{\small\texttt{r/c/l}}{\small )} & {\small horizontal alignment} & {\small\texttt{r}}\tabularnewline +{\small\texttt{rfont}} & {\small chars} & {\small font (}{\small\verb`\math<chars>`}{\small )} & \tabularnewline +{\small\texttt{rhead}} & {\small tokens} & {\small header} & \tabularnewline +{\small\texttt{rhnudge}} & {\small fp} & {\small nudge header }{\small{\small\verb`<fp>`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline +{\small\texttt{rhsize}} & {\small int (}{\small\texttt{-4}}{\small\ldots}{\small\texttt{5}}{\small )} & {\small font size relative to }{\small\verb`0=\normalsize`} & {\small\texttt{0}}\tabularnewline +{\small\texttt{rmath}} & {\small char (}{\small\texttt{s/t/d}}{\small )} & {\small script-, text-, displaystyle} & {\small\texttt{t}}\tabularnewline +{\small\texttt{rvar'}} & {\small tokens} & {\small 2nd row variable col. spec. } & \tabularnewline +{\small\texttt{rhead'}} & {\small tokens} & {\small header of 2nd row var. col. (if it exists) } & \tabularnewline +{\small\texttt{rhnudge'}} & {\small fp} & {\small nudge 2nd row var. col. header }{\small{\small\verb`<fp>`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline -{\small\texttt{rfrac}} & {\small int (}{\small\texttt{0}}{\small\ldots 5)} & {\small fraction form} & {\small\texttt{0}}\tabularnewline \bottomrule \end{tabular} \par\end{center} \end{table} -\subsubsection{Rounding: \texttt{rround}} - -After\index{rounding value!row variable|(}\index{rround@\texttt{rround}|(}\index{formatting!row variable values|(} -studying some of the previous tables, we might decide to adjust the -step size, say from $0.2$ to $0.25$. But changing \verb`rstep` -to the new value gives a disconcerting result. \texttt{numerica-tables} -uses a default rounding value of $1$ for the row variable and has -rounded $0.25$ down to $0.2$, then $0.2+0.25=0.45$ down to $0.4$, -then $0.4+0.25=0.65$ down to $0.6$, then $0.6+0.25=0.85$ down to -$0.8$, at which point it stops since $0.8+0.85>1$ which is the stopping -value. The rounded-down values are the values used for caclulating -the sines. The second table corrects matters by adjusting the row -variable rounding with \verb`rround=2`. -\begin{verbatim} - \tabulate[rvar=x,rstep=0.25,rstop=1] - { \sin x }[x=0][*] (Eh???) \quad - \tabulate[rvar=x,rstep=0.25,rstop=1,rround=2] - { \sin x }[x=0][*] -\end{verbatim} -$\Longrightarrow$ \tabulate[rvar=x,rstep=0.25,rstop=1] - { \sin x }[x=0][*](Eh???) \quad - \tabulate[rvar=x,rstep=0.25,rstop=1,rround=2] - { \sin x }[x=0][*]\index{rround@\texttt{rround}|)}\index{rounding value!row variable|)} - -\subsubsection{Font: \texttt{rfont}} - -In\index{rfont@\texttt{rfont}} the second table below bolding by -means of the setting \verb`rfont=bf` has been applied to emphasize -the distinction between the row variable values and the function values. -Possible values for this key are those characters that can be adjoined -to \verb`\math` to give a meaningful result. Thus other valid values -are \verb`it` (italic), \verb`sf` (sans serif), \verb`tt` (typewriter); -\verb`frak` (Fraktur); also \verb`rm` (roman) is available, but -that is the default.\index{formatting!row variable values|)} - -\subsubsection{Alignment: \texttt{ralign}} - -By\index{row variable column!alignment|(}\index{ralign@\texttt{ralign}} -default, the alignment of all columns is to the right, as in previous -examples. This lends itself to neat output when padding with zeros -is activated (the \verb`*` in the trailing argument) and when some -values are negative \textendash{} the minus signs interfere with -neat output in left or centred alignments.\index{function value formatting!sign handling} -But in a case like the second table in the last example, you might -prefer to centre the headers for both the row and function-value columns. -These alignments are independently set. For the row variable column -the default alignment is to the right \verb`ralign=r`; \verb`ralign=l` -(lowercase L) aligns entries in the row variable column to the left, -and \verb`ralign=c` centres entries in the row variable column. The -tables of the next example use a \verb`c` alignment to centre the -row variable column header. The third of those tables shows how minus -signs spoil the effect. - -\subsubsection{Row-variable header: \texttt{rhead}} - -In\index{rhead@\texttt{rhead}|(}\index{row variable header|(}\index{formatting!row variable header|(} -the second and third tables, the header for the row variable column -has also been bolded. The default header is the row variable symbol. -That can be replaced by giving a value to the key \verb`rhead`. I -have used \verb`rhead=\boldsymbol{x}` (rather than \verb`\mathbf{x}`) -in order to get an italicized bold symbol. Note that you do not need -to include math delimiters in the specification. It is assumed that -\verb`rhead` will sit between \verb`$ $` delimiters which are inserted -automatically by \verb`numerica-tables`. -\begin{verbatim} - \tabulate[rvar=x,rstep=0.25,rstop=1, - rround=2,ralign=c] - { \sin x }[x=0][*]\qquad - \tabulate[rvar=x,rstep=0.25,rstop=1,rround=2, - ralign=c,rfont=bf,rhead=\boldsymbol{x}] - { \sin x }[x=0][*]\qquad - \tabulate[rvar=x,rstep=0.25,rstop=0.5,rround=2, - ralign=c,rfont=bf,rhead=\boldsymbol{x}] - { \sin x }[x=-0.5][*] -\end{verbatim} -$\Longrightarrow$ \tabulate - [rvar=x,rstep=0.25,rstop=1, - rround=2,ralign=c] - { \sin x }[x=0][*]\qquad - \tabulate - [rvar=x,rstep=0.25,rstop=1, - rround=2,ralign=c,rfont=bf,rhead=\boldsymbol{x}] - { \sin x }[x=0][*]\qquad - \tabulate - [rvar=x,rstep=0.25,rstop=0.5,rround=2, - ralign=c,rfont=bf,rhead=\boldsymbol{x}] - { \sin x }[x=-0.5][*]\index{rfont@\texttt{rfont}}\medskip{} - -In these tables the row variable column has been given a centred alignment. -The third table shows what goes wrong when \emph{some} values in the -row variable column are negative. Better then is to use padding, a -right alignment (the default), and to use a phantom in the header. -The first table below does this. The second table incorporates kerning -into the header to achieve the same effect: -\begin{verbatim} - \tabulate[rvar=x,rstep=0.25,rstop=0.5,rround=2, - rfont=bf,rhead=\boldsymbol{x}\hphantom{0}] - { \sin x }[x=-0.5][*]\qquad - \tabulate[rvar=x,rstep=0.25,rstop=0.5,rround=2, - rfont=bf,rhead=\boldsymbol{x}\mkern 9 mu] - { \sin x }[x=-0.5][*] -\end{verbatim} -$\Longrightarrow$ \tabulate - [rvar=x,rstep=0.25,rstop=0.5,rround=2, - rfont=bf,rhead=\boldsymbol{x}\hphantom{0}] - { \sin x }[x=-0.5][*]\qquad - \tabulate - [rvar=x,rstep=0.25,rstop=0.5,rround=2, - rfont=bf,rhead=\boldsymbol{x}\mkern 9 mu] - { \sin x }[x=-0.5][*]\medskip{} - -(To my eye, aligning the $\boldsymbol{x}$ above the first column -of digits after the decimal point gives a better result than truly -centring it in the column; compare these examples with the first two -tables of the previous example.)\index{rhead@\texttt{rhead}}\index{formatting!row variable header|)} - -\subsubsection{Nudging~the~header: \texttt{rhnudge}} - -However\index{rhnudge@\texttt{rhnudge}|(}, you might prefer to avoid -inserting positioning commands into the actual row variable header, -obscuring its true content. You can avoid doing this by setting the -key \verb`rhnudge`. - -The first table below reverts to the default right alignment, avoids -any positioning commands in the row variable header, but instead nudges -it into position with the setting \verb`rhnudge=9`. For positive -nudge values, nudging works in the \emph{opposite} sense to the alignment. -The units for nudging are mu (math units, 18 to a quad), but only -a number \textendash{} generally an integer \textendash{} should be -specified; the `mu' is supplied by \verb`numerica-tables`. - -In the second table below the row variable takes single digit integer -values, while the row variable name now occupies more than one character. -With a right alignment the header would protrude out to the left. -By giving \verb`rhnudge` a \emph{negative} value (\verb`rhnudge=-12` -in the example) it is brought back to a centred position in the row -variable column. -\begin{verbatim} - \tabulate[rvar=x,rstep=0.25,rstop=0.5,rround=2, - rfont=bf,rhead=\boldsymbol{x},rhnudge=9] - { \sin x }[x=-0.5][4*]\qquad - \tabulate[rvar=x_{\text{int}},rstep=1,rstop=4, - rround=0,rfont=bf,rhnudge=-12, - rhead=\boldsymbol{x_{\text{int}}}] - { \sin x_{\text{int}} }[x_{\text{int}}=0][4*] -\end{verbatim} -$\Longrightarrow$ \tabulate - [rvar=x,rstep=0.25,rstop=0.5,rround=2, - rfont=bf,rhead=\boldsymbol{x},rhnudge=9] - { \sin x }[x=-0.5][4*]\qquad - \tabulate - [rvar=x_{\text{int}},rstep=1,rstop=4, - rround=0,rfont=bf,rhnudge=-12, - rhead=\boldsymbol{x_{\text{int}}}] - { \sin x_{\text{int}} }[x_{\text{int}}=0][4*]\index{row variable header|)}\index{rhnudge@\texttt{rhnudge}|)} - \subsubsection{Position in the table: \texttt{rpos}} \label{subsec:Row-var-col-pos}\index{row variable column!position in table}\index{rpos@\texttt{rpos}}By @@ -729,68 +794,41 @@ table. Its position is determined by the value of the key \texttt{rpos}: \item \verb`rpos=0`, suppressed (no row variable column); \item \verb`rpos=1`, first column (the default); \item \verb`rpos=2`, last column; -\item \verb`rpos=3`, first and last columns;\texttt{ } -\item \verb`rpos=4`, first and last columns, with the values in the last -column a user-defined function of the first; see §\ref{subsec:Second-row-var-col}; +\item \verb`rpos=3`, first and last columns (e.g. see §\ref{sec:Multiple-function-tables});\texttt{ } +\item \verb`rpos=4`, first column and a last column with values produced +by a user-defined function of the first; see §\ref{subsec:Second-row-var-col}; \item Any other integer acts like \verb`rpos=1`. \end{itemize} -An example with \verb`rpos=3` is given shortly below, §\ref{sec:Multiple-function-tables}. - -\subsubsection{\texttt{rvar'}, \texttt{rhead'}, \texttt{rhnudge'}} - -These settings become relevant only when \verb`rpos=4`; see §\ref{subsec:Second-row-var-col}. - -\subsubsection{Fraction-form values: rfrac} - -\label{subsec:rfracFraction-form-rvar}\index{row variable!fraction form|(}\index{fraction form!row variable values|(} -By giving the setting \verb`rfrac` an integer value between $1$ -and $5$ inclusive it is possible to render the row variable values -as fractions with no more than \verb`rround` digits in the denominator. -Initially \verb`rfrac=0`, which gives decimal output. -\begin{itemize} -\item \verb`rfrac=1` produces a slash fraction like $2/3$; -\item \verb`rfrac=2` produces a scalable \verb`\frac`-tion like $\frac{2}{3}$ -in textstyle and $\dfrac{2}{3}$ in displaystyle; -\item \verb`rfrac=3` produces a non-scalable \verb`\tfrac` like $\tfrac{2}{3}$; -\item \verb`rfrac=4` produces a non-scalable \verb`\dfrac` like $\dfrac{2}{3}$; -\item \verb`rfrac=5` produces a slash fraction like $\sfrac 23$ (by means -of the \verb`\sfrac`\index{sfrac@\texttt{\textbackslash sfrac}} -command) in text- and displaystyles if the \texttt{xfrac} package -is loaded and like $\scriptstyle 2/3$ if it isn't; when used as a -super- or sub-script, the fractions reduce in size like $e^{\sfrac23}$ -or $e^{{\scriptscriptstyle 2/3}}$. -\end{itemize} -The following table repeats an earlier table from §\ref{subsec:rdata-rfile-rverb}, -but more neatly, through using \verb`\sfrac` (with the \verb`rfrac=5` -call) instead of \verb`\tfrac`, and moving $\pi$ from the row variable -column into the formula: -\begin{verbatim} - \def\mydataii{1/2,1/3,2/3,1/4,3/4} - \tabulate[rdata=\mydataii,rpos=1, - rvar=k,rfrac=5,rhnudge=6,chnudge=18] - { k\pi }[k=0][6*] +When a table is transposed, these settings apply to the \emph{new} +row variable column (the old header row).\index{transpose@\texttt{transpose}!and rpos@and \texttt{rpos}} +For instance, with \verb`rpos=2` we get what was originally the header +row treated as the row variable column and placed on the right: +\begin{verbatim} + \tabulate[rspec={x,0.2,5},rpos=2,transpose,norules] + { \tan x }[x=0.2][*] \end{verbatim} -$\Longrightarrow$ \def\mydataii{1/2,1/3,2/3,1/4,3/4} - \tabulate[rdata=\mydataii,rpos=1, - rvar=k,rfrac=5,rhnudge=6,chnudge=18] - { k\pi }[k=0][6*]\index{fraction form!row variable values|)}\index{row variable!fraction form|)} +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},rpos=2,transpose,norules] + { \tan x }[x=0.2][*]\index{transpose@\texttt{transpose}!examples of use}\index{norules@\texttt{norules}} -\subsection{Adjoined multi-function tables} +\paragraph{Adjoined multi-function tables} \label{subsec:Adjoining-tables}\index{adjoined tables}\index{multi-function tables} -How might one tabulate multiple functions simultaneously? \emph{HMF} -has many, many examples where multiple functions (like the trigonometric -or the hyperbolic functions) are tabulated in separate columns of -the same table. - -With the settings described so far, one way is to adjoin single column -tables. In the tables below, which display as a single multi-columned -table, I have used three different\index{rpos@\texttt{rpos}} \verb`rpos` -settings (\verb`rpos=1` is implicit in the first). This is one way -to build a table that displays as multi-column. If you use this method, -note that the \verb`%` comment characters are essential at the end -of the last argument of the \verb`\tabulate` commands if you want -the tables to abut exactly. Omitting them results in a space between +By assigning different values to \verb`rpos` it is possible to include +multiple functions in the same table by adjoining distinct single +column tables. \emph{HMF} has many, many examples where multiple functions +(like the trigonometric or hyperbolic functions) are tabulated in +separate columns of the same table. + +\texttt{numerica-tables} has a systematic way of doing this, described +in §\ref{sec:Multiple-function-tables}, but with the settings discussed +to this point, the same effect can be achieved by the present more +naive means. The example below displays as a single multi-columned +table but is composed of three separate tables. I have used three +different\index{rpos@\texttt{rpos}} \verb`rpos` settings (\verb`rpos=1`, +the default setting, is implicit in the first) to achieve this and +ended all but the last \verb`\tabulate` command with the \LaTeX{} +comment character \verb`%`. These characters are essential if the +tables are to abut exactly. Omitting them results in a space between the tables. \begin{verbatim} \tabulate[rspec={x,0.2,6}] @@ -810,19 +848,223 @@ $\Longrightarrow$ \tabulate [rpos=2,rspec={x,0.2,6}] { \tan x }[x=0][*]\medskip{} -\noindent However, tabulating more than one function at a time is -too common a need to have to resort to a fudge like adjoining tables. -\verb`numerica-tables` offers a systematic way of doing this; see -§\ref{sec:Multiple-function-tables}. +It is possible to similarly join transposed tables (you need the \verb`headless` +setting), stacked one below the other, but again that effect can be +achieved more readily in a systematic way \textendash{} see §\ref{sec:Multiple-function-tables}. + +\subsubsection{Alignment: \texttt{ralign}} + +\label{subsec:RValignment:ralign}\index{row variable column!alignment|(}\index{ralign@\texttt{ralign}}By +default, the alignment of all columns including the row variable column +is to the right, as in the examples so far. This lends itself to clean +output when padding with zeros is activated (the \verb`*` in the +trailing argument) or when some values are negative, since minus signs +can interfere with alignment of digits in left or centred alignments. +In the first example below I present an earlier table where the row +variable values were a mix of powers of $10$ multiplied out and in +exponent form. In that table (see §\ref{subsec:rdata-rfile-rverb}) +with its right alignment the eye stumbled over the exponents. In left +alignment, it is a column of `$1$'s which is aligned and the alignment +is `smooth'. Left alignment is achieved by setting \verb`ralign=l` +(a lowercase `L'). In the second table I have chosen a centred alignment +for the row variable column, \verb`ralign=c` (and once more used +\verb`calign=c` for the function-value column). The other possible +setting for \verb`ralign` is the default right alignment, \verb`ralign=r`. +\begin{verbatim} + \tabulate[rdata={10,100,1000,10^4,10^5,10^6,10^7}, + rverb=1,rvar=n,ralign=l] + { n/\ln n }[0]\qquad + \tabulate[rspec={x,0.01,7},rround=2,ralign=c,calign=c] + { \sin x }[x=1.51][*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rdata={10,100,1000,10^4,10^5,10^6,10^7}, + rverb=1,rvar=n,ralign=l] + { n/\ln n }[0]\qquad + \tabulate[rspec={x,0.01,7},rround=2, + ralign=c,calign=c] + { \sin x }[x=1.51][*] + +\subsubsection{Font: \texttt{rfont}} + +\index{rfont@\texttt{rfont}}\index{formatting!row variable column|(}You +may wish to distinguish the values in the row variable column from +the values in the function-value column typographically, for instance +by bolding them. This is effected through the setting \verb`rfont`. +Possible values for this setting are \verb`rm` (roman), \verb`bf` +(bold face), \verb`it` (italic), \verb`sf` (sans serif), and \verb`tt` +(typewriter). For instance, \verb`rfont=bf` applies \verb`\mathbf` +to each row variable value, \verb`rfont=tt` applies \verb`\mathtt` +to each row variable value, and so on. If you set \verb`rfont=xx` +when \verb`\mathxx` has not been defined, then a \LaTeX{} error will +result. Equally, something like \verb`rfont=cal` (where \verb`\mathcal` +\emph{has} been defined) should not cause error but will give unexpected +results if the digits $0$ to $9$ are not part of the font. In the +next example, I repeat the last example in `vertical' form but with +the row variable column values bolded to draw attention to their placement +on the right \textendash{} but note that the \verb`rfont` setting +has no effect on the row variable header. +\begin{verbatim} + \tabulate[rspec={x,0.2,5},rpos=2,rfont=bf] + { \tan x }[x=0.2][*] \qquad + \tabulate[rspec={x,0.2,5},rpos=2,rfont=bf, + rhead=\boldsymbol{x}\phantom{.}] + { \tan x }[x=-0.4][*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},rpos=2,rfont=bf] + { \tan x }[x=0.2][*] \qquad + \tabulate[rspec={x,0.2,5},rpos=2,rfont=bf, + rhead=\boldsymbol{x}\phantom{.}] + { \tan x }[x=-0.4][*] + +\subsubsection{Row variable header: \texttt{rhead}} + +\label{subsec:RVheader:rhead}\index{rhead@\texttt{rhead}|(}\index{row variable header|(}The +default header is the row variable symbol. In the second table that +has been changed by giving a value to the key \verb`rhead`. I have +used \verb`rhead=\boldsymbol{x}` (rather than \verb`\mathbf{x}`) +in order to get an italicized bold symbol. I've also added a \verb`\phantom{.}` +to \verb`rhead` in order to push the (by default) right-aligned header +leftwards and give a more centred appearance to its positioning. Math +delimiters (\verb`$ $`) are inserted automatically by \texttt{numerica-tables}. +If you do not want any header at all for the row variable column, +entering \verb`rhead= ` , an empty setting, will achieve this \textendash{} +at least from version 3.2 of \texttt{numerica-tables}. In earlier +versions you needed to set \verb`rhead` to (e.g.) a space of some +kind like \verb`\ ` (an inter-word space). \index{rhead@\texttt{rhead}|)}\index{formatting!row variable header|)} + +\subsubsection{Nudging the header: \texttt{rhnudge}} + +\label{subsec:RVnudging:rhnudge}Choosing a centred alignment for +the header to the row variable column will generally not be appropriate +when negative values are involved. In an earlier table I used a phantom +to get around this. Another way which avoids obscuring the true content +of the \verb`rhead` setting with positioning commands is to use the +setting \verb`rhnudge`\index{rhnudge@\texttt{rhnudge}|(}. + +Both tables below have the default right alignment of the row variable +column and use \verb`rhnudge` to adjust the header position. For +positive values, \verb`rhnudge` works in the \emph{opposite} sense +to the alignment, to the left for right alignment, and to the right +otherwise. The units for nudging are mu (math units, 18 to a quad), +but only a number should be specified; the `mu' is supplied by +\texttt{numerica-tables}. +\begin{verbatim} + \tabulate[rvar=x,rstep=0.25,rstop=0.5,rround=2, + rfont=bf,rhead=\boldsymbol{x},rhnudge=9] + { \sin x }[x=-0.5][4*]\qquad + \tabulate[rvar=x_{\text{int}},rstep=1,rstop=4, + rround=0,rfont=bf,rhnudge=-12, + rhead=\boldsymbol{x_{\text{int}}}] + { \exp x_{\text{int}} }[x_{\text{int}}=0][4*] +\end{verbatim} +$\Longrightarrow$ \tabulate + [rvar=x,rstep=0.25,rstop=0.5,rround=2, + rfont=bf,rhead=\boldsymbol{x},rhnudge=9] + { \sin x }[x=-0.5][4*]\qquad + \tabulate + [rvar=x_{\text{int}},rstep=1,rstop=4, + rround=0,rfont=bf,rhnudge=-12, + rhead=\boldsymbol{x_{\text{int}}}] + { \exp x_{\text{int}} }[x_{\text{int}}=0][4*]\medskip{} + +\noindent In the second table the row variable takes single digit +integer values, while the row variable name occupies more than one +character. With a right alignment the header would protrude out to +the left. By giving \verb`rhnudge` a \emph{negative} value (\verb`rhnudge=-12` +in the example) it is brought back to a centred position in the row +variable column. \index{rhnudge@\texttt{rhnudge}|)}\index{formatting!row variable column|)} + +\subsubsection{Font size of the header: \texttt{rhsize}} + +The font size of the header can be changed by assigning an integer +value to the key \verb`rhsize`\index{rhsize@\texttt{rhsize}}. Valid +values range from $-4$, corresponding to \verb`\tiny`, through $0$, +corresponding to \verb`\normalsize`, to $5$, corresponding to \verb`\Huge`. +The actual values in points differ depending as \verb`\normalsize` +is $10$pt, $11$pt, or $12$pt, but range from $5$pt for \verb`\tiny` +when \verb`\normalsize` is $10$pt, to $24.88$pt for \verb`\Huge` +in all three cases. See also §\ref{subsec:chsize}. + +\subsubsection{\texttt{rmath}} + +\label{subsec:rmath}By\index{rmath@\texttt{rmath}} default the row +variable column is displayed in textstyle (\verb`\textstyle`). By +setting \verb`rmath` to \verb`s`, \verb`t` or \verb`d` it is possible +to explicitly choose \verb`\scriptstyle`, \verb`\textstyle` or \verb`\displaystyle` +for its display. \verb`rmath` is relevant when one might want to +display more complicated structures than numbers in the row variable +column, e.g. verbatim items when using \verb`rdata` or \verb`rfile`, +or in transposed multi-function tables; see the example in §\ref{sec:Multiple-function-tables}. +Returning to an earlier example, suppose the macro \verb`\mydata` +contains a list of fractions, not in slash form but \verb`\frac`-form:\index{arraystretch@\texttt{\textbackslash arraystretch}} +\begin{verbatim} + \def\mydata{\frac14,\frac13,\frac12,\frac23,\frac34,1} + \tabulate[rdata=\mydata,rverb=1,rvar=k,calign=c] + { k\pi }[k=0][6*]\qquad + \renewcommand\arraystretch{1.2} + \tabulate[rdata=\mydata,rverb=1,rvar=k,calign=c] + { k\pi }[k=0][6*]\qquad + \renewcommand\arraystretch{1} + \tabulate[rdata=\mydata,rverb=1,rvar=k,rmath=s,calign=c] + { k\pi }[k=0][6*] +\end{verbatim} +$\Longrightarrow$ \def\mydata{\frac14,\frac13,\frac12,\frac23, + \frac34,1} + \tabulate[rdata=\mydata,rverb=1,rvar=k,calign=c] + { k\pi }[k=0][6*]\qquad + \renewcommand\arraystretch{1.2} + \tabulate[rdata=\mydata,rverb=1,rvar=k,calign=c] + { k\pi }[k=0][6*]\qquad + \renewcommand\arraystretch{1} + \tabulate[rdata=\mydata,rverb=1,rvar=k,rmath=s,calign=c] + { k\pi }[k=0][6*]\medskip{} + +\noindent The first table is unacceptable; the second (with \verb`\arraystretch` +set to $1.2$) and the third (with \verb`rmath=s`) are at least readable +but neither is really satisfactory. Better is to use \verb`\sfrac`, +either directly or by the `semi-verbatim' \verb`rverb` setting +discussed earlier in §\ref{subsec:rfracFraction-form-rvar}.\index{fraction form!row variable values|)}\index{row variable!fraction form|)}\index{row variable!verbatim values|)}\index{rverb@\texttt{rverb}|)}\index{rdata@\texttt{rdata}|)} + +\subsubsection{\texttt{rvar'}, \texttt{rhead'}, \texttt{rhnudge'}} + +These settings become relevant only when \verb`rpos=4`; see §\ref{subsec:Second-row-var-col}.\index{row variable header|)} \section{Column-variable settings} -\label{sec:Column-variable-settings} -\begin{table} +\label{sec:Column-variable-settings}When a function of \emph{two} +variables is tabulated, we generally think of one variable as the +primary variable and the other as a parameter. To tabulate such a +function, we use the primary variable as the row variable and create +a succession of adjacent columns of function values according to the +different values of the parameter. In this document I call this second +variable the \emph{column} variable (\verb`cvar`). Just as with the +row variable, to create a table we need its start value (specified +in the vv-list), and its step (\verb`cstep`) and stop (\verb`cstop`) +values\index{column variable!start, step, stop values|(}. +\begin{verbatim} + \tabulate[rspec={x,0.2,5},cvar=k,cstep=2,cstop=9] + { \sin kx }[k=3,x=0.2][4*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},cvar=k,cstep=2,cstop=9] + { \sin kx}[k=3,x=0.2][4*] \medskip{} + +In \index{cvar@\texttt{cvar}|(} this example \verb`cvar=k` is the +column variable. I have chosen a step size \verb`cstep=2`\index{cstep@\texttt{cstep}|(} +and a stop value \verb`cstop=9`\index{cstop@\texttt{cstop}|(}. The +start value ({\ttfamily\verb`k=3`}) is specified in the +vv-list\index{vv-list!row/col.variables@row/col.\ variables}\index{column variable!initial value in vv-list}. +Although in the example these values are numbers, all three values +could be \LaTeX{} expressions that evaluate to numbers. In particular, +the expressions for step and stop values may include the row and column +variables (in the example $x$ and $k$) which are assigned their +\emph{initial} vv-list values. + +\begin{table}[t] +\centering \caption{Column-variable specification} \centering{}\begin{center} -\begin{tabular}{ll>{\raggedright}p{4cm}l} +\begin{tabular}{ll>{\raggedright}p{3cm}l} \toprule {\small key} & {\small type} & {\small meaning} & {\small comment}\tabularnewline \midrule @@ -831,453 +1073,591 @@ too common a need to have to resort to a fudge like adjoining tables. {\small\texttt{cstop}} & {\small real num.} & {\small stop value} & {\small either }{\small\texttt{cstop}}\tabularnewline {\small\texttt{cols}} & {\small int} & {\small number of columns} & {\small or }{\small\texttt{cols}}\tabularnewline {\small\texttt{cspec}} & {\small comma list} & {\small\texttt{\{cvar,cstep}}{\small ,}{\small\texttt{cols\}}} & {\small short form spec.}\tabularnewline +{\small\texttt{chround}} & {\small int} & {\small col. var. rounding} & {\small default: }{\small\texttt{0}}\tabularnewline \bottomrule \end{tabular} \par\end{center} \end{table} -When a function of \emph{two} variables is being tabulated, we generally -think of one variable as the primary variable and the other as a parameter. -To tabulate such a function, one way to proceed, as we have seen, -is to create and adjoin separate tables, one per parameter value, -but that is clumsy. A more systematic procedure is to specify, in -addition to the row variable, a \emph{column} variable and its start, -step and stop values\index{column variable!start, step, stop values|(}. - -In \index{cvar@\texttt{cvar}|(} the following example \verb`cvar=k` -is the column variable. I have chosen a step size \verb`cstep=2`\index{cstep@\texttt{cstep}|(} -and a stop value \verb`cstop=9`\index{cstop@\texttt{cstop}|(}. As -with the row variable, the start value ({\ttfamily\verb`k=3`}) -of the column variable is specified in the vv-list\index{vv-list!row/col. variable initial values}\index{column variable!initial value in vv-list}. -Although in the example these values are numbers, all three values -could be \LaTeX{} expressions that evaluate to numbers. In particular, -the expressions for step and stop values may include the row and column -variables (in the example $x$ and $k$) which are assigned their -initial vv-list values. Note also the setting for \verb`rhead`\index{rhead@\texttt{rhead}} -which shows the reader of the table that the numerical values displayed -in the column headers are values of \verb`k`. This usage occurs throughout -\emph{HMF}. +\texttt{numerica-tables} has automatically inserted the row variable +header which now has a double function: it serves also as the `header' +of the header row. As you can see $x\mkern1mu\backslash\mkern1mu k$ +has been inserted into this cell. From version 3.2 of \texttt{numerica-tables} +the automatic insertion of this form of header in a multi-column table +occurs if \verb`rhead`\index{rhead@\texttt{rhead}} is omitted. (If +you want nothing to show, put \verb`rhead= `, an empty setting.) +This form of header above the row variable column occurs throughout +\emph{HMF}. In the present example it shows the reader that the numerical +values displayed in the row variable column are values of $x$ and +the values in the column headers are values of $k$. + +As with the row variable, rather than using a stop value,\texttt{ +}{\ttfamily\verb`cstop`}, you can specify the number of +columns, \verb`cols`\index{cols@\texttt{cols}|(}, explicitly. I +could have replaced {\ttfamily\verb`cstop=9`} with \verb`cols=4` +to get the same result. \verb`cols` specifies the number of \emph{function-value} +columns; the row variable column is ignored for this count. A quirk +of using \verb`cols` is that it is then possible to have a \emph{zero} +step size, \verb`cstep=0`.\index{cstep@\texttt{cstep}!zero value} +(A similar comment applies to \verb`rows` and \verb`rstep`\index{rstep@\texttt{rstep}!zero value}.) +For an instance where this is useful, see the last example in §\ref{subsec:Nudgingtheheaders:-chnudge}. + +Again as with the row variable, the column specification can be condensed +into a comma list with the key \verb`cspec`\index{cspec@\texttt{cspec}|(}. +This is a $3$-element comma list of the form \verb`{cvar,cstep,cols}` +(and as for \verb`rspec`, with an optional fourth element, \verb`chround`; +see §\ref{subsec:chround}). Thus, for the preceding table I could +also have written \begin{verbatim} - \tabulate[rspec={x,0.2,6},rhead=x\backslash k, - cvar=k,cstep=2,cstop=9] - { \sin kx }[k=3,x=0][*] -\end{verbatim} -$\Longrightarrow$ \tabulate - [rspec={x,0.2,6},rhead=x\backslash k, - cvar=k,cstep=2,cstop=9] - { \sin kx }[k=3,x=0][*] \medskip{} - -Again, as with the row variable, rather than using an explicit stop -value\texttt{ }{\ttfamily\verb`cstop`}, you might prefer -to specify the number of columns, \verb`cols`\index{cols@\texttt{cols}|(}, -explicitly. I could have replaced {\ttfamily\verb`cstop=9`} -with \verb`cols=4` to get the same result. Note that the number of -columns specified here is the number of \emph{function-value} columns; -the row variable column is ignored for this count. - -It is worth pointing out explicitly that if \verb`cols` is specified, -then it is possible to have a \emph{zero} step size, \verb`cstep=0`. -(A similar comment applies to \verb`rows` and \verb`rstep`.) - -And again, as with the row variable, it is possible to condense the -specification into a comma list with the key \verb`cspec`\index{cspec@\texttt{cspec}|(}. -This is a $3$-element comma list of the form \verb`{cvar,cstep,cols}`. -Thus, for the preceding table I could also have written -\begin{verbatim} - \tabulate[rspec={x,0.2,6},rhead=x\backslash k, - cvar=k,cstep=2,cols=4] - { \sin kx }[k=3,x=0][*] + \tabulate[rspec={x,0.2,5},cvar=k,cstep=2,cols=4] + { \sin kx }[k=3,x=0.2][4*] \end{verbatim} or more succinctly \begin{verbatim} - \tabulate[rspec={x,0.2,6},rhead=x\backslash k, - cspec={k,2,4}] - { \sin kx }[k=3,x=0][*] + \tabulate[rspec={x,0.2,5},cspec={k,2,4}] + { \sin kx }[k=3,x=0.2][4*] \end{verbatim} and produced the same table. -As with the row equivalents, \verb`cstep`, \verb`cstop` and \verb`cols` -can all be \LaTeX{} expressions. Again like the row equivalents, the -first two are evaluated \emph{after }the vv-list and so may depend -not only on numbers and constants but also the initial values of the -row and column variables, which are given those values in the vv-list.\index{vv-list!row/col. variable initial values} +Like their row equivalents, \verb`cstep`, \verb`cstop` and \verb`cols` +can all be \LaTeX{} expressions, and like those equivalents, the first +two are evaluated \emph{after }the vv-list and so may depend not only +on numbers and constants but also on the \emph{initial} values of +the row and column variables, which are assigned in the vv-list.\index{vv-list!row/col.variables@row/col.\ variables} \verb`cols` is evaluated \emph{before }the vv-list; it may be a \LaTeX{} expression but cannot depend on the row or column variable.\index{cvar@\texttt{cvar}|)}\index{cstep@\texttt{cstep}|)}\index{cstop@\texttt{cstop}|)}\index{cols@\texttt{cols}|)}\index{cspec@\texttt{cspec}|)}\index{column variable!start, step, stop values|)} +Multi-column tables can be transposed: simply add \verb`transpose` +to the settings option. I have done this for the last example. Transposition +automatically changes the positions of $k$ and $x$ in the new row +variable header:\index{transpose@\texttt{transpose}!examples of use} +\begin{verbatim} + \tabulate[rspec={x,0.2,5},cvar=k,cstep=2,cstop=9, + transpose]{ \sin kx }[k=3,x=0.2][4*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},cvar=k,cstep=2,cstop=9, + transpose] + { \sin kx}[k=3,x=0.2][4*] \medskip{} + +\noindent To my eye, this table looks odd. Admittedly, the oddness +is exaggerated by the new row variable column being a column of integers; +\verb`ralign=c` would help there. But the underlying problem remains: +we expect the coarser graining in the column variable and its header +row, and the finer graining in the row variable and its column. Transposition +has confounded this expectation. + +\subsection{Rounding: \texttt{chround}} + +\label{subsec:chround}\index{chround@\texttt{chround}|(}\index{rounding value!header row|(}Generally +the column variable increments in larger steps than the row variable. +Often these are integer steps. For that reason the default rounding +value of the column variable is $0$ (as against $1$ for the row +variable), but it can be changed by assigning a different value to +the key \verb`chround`. + +For transposed tables, rounding is applied \emph{before }transposition. +The function values displayed in the body of the table are calculated +for the \emph{displayed} values of the row and column variables. Transposition +doesn't change the displayed function values and therefore shouldn't +change the displayed row and column variable values, irrespective +of their placement in the table. In the last (transposed) example +the `new' column headers show that the original row variable rounding +has been retained. As an essential part of the column variable specification, +\verb`chround` can be included as an optional fourth element in \verb`cspec`\index{cspec@\texttt{cspec}}. + +Since the default column variable rounding is $0$, any non-integer +increment to that variable needs an explicit value to be assigned +to \verb`chround`. For instance if $k$ increments by, say, $0.25$, +then \verb`chround=2` will need to be entered in the settings option +of the \verb`\tabulate` command, or \verb`2` entered as the fourth +element of \verb`cspec`: +\begin{verbatim} + \tabulate[rspec={x,0.2,5},ralign=c, + cspec={k,0.25,4,2}] + { \sin kx }[k=3,x=0.2][*4] +\end{verbatim} +$\Longrightarrow$ \tabulate + [rspec={x,0.2,5},ralign=c, + cspec={k,0.25,4,2}] + { \sin kx }[k=3,x=0.2][*4]\index{rounding value!header row|)}\index{chround@\texttt{chround}|)}\index{formatting!column variable header} + \subsection{Column header formatting} -\label{subsec:Column-header-formatting}\index{formatting!column variable header|(}\index{column variable header|(} -\begin{table}[b] +\label{subsec:Column-header-formatting}\index{formatting!column variable header|(}\index{column variable header!formatting|(} +\begin{table}[t] +\centering \centering{}\caption{Formatting the column variable header} \begin{center} +{\small{}% \begin{tabular}{ll>{\raggedright}p{4cm}l} \toprule -{\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline +{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline \midrule -{\small\texttt{chstyle}} & {\small int (0\ldots 4)} & {\small header style} & {\small\texttt{0}}\tabularnewline -{\small\texttt{ctitle}} & {\small tokens} & {\small single col. alt. header} & \tabularnewline -{\small\texttt{chead}} & {\small tokens} & {\small user-defined header } & \tabularnewline -{\small\texttt{calign}} & {\small char (r/c/l)} & {\small column alignment} & {\small\texttt{r}}\tabularnewline -{\small\texttt{chnudge}} & {\small int} & {\small nudge header }{\small{\small\verb`int`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline -{\small\texttt{chround}} & {\small int} & {\small{} rounding} & {\small\texttt{0}}\tabularnewline +{\small\texttt{chstyle}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small header style} & {\small\texttt{0}}\tabularnewline +{\small\texttt{chead}} & {\small token(s)} & {\small user-defined col. var. header} & \tabularnewline +{\small\texttt{calign}} & {\small char (}{\small\texttt{r}}{\small /}{\small\texttt{c}}{\small /}{\small\texttt{l}}{\small )} & {\small column alignment} & {\small\texttt{r}}\tabularnewline +{\small\texttt{chnudge}} & {\small fp} & {\small nudge header }{\small{\small\verb`chnudge`}}{\small{} +mu} & {\small\texttt{0}}\tabularnewline +{\small\texttt{chfont}} & {\small chars} & {\small font (}{\small\verb`\math<chars>`}{\small )} & \tabularnewline +{\small\texttt{chmath}} & {\small char (}{\small\texttt{s}}{\small /}{\small\texttt{t}}{\small /}{\small\texttt{d}}{\small )} & {\small script-, text- or displaystyle} & {\small\texttt{t}}\tabularnewline +{\small\texttt{chsize}} & {\small int (}{\small\texttt{-4}}{\small\ldots}{\small\texttt{5}}{\small )} & {\small font size relative to }{\small\verb`0 => \normalsize`} & {\small\texttt{0}}\tabularnewline \bottomrule -\end{tabular} +\end{tabular}} \par\end{center} \end{table} -There are four built-in style settings for the header to the column -variable (or function-value) columns (the `ch' prefix evoking `column -header'). If these don't meet your needs or otherwise satisfy, then -it is possible to define your own header to the function value columns -using the key \verb`chead`. First I discuss the built-in styles. +Column header formatting depends on whether the table has a single +column of function values, or multiple columns of function values. +For the latter there are a number of built-in style settings for the +header, accessed by assigning a value (\verb`0`, \verb`1`, \verb`2`, +\verb`3`) to the setting \verb`chstyle`. This can (sometimes) be +adjusted with the settings \verb`chfont`, \verb`chnudge` and \verb`chmath`. +If these built-in styles don't satisfy then it is possible to define +your own header to the function-value columns by using the setting +\verb`chead`. \subsubsection{Single-column header} -When\index{column variable header!single column case} there is only +\index{column variable header!single column case}When there is only one column of function values, the function being tabulated is by -default set as the header to the column. This corresponds to setting -\verb`ctitle=*`\index{ctitle@\texttt{ctitle}!* setting@\texttt{{*}} setting} -(see §\ref{subsec:Title:-ctitle-setting} below). You may want some -other header. Then give \verb`ctitle`\index{ctitle@\texttt{ctitle}} -some other value (although note that giving it the value \texttt{{*}{*}}\index{ctitle@\texttt{ctitle}!** setting@\texttt{{*}{*}} setting} -will set both the function and the vv-list as the header; again see -§\ref{subsec:Title:-ctitle-setting}). Whatever value you set, it -will be typeset between math delimiters (\verb`$` signs) and can -be nudged (see §\ref{subsec:Nudgingtheheaders:-chnudge}) left or -right to fine-tune its position in the column. (If you want an asterisk -as the header, you will need to place it between \emph{two} pairs -of braces, \verb`ctitle={{*}}`, to prevent it being misinterpreted -as the default setting.) - -If you want some more complicated header, perhaps not constrained -by the \verb`$` delimiters, then give \verb`chead`\index{chead@\texttt{chead}} -a value. This key I discuss below in §\ref{subsec:chead}. \verb`chead` -is entirely up to the user to specify, including any math delimiters -and positioning (nudging) of elements. - -If both \verb`ctitle` and \verb`chead` are given, the \verb`chead` -value prevails. - -\subsubsection{Multi-column header: \texttt{chstyle}} - -\label{subsec:Headerstyle:-multi-column-case}\index{column variable header!multi-column case|(}\verb`chstyle=0`\index{chstyle@\texttt{chstyle}|(}\index{column variable header!multi-column case!chstyle@\texttt{chstyle}|(} -which is the default gives a header of the form displayed in the last -example, with only the column variable \emph{value} at the head of -each column. This style generally requires the row variable header -to indicate what the values denote, as in the example above where -\verb`rhead=x\backslash k`, the backslash separating row from column -variable. \emph{HMF} contains a multitude of instances of this style; -see Tables~9.7, 17.5, 21.1, 24.3, 27.4, etc. for examples. - -\verb`chstyle=1` changes the header of the \emph{first} function -value column to the form \emph{variable=value} \textendash{} in the -example below, to $k=1$. This may be an appropriate choice when a -small rounding value is being used and the resulting columns are narrow. -I can find only one real instance in \emph{HMF}, Table~26.7. In the -example I have used the \verb`rdata` setting to collect an assortment -of nonsense values and for some weird reason wish to tabulate the -sines of multiples of these oddballs. Note that the row variable setting -\verb`rhead` (producing $X\backslash k$) is no longer needed since -the column variable is now explicitly indicated. (But the table is -lacking a title \textendash{} what on earth are we calculating with -this strange group of numbers?) -\begin{verbatim} - \tabulate[rdata={-e^2,1.234e2,3.1416,\pi/\gamma,1/9}, - ^,rvar=X,rverb=1,cspec={k,1,3},chstyle=1] - { \sin kX }[k=1][3*] -\end{verbatim} -$\Longrightarrow$ \tabulate[rdata={-e^2,1.234e2,3.1416,\pi/\gamma,1/9}, - ^,rvar=X,rverb=1,cspec={k,1,3},chstyle=1] - { \sin kX }[k=1][3*]\medskip{} +default set as the header to the column. This can be nudged left or +right by giving a numerical value to \verb`chnudge`. It can be set +in script-, text- or displaystyle (see the example) by setting \verb`chmath`\index{chmath@\texttt{chmath}} +to \verb`s`, \verb`t` (the default) or \verb`d`. \verb`chfont`, +however, has no effect in the single column case. +\begin{verbatim} + \tabulate[rspec={x,0.25,5},rround=2,chmath=s] + { \frac{2}{\sqrt\pi} e^{-x^2} }[x=0][*] \qquad + \tabulate[rspec={x,0.25,5},rround=2] + { \frac{2}{\sqrt\pi} e^{-x^2} }[x=1][*] \qquad + \tabulate[rspec={x,0.25,5},rround=2,chmath=d] + { \frac{2}{\sqrt\pi} e^{-x^2} }[x=2][*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rspec={x,0.25,5},rround=2,chmath=s] + { \frac{2}{\sqrt\pi} e^{-x^2} }[x=0][*]\qquad + \tabulate[rspec={x,0.25,5},rround=2] + { \frac{2}{\sqrt\pi} e^{-x^2} }[x=1][*]\qquad + \tabulate[rspec={x,0.25,5},rround=2,chmath=d] + { \frac{2}{\sqrt\pi} e^{-x^2} }[x=2][*]\medskip{} + +You may want some other header to the function-value column, one of +your own choosing. In that case give \verb`chead`\index{chead@\texttt{chead}} +a value. You are responsible for the entire content, including any +math environment; see the examples at §\ref{subsec:Fraction-form-output}. + +\subsubsection{Multi-column header} + +\label{subsec:Headerstyle:-multi-column-case}\index{column variable header!multi-column|(}The +default header style lists the column variable \emph{value} at the +head of each function-value column. Implicitly it corresponds to \verb`chstyle=0`. +\begin{verbatim} + \tabulate[rspec={x,0.2,5},cvar=k,cstep=2,cstop=9] + { \sin kx }[k=3,x=0.2][4*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},cvar=k,cstep=2,cstop=9] + { \sin kx}[k=3,x=0.2][4*]\medskip{} + +\noindent Unless \verb`rhead` is explicitly specified, \texttt{numerica-tables} +will automatically insert a row variable header, like $x\:\!\backslash\:\negmedspace k$ +in the example, where the backslash separates row from column variable. +(But see also §\ref{subsec:diagbox,-slashbox} below.) \emph{HMF} +contains a multitude of instances of this style; see Tables~9.7, +17.5, 21.1, 24.3, 27.4, etc. for examples. As noted, you can override +the default assignment to \verb`rhead` with your own specification, +but for \verb`chstyle > 0` this is no longer true.\index{rhead@\texttt{rhead}} + +\subsubsection{\texttt{diagbox}, \texttt{slashbox}} + +\label{subsec:diagbox,-slashbox}Rather than having $x\:\!\backslash\:\negmedspace k$ +or something similar automatically inserted, you might be tempted +to use \verb`\diagbox`\index{diagbox@\texttt{\textbackslash diagbox}} +from the \texttt{diagbox} package, or \verb`\backslashbox`\index{backslashbox@\texttt{\textbackslash backslashbox}} +from the earlier \texttt{slashbox}\index{slashbox package@\texttt{slashbox} package} +package, say \verb`rhead=\diagbox{x}{k}`. Because \verb`rhead` wraps +its contents in math delimiters (\verb`$` signs) this causes a \LaTeX{} +error. The solution is to \emph{add} \verb`$` signs: \verb`rhead=$\diagbox{x}{k}$` +or, since $x$ and $k$ should be in math-italic, \verb`rhead=$\diagbox{$x$}{$k$}$`.\index{rhead@\texttt{rhead}} +You might therefore try +\begin{verbatim} + \tabulate[rspec={x,0.2,5},ralign=c, + cvar=k,cstep=2,cstop=9, + rhead=$\diagbox{$x$}{$k$}$] + { \sin kx }[k=3,x=0.2][4*] +\end{verbatim} +but the diagonal box is far too large. One solution is to use the +optional first argument of the \verb`\diagbox` command. In that argument +one can specify an explicit \verb`height` and \verb`width` but I +prefer to work solely with the settings available in \texttt{numerica-tables}. +In that spirit, put \verb`rhsize=-4`, its smallest value. Alas this +also shrinks the displayed size of the variables $x$ and $k$. To +counteract that, explicitly specify \verb`\small$x$` and \verb`\small$k$` +for the mandatory \verb`\diagbox` arguments, the \verb`\small` overriding +the \verb`rhsize` setting to give a presentable result: +\begin{verbatim} + \tabulate[rspec={x,0.2,5},ralign=c,rhsize=-4,cspec={k,2,4}, + rhead=$\diagbox{\small$x$}{\small$k$}$] + { \sin kx }[k=3,x=0.2][4*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},ralign=c,rhsize=-4,cspec={k,2,4}, + rhead=$\diagbox{\small$x$}{\small$k$}$] + { \sin kx }[k=3,x=0.2][4*] + +However, the \texttt{booktabs} manual admonishes against the use of +vertical rules and double rules and generally advises to use as few +rules as possible. By adding a line (admittedly \emph{diagonal}, so +only half sinning) the \texttt{diagbox} or \texttt{slashbox} packages +are working in the opposite direction, more suited to tables in the +older `grid' style. In the present document I shall use $x\:\!\backslash\:\negmedspace k$ +and make no further use of either \verb`\diagbox` or \verb`\backslashbox`. + +\subsubsection{Built-in styles: \texttt{chstyle}} + +There\index{chstyle@\texttt{chstyle}|(} are some built-in header +styles, depending on the value of the setting \verb`chstyle`. No +value for this setting corresponds to \verb`chstyle=0`, which results +in column variable values at the heads of function-value columns and +something like $x\:\!\backslash\:\negmedspace k$ at the head of the +row variable column, as previously described. + +\verb`chstyle=1` changes the header of the \emph{first} function-value +column to the form \emph{variable=value}, e.g. $k=3$ in the example +below. This may be apt when a small rounding value is being used and +the resulting columns are narrow. I can find only one real instance +in \emph{HMF}, Table~26.7. Note that the row variable header, inserted +automatically, simplifies in this case to $x$ rather than $x\:\!\backslash\:\negmedspace k$. +\begin{verbatim} + \tabulate[rspec={x,0.2,5},cspec={k,2,3},chstyle=1] + { \sin kx }[k=3,x=0.2][3*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},cspec={k,2,3},chstyle=1] + { \sin kx }[k=3,x=0.2][3*]\medskip{} \verb`chstyle=2` changes the header of all function-value columns to the form \emph{variable=value}.\emph{ }In \emph{HMF} examples are -Tables~7.4, 7.9, 10.10, 16.6, etc. Again, the row variable setting -\verb`rhead` no longer needs the \verb`\backslash k` part since -the column variable is now explicitly indicated. +Tables~7.4, 7.9, 10.10, 16.6, etc. Again, the row variable header +is inserted automatically and displays the row variable. \begin{verbatim} - \tabulate[rspec={x,0.2,6}, - cspec={k,2,3},chstyle=2] - { \sin kx }[k=3,x=0][3*] + \tabulate[rspec={x,0.2,5},cspec={k,2,3},chstyle=2] + { \sin kx }[k=3,x=0.2][4*] \end{verbatim} -$\Longrightarrow$ \tabulate[rspec={x,0.2,6}, - cspec={k,2,3},chstyle=2] - { \sin kx }[k=3,x=0][3*]\medskip{} +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},cspec={k,2,3},chstyle=2] + { \sin kx }[k=3,x=0.2][4*]\medskip{} \verb`chstyle=3` fills each column variable header with the expression being tabulated but with the column variable replaced by its respective -values. See \emph{HMF} Tables~5.4, 8.1, 9.1, 19.1, etc. for examples. -Note that if the column variable value is \verb`1`, the \verb`1` -will be displayed: +values. See \emph{HMF} Tables~5.4, 8.1, 9.1, 19.1, etc. for examples. \begin{verbatim} - \tabulate[rspec={x,0.2,6}, - cspec={k,2,3},chstyle=3] - { \sin kx }[k=1,x=0][4*] + \tabulate[rspec={x,0.2,5},cspec={k,2,3},chstyle=3] + { \sin kx }[k=1,x=0.2][4*] \end{verbatim} -$\Longrightarrow$ \tabulate - [rspec={x,0.2,6}, - cspec={k,2,3},chstyle=3] - { \sin kx }[k=1,x=0][4*]\medskip{} - -In this last example you may not want the \verb`1` displayed. To -achieve that effect put \verb`chstyle=4`. This results in a header -as for \verb`chstyle=3` but if the column variable takes the value -\verb`1`, it has an empty replacement: -\begin{verbatim} - \tabulate[rspec={x,0.2,6}, - cspec={k,2,3},chstyle=4] - { \sin kx }[k=1,x=0][4*] -\end{verbatim} -$\Longrightarrow$ \tabulate - [rspec={x,0.2,6}, - cspec={k,2,3},chstyle=4] - { \sin kx }[k=1,x=0][4*]\index{column variable header!multi-column case!chstyle@\texttt{chstyle}|)}\index{chstyle@\texttt{chstyle}|)} +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},cspec={k,2,3},chstyle=3] + { \sin kx }[k=1,x=0.2][4*]\medskip{} + +Notable here is that the column variable takes the value $1$, and +the $1$ is displayed ($\sin1x$) where it would normally be suppressed. +A similar situation can arise when the column variable takes the value +$0$. The legacy setting \verb`chstyle=4` will, in the present instance, +display $\sin1x$ as $\sin x$ but is deprecated \textendash{} given +the variety of ways that $1$ (and $0$) can appear in formulas, it +seems more straightforward in to set \verb`chstyle=0` (or omit it) +and \ldots{} \subsubsection{User-defined header: \texttt{chead}} -\label{subsec:chead}\index{chead@\texttt{chead}|(}If the function -in the last example were, for instance, $k+\sin kx$, then neither -replacing $k$ by \verb`1` nor an empty replacement would be appropriate. -In that case the only recourse is to use the \verb`chead` key. Users -can assign whatever value they like to \verb`chead`. The assignment -must contain the correct number of tab characters (\verb`&`) for -the \emph{column variable columns only}. It is a header only to the -function-value columns. The user will need to insert \verb`$` signs -or other math delimiters as appropriate. This differs from the practice -for \verb`rhead`, but \verb`chead` is potentially far more complicated. -Thus for $k+\sin kx$, -\begin{verbatim} - \tabulate[rspec={x,0.2,6},cspec={k,2,3}, - chead=$1+\sin x$&$3+\sin3x$&$5+\sin 5x$] - { k+\sin kx }[k=1,x=0][4*] +\ldots{} form \verb`chead`\index{chead@\texttt{chead}} oneself, with +the relevant math delimiters (\verb`$`) and tabbing ampersands (\verb`&`). +(Always, to use \verb`chead`, \verb`chstyle` must be omitted or +set to zero.) In the example, +\begin{verbatim} + chead=$\sin x$&$\sin3x$&$\sin5x$ \end{verbatim} -$\Longrightarrow$ \tabulate - [rspec={x,0.2,6}, - cspec={k,2,3}, - chead=$1+\sin x$&$3+\sin3x$&$5+\sin 5x$] - { k+\sin kx }[k=1,x=0][4*]\medskip{} +would do. It is a header for the function-value columns; a header +for the row variable column is not included. Note that there is also +no end of row mark which is inserted automatically. -\noindent Non-empty content for the \verb`chead` key overrides any -\verb`chstyle` setting and, in the case of a table with only a single -function-value column, overrides any \verb`ctitle` setting.\index{chead@\texttt{chead}|)} +Another way to tackle the particular issue in the present case is +to form a multi-function table (see §\ref{sec:Multiple-function-tables}) +of, say, $\sin x$, $\sin3x$, $\sin5x$. \subsubsection{Alignment: \texttt{calign}} -The\index{calign@\texttt{calign}|(} function-value columns are aligned -right, \verb`calign=r`, by default. Also available are \verb`calign=c` -for centred alignment and \verb`calign=l` (lowercase L) for left -alignment. Using centred alignment with {\ttfamily\verb`chstyle=2`} -in a previous example table gives -\begin{verbatim} - \tabulate[rspec={x,0.2,6},ralign=c, - cspec={k,2,3},chstyle=2,calign=c] - { \sin kx }[k=3,x=0][*] -\end{verbatim} -$\Longrightarrow$ \tabulate - [rspec={x,0.2,6},rhead=x,ralign=c, - cspec={k,2,3},chstyle=2,calign=c] - { \sin kx }[k=3,x=0][*]\medskip{} - -\noindent The first column of function values looks better, but the -minus signs spoil the effect in the others. Handling signs in tables -is discussed below; see §\ref{subsec:Signs}\index{calign@\texttt{calign}|)}. +\label{subsec:Alignment:-calign}\index{calign@\texttt{calign}}By +default the function-value columns are aligned right, \verb`calign=r`. +Also available are \verb`calign=c` for centred alignment and \verb`calign=l` +(lowercase L) for left alignment. A table containing both positive +and negative values is best set with right alignment and number padding +(\verb`*` in the trailing optional argument) to avoid the misalignment +of digits in columns caused by the presence of minus signs. (Handling +signs in tables is discussed later; see §\ref{subsec:Signs}.) Varying +numbers of digits \emph{before} the decimal point may be best handled +with the t-notation of §\ref{subsec:t-option}. Examples where a centred +alignment is helpful, centring the header above uniform columns of +figures (same number of digits before and after the decimal point) +have already been given; see §\ref{subsec:rfracFraction-form-rvar} +and §\ref{subsec:RValignment:ralign}. \subsubsection{Nudging header entries: \texttt{chnudge}} -\label{subsec:Nudgingtheheaders:-chnudge}In\index{chnudge@\texttt{chnudge}|(} -left or right alignment it is possible to nudge the column headers -in the opposite direction by giving a numerical value to the the key -\verb`chnudge`. The header is moved by the specified number of mu -(math units; 18 to a quad). Note that the `mu' does not need to -be written. {\ttfamily\verb`numerica-tables`} provides -that. In the next example I have chosen \verb`chnudge=12` to nudge -the column headers to the left to give a centred effect to the header -but leaving the function values with their (potentially) awkward minus -signs right aligned. -\begin{verbatim} - \tabulate[rspec={x,0.2,6},ralign=c, - cspec={k,2,3},chstyle=2,chnudge=12] - { \sin kx }[k=3,x=0][*] +\label{subsec:Nudgingtheheaders:-chnudge}\index{chnudge@\texttt{chnudge}|(}As +with the row variable header, it is possible to nudge the column headers +to left or right. A positive nudge value shifts the header in the +\emph{opposite} sense to the alignment by the specified number of +mu (math units; 18 to a quad), to the left in a right alignment and +to the right otherwise. In version 3.2 of \texttt{numerica-tables, +t}he key \verb`chnudge` is a comma list of nudge values (as against +a single value in earlier versions). As with \verb`rhnudge`, only +numbers should be supplied; \texttt{numerica-tables} inserts the unit +(\verb`mu`). Often a single value is all that is required. For instance, +in the next example \verb`chnudge=9` suffices to nudge the column +headers to the left but leave the function values (with their potentially +awkward minus signs) right aligned. +\begin{verbatim} + \tabulate[rspec={x,0.2,5},ralign=c, + cspec={k,2,3},chstyle=2,chnudge=9] + { \sin kx }[k=3,x=0.2][*] \end{verbatim} $\Longrightarrow$ \tabulate - [rspec={x,0.2,6},ralign=c, - cspec={k,2,3},chstyle=2,chnudge=12] - { \sin kx }[k=3,x=0][*]\medskip{} - -The \verb`chnudge` value does not need to be positive. Negative nudges -can be useful when a column header is \emph{longer} than the rounded -function values. In the second example below, I've reduced the rounding -value for function values to $3$, and chosen an initial $k$ value -of $100$ to ensure this circumstance. To centre the column headers -I have used \verb`chnudge=-9`. + [rspec={x,0.2,5},ralign=c, + cspec={k,2,3},chstyle=2,chnudge=9] + { \sin kx }[k=3,x=0.2][*]\medskip{} + +But if the values of $k$ in this example had different numbers of +digits, no single \verb`chnudge`\index{chnudge@\texttt{chnudge}!comma list of values} +value would do. In that case give \verb`chnudge` a comma-list of +values, potentially one for each function-value column. If the comma +list has fewer entries than the number of columns, the last entry +is used for all `left over' columns. For instance, in the following +example, \verb`chnudge` is given three values, the third of which +is also used for the fourth function-value column: \begin{verbatim} - \tabulate[rspec={x,0.2,6},ralign=c, - cspec={k,2,3},chstyle=2,chnudge=-9] - { \sin kx }[k=100,x=0][3*] + \tabulate[rspec={x,0.2,5},ralign=c,cspec={k,50,3}, + chstyle=2,chnudge={13.5,9,4.5}] + { \cos kx }[k=1,x=0.2][*] \end{verbatim} -$\Longrightarrow$ \tabulate - [rspec={x,0.2,6},ralign=c, - cspec={k,2,3},chstyle=2,chnudge=-9] - { \sin kx }[k=100,x=0][3*]\index{chnudge@\texttt{chnudge}|)} - -\subsubsection{Rounding: \texttt{chround}} - -In\index{chround@\texttt{chround}|(}\index{rounding value!header row|(} -the examples so far, the column variable has incremented in integer -steps. The default rounding value for the column variable is $0$ -(for the row variable it is $1$), so if it increments by some non-integer -amount, the result will be confusing \textendash{} if $k$ incremented -by, say, $0.25$, starting from $k=3$, then the next column would -also have a header $k=3$ (since $3.25$ with a rounding value $0$ -rounds to $3$). The appropriate key to remedy this state of affairs -is {\ttfamily\verb`chround`}. For a step size of $0.25$ -the appropriate setting is {\ttfamily\verb`chround=2`}. -\begin{verbatim} - \tabulate[rspec={x,0.2,6},ralign=c, - cspec={k,0.25,3},chstyle=2,chround=2] - { \sin kx }[k=3,x=0][*] +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},ralign=c,cspec={k,50,4}, + chstyle=2,chnudge={13.5,9,4.5}] + { \cos kx }[k=1,x=0.2][*]\medskip{} + +\emph{Negative} nudges can be useful when a column header is longer +than the displayed function values. Just for the pleasure of providing +an instance where a \emph{zero} step value\index{cstep@\texttt{cstep}!zero value} +is useful, the following small table allows an immediate comparison +of a negative nudge with no nudge. I have adjusted the separation +of the columns with \verb`\tabcolsep`\index{tabcolsep@\texttt{\textbackslash tabcolsep}} +to provide a more adequate separation of the headers: +\begin{verbatim} + \setlength\tabcolsep{9pt} + \tabulate[rspec={n,100,5},ralign=c,rround=0, + cspec={k,0,2},chstyle=3,chnudge={-21,0}] + { (1+k/n)^n }[k=1,n=100][3*] + \setlength\tabcolsep{6pt} \end{verbatim} -$\Longrightarrow$ \tabulate - [rspec={x,0.2,6},rhead=x,ralign=c, - cspec={k,0.25,3},chstyle=2,chround=2] - { \sin kx }[k=3,x=0][*]\index{column variable header|)}\index{rounding value!header row|)}\index{column variable header!multi-column case|)}\index{chround@\texttt{chround}|)}\index{formatting!column variable header} +$\Longrightarrow$ \setlength\tabcolsep{9pt} + \tabulate[rspec={n,100,5},ralign=c,rround=0, + cspec={k,0,2},chstyle=3,chnudge={-21,0}] + { (1+k/n)^n }[k=1,n=100][3*] + \setlength\tabcolsep{6pt}\index{chnudge@\texttt{chnudge}|)} + +\subsubsection{Math style: \texttt{chmath}} + +The \verb`chmath` setting is a comma list of the values \verb`s`, +\verb`t`, or \verb`d` presenting the header of the corresponding +function-value column in either scriptstyle, textstyle or displaystyle +respectively. The setting is initialized to \verb`t`. If there are +more function-value columns in the table than entries in \verb`chmath` +the last value in \verb`chmath` is used for the extra columns. See +the first table in §\ref{sec:Multiple-function-tables} (on multi-function +tables) for an example of this setting's use. + +\subsubsection{Font size: \texttt{chsize}} + +\label{subsec:chsize}The font size of the header row entries can +be changed with the setting \verb`chsize`\index{chsize@\texttt{chsize}} +which is a comma list of integer values, each integer in the range +$-4$ to $5$ corresponding to a \LaTeX{} font size command according +to the scheme + +\begin{tabular}{rlcrl} +$-4$ & \verb`\tiny` & & $1$ & \verb`\large`\tabularnewline +$-3$ & \verb`\scriptsize` & & $2$ & \verb`\Large`\tabularnewline +$-2$ & \verb`\footnotesize` & & $3$ & \verb`\LARGE`\tabularnewline +$-1$ & \verb`\small` & & $4$ & \verb`\huge`\tabularnewline +$0$ & \verb`\normalsize` & & $5$ & \verb`\Huge`\tabularnewline +\end{tabular}\medskip{} + +\noindent Should there be more columns in the table than entries in +\verb`chsize`, then the last value in \verb`chsize` is used for +the extra columns. In the following example, the font size has been +tweaked by means of the setting \verb`chsize={1,-1,1}` to bring the +character size of the column headings into greater conformity: +\begin{verbatim} + \tabulate[ff,rspec={x,0.5,4},chsize={1,-1,1}] + { \frac{ e^{2x}-1}{ e^{2x}+1},\tanh x, + \frac{ \sinh x}{\cosh x}} [x=0.5] [4*] +\end{verbatim} +$\Longrightarrow$ \tabulate[ff,rspec={x,0.5,4},chsize={1,-1,1}] + { \frac{ e^{2x}-1}{ e^{2x}+1}, \tanh x , + \frac{ \sinh x}{\cosh x}} [x=0.5] [4*]\index{column variable header!formatting|)}\index{column variable header!multi-column|)} \section{Multiple functions in a single table} \label{sec:Multiple-function-tables}As\index{multi-function tables|(} -already noted in §\ref{subsec:Adjoining-tables}, tabulating more -than one function at a time is too common a need to have to resort -to a fudge like adjoining tables. The systematic way of handling this -task is to enter the functions in the main argument of a \verb`\tabulate` -command separated by a specified mark then alert \verb`\tabulate` -that this has happened with the \verb`ff`\index{ff@\texttt{ff}} -key in the settings option. - -By default the multi-function delimiter\index{multi-function tables!function delimiter} -is the comma if the decimal point is a dot (or period), or the semicolon\index{semicolon and decimal comma} -if the decimal point is a comma\index{decimal comma} (\texttt{numerica} -loaded with the \verb`comma` package option). If you are content -with the default delimiter then it suffices to enter \verb`ff` in -the settings option. If not, then enter \verb`ff=<mark>` there. For -example \verb`ff=|` would make the `pipe' character \verb`|` the -multi-formula delimiter. If the \verb`ff` key is overlooked then -multiple formulas in the main argument of \verb`\tabulate` will almost -certainly cause a \LaTeX{} error. - -In the following example, using the default comma, note first the -\verb`ff` setting, and then the \verb`o` setting indicating that -the arguments of the trig functions are in degrees and (just to amuse -myself) I have put the row variable column on both sides with the -\verb`rpos=3`\index{rpos@\texttt{rpos}} setting: -\begin{verbatim} - \tabulate[ff,o,rpos=3,rround=0, - rvar=\theta,rstep=10,rstop=90] - { \sin \theta, \cos \theta }[\theta=0][*] -\end{verbatim} -$\Longrightarrow$ \tabulate[ff,o,rpos=3,rround=0, - rvar=\theta,rstep=10,rstop=90] - { \sin \theta, \cos \theta }[\theta=0][*]\medskip{} - -The tables suggest a space saving possibility: since $\sin$ and $\cos$ -are complementary functions\index{complementary functions} ($\cos\theta=\sin(90-\theta)$), -the values in the bottom half of the table duplicate values in the -top half, only with the columns reversed. This is the reason for the -space saving \verb`rpos=4`\index{rpos@\texttt{rpos}} setting (§\ref{subsec:Second-row-var-col}) -which enables complementary functions to be tabulated in `half tables' -(\emph{HMF} Tables 4.10\textendash 4.12 are examples for the trigonometric -functions). - -In the next example, the row variable column is again duplicated, -left and right, with the \verb`rpos=3` setting, and a centred alignment -is used for the function values. Because $\sin$ and $\cos$ are complementary, -I have stopped the table at \verb`rstop=45` since continuing to \verb`90` -would simply give a mirror reflection of the preceding values. To -accommodate the use of the comma in \verb`\max` and \verb`\min`\index{n-ary@$n$-ary functions} -I have stipulated \verb`ff=|`\index{ff@\texttt{ff}}. Visually, the -table has an unsatisfactory, sprawling appearance \textendash{} which -directs attention to how tables might be titled (§\ref{subsec:Title:-ctitle-setting})\index{rpos@\texttt{rpos}}: -\begin{verbatim} - \tabulate[ff=|,o,rpos=3,rround=0, - rvar=\theta,rstep=10,rstop=45,calign=c] - { \max(\sin \theta,\cos \theta)| - \min(\sin \theta,\cos \theta) } - [\theta=0][*] -\end{verbatim} -$\Longrightarrow$ \tabulate[ff=|,o,rpos=3,rround=0, - rvar=\theta,rstep=10,rstop=45,calign=c] - { \max(\sin \theta,\cos \theta)| - \min(\sin \theta,\cos \theta) } - [\theta=0][*] \index{multi-function tables|)} +noted in §\ref{subsec:Adjoining-tables}, using adjoining tables in +order to tabulate more than one function at a time is `clunky'. +Much better is to enter the functions in the main argument of a \verb`\tabulate` +command separated by a specified mark, then alert \verb`\tabulate` +that this has happened by setting \texttt{numerica}'s \verb`ff`\index{ff@\texttt{ff}} +key. + +The default multi-function delimiter\index{multi-function tables!function delimiter} +is the comma if the decimal point is a dot (period), or the semicolon\index{decimal comma!semicolon use} +if it is a comma (\texttt{numerica} loaded with the \verb`comma` +package option\index{comma package option@\texttt{comma} package option}). +With the default delimiter it suffices to enter \verb`ff` in the +settings option; otherwise enter \verb`ff=<mark>` there. For example +\verb`ff=|` makes the `pipe' character the multi-formula delimiter. +If the \verb`ff` key is overlooked then multiple formulas in the +main argument will almost certainly cause a \LaTeX{} error. + +The following example (of the so-called `Einstein functions') illustrates +both a multi-function table using the default comma as separator, +and the use of \verb`chmath`\index{chmath@\texttt{chmath}} to shrink +the function in the last column. If not shrunk, $\ln(1-e^{-x})$ appears +too large compared to the symbols in the fractions in the middle columns: +\begin{verbatim} + \tabulate[ff,rspec={x,0.3,4},rround=2,rhnudge=9, + chnudge={0,9,0},chmath={t,t,s}] + { \frac{x^2e^x}{(e^x-1)^2}, \frac{x}{e^x-1}, + \ln(1-e^{-x}) }[x=0.15][5*] +\end{verbatim} +$\Longrightarrow$ \tabulate[ff,rspec={x,0.3,4},rround=2,rhnudge=9, + chnudge={0,9,0},chmath={t,t,s}] + { \frac{x^2e^x}{(e^x-1)^2}, \frac{x}{e^x-1}, + \ln(1-e^{-x}) }[x=0.15][5*] \medskip{} + +In the next example I transpose this table, not because it is a good +example of a table that might benefit from this, but because it illustrates +a problem that can arise when a multi-function table is transposed. +On transposition the current header row becomes the row variable column: +\index{transpose@\texttt{transpose}!examples of use} +\begin{verbatim} + \tabulate[ff,rspec={x,0.3,4},rround=2,ralign=l, + chnudge={0,9},rules=hB,chstyle=1,transpose] + { \frac{x^2e^x}{(e^x-1)^2}, \frac{x}{e^x-1}, + \ln(1-e^{-x}) }[x=0.15][5*] +\end{verbatim} +$\Longrightarrow$ \tabulate[ff,rspec={x,0.3,4},rround=2,ralign=l, + chnudge={0,9},rules=hB,chstyle=1,transpose] + { \frac{x^2e^x}{(e^x-1)^2}, \frac{x}{e^x-1}, + \ln(1-e^{-x}) }[x=0.15][5*] \medskip{} + +\noindent What is immediately noticeable is the disparity in size +of the characters forming $\ln(1-e^{-x})$ compared to those in the +fractions above it. This was the reason for the \verb`chmath={t,t,s}` +setting in the original table, but \verb`rmath`\index{rmath@\texttt{rmath}} +is a single value applying to the \emph{whole} row variable column, +so that solution is not available. Two possibilities come to mind. +One is to force the function being tabulated into scriptstyle by +tabulating +\begin{verbatim} + \vphantom{\frac11}\scriptstyle\ln(1-e^{-x}) +\end{verbatim} +(The phantom ensures the spacing between the rows is the same.) This +new function produces exactly the same values as the previous one +did. The other possibility is to set \verb`rmath=d`, forcing all +members of the row variable column, in particular the fractions, into +displaystyle. With this setting the \verb`\scriptstyle` can be omitted +(but the phantom retained). + +Another example, now using \verb`ff=|` as the function separator, +places the row variable column on both sides of the table (\verb`rpos=3`)\index{rpos@\texttt{rpos}} +and uses the \verb`o` setting (see \texttt{numerica.pdf}) to indicate +that arguments of the trig functions are in degrees: +\begin{verbatim} + \tabulate[ff=|,o,rpos=3,rround=0,chnudge=9, + rvar=\theta,rstep=15,rstop=90] + { \sin \theta | \cos \theta }[\theta=0][*] +\end{verbatim} +$\Longrightarrow$ \tabulate[ff=|,o,rpos=3,rround=0,chnudge=9, + rvar=\theta,rstep=15,rstop=90] + { \sin \theta |\cos \theta }[\theta=0][*]\medskip{} -\section{Whole-of-table formatting} +The table suggests a space saving possibility: since $\sin$ and $\cos$ +are complementary functions\index{complementary functions} ($\sin(90-\theta)=\cos\theta$), +values in the bottom half of the table duplicate values in the top +half, only with the columns reversed. This is the reason for the \verb`rpos=4`\index{rpos@\texttt{rpos}} +setting discussed in §\ref{subsec:Second-row-var-col}, which enables +complementary functions to be tabulated in `half tables'.\index{multi-function tables|)} -\label{sec:Whole-of-table-formatting}There\index{formatting!title, subtitle, footer|(} -are a number of settings pertaining to the appearance of the table -as a whole, things like the position of the row variable column, division -of the function values into blocks to aid readability, the presence -of horizontal rules or of a collective column title or of a footer -row. I discuss these here. +\section{Whole-of-table formatting} -\begin{table} -\centering{}\caption{\protect\label{tab:Table-formatting-settings}Table formatting} +\label{sec:Whole-of-table-formatting} +\begin{table}[t] +\centering +\centering{}\caption{\label{tab:Table-formatting-settings}\label{tab:Table-formatting}Table +formatting} \begin{center} \begin{tabular}{ll>{\raggedright}p{4cm}l} \toprule {\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline \midrule -{\small\texttt{ctitle}} & {\small token(s)} & {\small collective title for function-value columns} & \tabularnewline - & {\small token(s)} & {\small subtitle row for} {\small function- value cols} & \tabularnewline -{\small\texttt{header}} & {\small int (}{\small\texttt{0/1}}{\small )} & {\small suppress/show header row} & {\small\texttt{1}}\tabularnewline +{\small\texttt{ctitle}} & {\small token(s)} & {\small collective title for columns} & \tabularnewline +{\small\texttt{csubttl}} & {\small token(s)} & {\small subtitle row for} {\small function- value cols} & \tabularnewline +{\small\texttt{calign}} & {\small char(}{\small\texttt{r}}{\small /}{\small\texttt{c}}{\small /}{\small\texttt{l}}{\small )} & {\small column alignment} & {\small\texttt{r}}\tabularnewline +{\small\texttt{headless}} & & {\small suppress header row} & \tabularnewline {\small\texttt{foot}} & {\small token(s)} & {\small table-wide footer row} & \tabularnewline {\small\texttt{rules}} & {\small char(s)} & {\small horizontal rule spec.} & {\small\texttt{ThB}}\tabularnewline +{\small\texttt{norules}} & & {\small cancel all rules} & \tabularnewline {\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small row variable col. position(s)} & {\small\texttt{1}}\tabularnewline {\small\texttt{rbloc}} & {\small integer comma list} & {\small row block specification} & \tabularnewline +{\small\texttt{rblocsep}} & {\small length} & {\small extra space between row blocks} & {\small\texttt{1ex}}\tabularnewline +{\small\texttt{transpose}} & & {\small show funct. vals in rows} & \tabularnewline {\small\texttt{valign}} & {\small char (}{\small\texttt{t}}/{\small\texttt{m}}/{\small\texttt{b}}{\small )} & {\small vertical alignment of table relative to text baseline} & {\small\texttt{m}}\tabularnewline \bottomrule \end{tabular} \par\end{center} \end{table} - +There\index{formatting!title, subtitle, footer|(} are a number of +settings that affect the appearance of the table as a whole, things +like the position of the row variable column, the grouping of function +values in a column into blocks to aid readability, the presence of +horizontal rules, the use of a collective column title, or of a footer +row, or display of the table `horizontally' in rows rather than +columns. I discuss these here (Table~\ref{tab:Table-formatting}). \subsection{Title for function-value columns: \texttt{ctitle}} \label{subsec:Title:-ctitle-setting}\index{ctitle@\texttt{ctitle}|(}The\index{title row|(} -function-value columns have individual headers, formatted in the various -ways provided by the settings discussed in previous sections, but -it can also be helpful to have a collective title for these columns. -We saw the need in the last example. The need is met with the \verb`ctitle` -key. This can be set to whatever you like (e.g. \verb`ctitle=\text{Fred}`) -but, to more purpose, I shall use the setting to improve the look -of the table in the last example. Note that the content of the \verb`ctitle` -key is enclosed in braces. This is to shield the commas there from -being misinterpreted as item separators in the settings option. For -an analogous reason I have specified the semicolon (\verb`ff=;`)\index{ff@\texttt{ff}} -as the function separator in the main argument:\index{rpos@\texttt{rpos}}\index{n-ary@$n$-ary functions} -\begin{verbatim} - \tabulate[ff=;,o,rpos=3,rround=0, - rvar=\theta,rstep=10,rstop=45, - ctitle={\max,\min(\sin\theta,\cos\theta)}, - chead=max\hphantom{00} & min\hphantom{00}] - { \max(\sin \theta,\cos \theta); - \min(\sin \theta,\cos \theta) } [\theta=0][*] -\end{verbatim} -$\Longrightarrow$ \tabulate[ff=;,o,rpos=3,rround=0, - rvar=\theta,rstep=10,rstop=45, - ctitle={\max,\min(\sin\theta,\cos\theta)}, - chead=max\hphantom{00} & min\hphantom{00}] - { \max(\sin \theta,\cos \theta); - \min(\sin \theta,\cos \theta) } - [\theta=0][*]\medskip{} - -\noindent Now it is clearer what is being tabulated and the wide separation -of the function-value columns is reduced. - -There are two built-in values for the \verb`ctitle` key: \verb`ctitle=*`, +function-value columns have individual headings, formatted in the +various ways provided by the settings already discussed, but it can +also be helpful to have a collective title for these columns. For +instance our table of Einstein functions in §\ref{sec:Multiple-function-tables} +would have benefited from this. The need is met with the \verb`ctitle` +key. This can be set to whatever you like (e.g. \verb`ctitle=Fred`), +taking care to shield commas with braces. By default, the title is +set between math delimiters (\verb`$` signs). For that reason in +the example I have placed it inside a \verb`\text` command: +\begin{verbatim} + \tabulate[ff,rspec={x,0.15,6},rround=2,rhnudge=9, + chnudge={0,9,0},chmath={t,t,s}, + ctitle=\text{Einstein functions}] + { \frac{x^2e^x}{(e^x-1)^2}, \frac{x}{e^x-1}, + \ln(1-e^{-x}) }[x=0.15][5*] +\end{verbatim} +$\Longrightarrow$ \tabulate[ff,rspec={x,.15,6},rround=2,rhnudge=9, + chnudge={0,9,0},chmath={t,t,s}, + ctitle=\text{Einstein functions}] + { \frac{x^2e^x}{(e^x-1)^2}, + \frac{x}{e^x-1},\ln(1-e^{-x}) }[x=0.15][5*] + +\medskip{} + +There are two built-in values for the \verb`ctitle` key: \verb`ctitle=*`,\index{ctitle@\texttt{ctitle}!{*},{*}{*} settings} which forms the title from the function being tabulated, and \verb`ctitle=**` which uses the function and vv-list for the title. Obviously these, particularly the latter, could easily become too long to be useful. -An example of \verb`ctitle=**` is presented below in §\ref{subsec:Rules:-rules-setting}. -In the following example, inclusion of the vv-list in the title is -pointless since the variables there form the row and column variables -of the table; \verb`ctitle=*` is all that is needed: +An example of \verb`ctitle=**` is presented later in §\ref{subsec:Rules:-rules-setting}, +but inclusion of the vv-list in the title in the next example would +be pointless since the row and column variables of the table are the +only members of the vv-list; \verb`ctitle=*` suffices: \begin{verbatim} \tabulate[rspec={n,1,5},rround=0, cspec={m,1,4},chstyle=2,ctitle=*] @@ -1289,266 +1669,351 @@ $\Longrightarrow$ \tabulate[rspec={n,1,5},rround=0, \subsection{Between header \& title: \texttt{csubttl}} -Some\index{csubttl@\texttt{csubttl}}\index{title row!subtitle row} -tables need to fit more header or title material into their rows than -can be comfortably accommodated in either row alone. For examples, -see \emph{HMF} Tables 7.9 (error function for complex arguments), -17.7 (Jacobian zeta function), 21.1 (eigenvalues of spheroidal wave -functions), and 26.7 (probability integrals). One way of handling -this problem is to resort to more complicated environments in header -and title rows. Another, more direct way, is to insert a row between -the header and title rows. - -I have chosen \verb`csubttl`, a contraction of `column variable -subtitle' for the key name. (In version 2 of \texttt{numerica-tables} -the name \verb`cmidrow`\index{cmidrow@\texttt{cmidrow }|see{\texttt{csubttl}}} -was used; that will still work but \verb`csubttl` gives a clearer -indication of what the key does.) The initial `c' emphasizes that -like \verb`chead` and \verb`ctitle` it is constrained to the span -of the column variable (or function-value) columns only. The content -of \verb`csubttl` is entirely the responsibility of the user, including -insertion of the necessary number of tab characters, \verb`&`, and -any math delimiters required. -\begin{verbatim} - \tabulate[ff,rspec={x,1,7},rround=0,chnudge=18, +\label{subsec:subtitle-row}Some\index{csubttl@\texttt{csubttl}}\index{title row!subtitle row} +tables need more header or title material than can be comfortably +accommodated in either row alone. For examples, see \emph{HMF} Tables +7.9, 17.7, 21.1, and 26.7. One way of handling this problem is to +resort to more complicated environments in header and title rows. +Another, more direct way, is to insert a row between the header and +title rows by means of the key \verb`csubttl`, a contraction of `c(olumn +variable) subtitle'. (In version 2 of \texttt{numerica-tables} the +name \verb`cmidrow`\index{cmidrow@\texttt{cmidrow }|see{\texttt{csubttl}}} +was used; that is still available, but deprecated.) Like \verb`chead` +and \verb`ctitle`, \verb`csubttl` is limited to the span of the +column variable (or function-value) columns only. The content of \verb`csubttl` +is entirely up to the user, including insertion of sufficient \verb`&` +characters and math delimiters (if required). + +An example where the subtitle row serves an explanatory role is shown +in the next table. I have used \verb`\sfrac` from the \texttt{xfrac} +package for neater fractions: +\begin{verbatim} + \tabulate[ff,rspec={x,1,6,0},chnudge={22.5,18}, ctitle=\text{Hyperbolic functions}, - csubttl=\multicolumn{3}{c}{$\sinh x=\tfrac12 - (e^x-e^{-x}),\ \cosh x=\tfrac12(e^x+e^{-x})$}] - { \tfrac12e^x, \sinh x, \cosh x } - [{x}=0][8*] + csubttl=\multicolumn{3}{c}{\small$\sinh x=\sfrac12 + (e^x-e^{-x}),\ \cosh x=\sfrac12(e^x+e^{-x})$}] + { \sfrac12\,e^x, \sinh x, \cosh x } [{x}=0][8*] \end{verbatim} -$\Longrightarrow$ \tabulate[ff,rspec={x,1,7},rround=0,chnudge=18, +$\Longrightarrow$ \tabulate[ff,rspec={x,1,6,0},chnudge={22.5,18}, ctitle=\text{Hyperbolic functions}, - csubttl=\multicolumn{3}{c}{$\sinh x=\tfrac12 - (e^x-e^{-x}),\ \cosh x=\tfrac12(e^x+e^{-x})$}] - { \tfrac12e^x, \sinh x, \cosh x } - [{x}=0][8*]\index{ctitle@\texttt{ctitle}|)}\index{title row|)} + csubttl=\multicolumn{3}{c}{\small$\sinh x=\sfrac12 + (e^x-e^{-x}),\ \cosh x=\sfrac12(e^x+e^{-x})$}] + { \sfrac12\,e^x, \sinh x, \cosh x } [{x}=0][8*]\index{ctitle@\texttt{ctitle}|)}\index{title row|)} +\begin{verbatim} + +\end{verbatim} \subsection{Suppress/show header row} -\label{subsec:Header-row-suppress}Usually\index{headless@\texttt{headless}} +\label{subsec:Header-row-suppress}By default\index{headless@\texttt{headless}} the header row in a table is shown. It carries essential information -as to the table's contents. However, there are occasions when it should -be suppressed. An example where this is appropriate is given in §\ref{subsec:rdata-rfile-rverb} +as to the table's contents, but there are occasions when it should +be suppressed. An example occurs in §\ref{subsec:rdata-rfile-rverb} where a table listing fractions of $\pi$ and their values is shown. -As in that example, to suppress the header enter the setting \verb`headless=1`. -(Otherwise tables default to \verb`headless=0`.) - -\subsection{Footer row: \texttt{foot} } - -Some\index{foot@\texttt{foot}}\index{footer row} tables have a footer -row and \verb`numerica-tables` allows such a row to be inserted, -but its entire content, with one exception, is the responsibility -of the user, including insertion of the necessary number of tab characters -\verb`&`. This will usually be $1$ less than the total number of -columns (including row variable columns) in the table \textendash{} -or some adjustment thereof if you use \verb`\multicolumn`. (\emph{HMF} -uses the footer mainly for cryptic descriptions of the accuracy and -needs of interpolation methods.) +The header there serves no purpose. To suppress it, use the \verb`headless`\footnote{There was a mix-up in the documentation of version 3.1 of \texttt{numerica-tables} +between what was entered in the table of whole-of-table settings and +what was described in this section.} setting. \index{norules@\texttt{norules}} +\begin{verbatim} + \def\mydata{\sfrac14\,\pi,\sfrac13\,\pi,\sfrac12\,\pi, + \sfrac23\,\pi,\sfrac34\,\pi,\pi} + \tabulate[rdata=\mydata,rverb=1,rvar=k,headless, + norules]{ k }[*] +\end{verbatim} +$\Longrightarrow$ \def\mydata{\sfrac14\,\pi,\sfrac13\,\pi,\sfrac12\,\pi, + \sfrac23\,\pi,\sfrac34\,\pi,\pi} + \tabulate[rdata=\mydata,rverb=1,rvar=k, headless, + norules]{ k }[*] + +\subsection{Footer row: \texttt{foot}} + +\label{subsec:Footer-row:-foot}Some\index{foot@\texttt{foot}}\index{footer row} +tables have a footer row and \verb`numerica-tables` allows such a +row to be inserted, but its entire content, with two exceptions, is +the responsibility of the user, including insertion of the necessary +number of tab characters \verb`&`. This will usually be $1$ less +than the total number of columns (including row variable columns) +in the table \textendash{} or some adjustment thereof if you use \verb`\multicolumn`. +(\emph{HMF} uses the footer mainly for cryptic descriptions of the +accuracy and needs of interpolation methods.) You can put into the footer what you wish with the setting \verb`foot=<tokens>`. -The one exception is when \verb`foot=*`\index{foot@\texttt{foot}}. -This will fill the footer with the header, but with the items of the -header presented in \emph{reversed} order \textendash{} the last item -first, and so on. This is useful for tabulating complementary functions -like the sine and cosine or, more generally, $f(x)$ and $g(x)$ where -$g(x)=f(k-x)$ for some constant $k$. Values for the complementary +There are two exceptions: +\begin{itemize} +\item when \verb`foot=''`\index{foot@\texttt{foot}} (two single quote +marks), suggesting ditto marks, the footer row is set equal to the +header row with one change, the backslash in row variable headers +of the form $n\:\!\backslash\:\!m$ are converted to forward slashes, +e.g. $n\:\!/\:\!m$; +\item when \verb`foot=*`\index{foot@\texttt{foot}}, the footer is set +equal to the header in \emph{reversed} order \textendash{} the last +item first, and so on. This is useful for tabulating complementary +functions like sine and cosine or, more generally, $f(x)$ and $g(x)$ +where $g(x)=f(k-x)$ for some constant $k$. Values for the complementary function are read from the bottom up and require a reversed row variable -column on the right of the table; see §\ref{subsec:Second-row-var-col}. - -\noindent{}% -\noindent\begin{minipage}[t]{1\columnwidth}% -\begin{shaded}% +column on the right of the table; see the third example in §\ref{subsec:Second-row-var-col}. +\end{itemize} -\subsubsection{Footer functions} +\paragraph{Footer functions} -In\index{footer functions} previous versions of \texttt{numerica-tables} -it was possible to perform certain simple operations on columns \textendash{} -calculate the sum, the average and maximum and minimum values. This -is no longer so in version 3. Not only does it seem tangential to -the primary function of the \verb`\tabulate` command but it was also -acutely dependent on the format of the numbers being operated on. -A simple change in the number-format option could cause a \LaTeX{} -error.\end{shaded}% -\end{minipage}\index{formatting!title, subtitle, footer|)} +In\index{footer functions} versions of \texttt{numerica-tables} before +version 3 it was possible to perform certain simple operations on +columns \textendash{} calculate the sum, the average and maximum and +minimum values. This is no longer so in version 3. Not only does it +seem tangential to the primary function of the \verb`\tabulate` command +but it was also acutely dependent on the format of the numbers being +operated on \textendash{} a change in the number-format option could +cause a \LaTeX{} error.\index{formatting!title, subtitle, footer|)} \subsection{Horizontal rules: \texttt{rules}} -\label{subsec:Rules:-rules-setting}\index{rules@\texttt{rules}|(}\index{rules|(}\index{formatting!rules|(} -\begin{table}[!b] -\centering{}\caption{\protect\label{tab:Rules}Rules. {\small (In the \textquoteleft span\textquoteright{} -column, \textquoteleft f-v\textquoteright =function-value; \textquoteleft r-v\textquoteright =row -variable; \textquoteleft$<$~table\textquoteright{} indicates that -the rule spans the table but is trimmed at each end.)}} +\label{subsec:Rules:-rules-setting}\index{rules@\texttt{rules}|(}\index{formatting!rules|(}The +\texttt{booktabs} \index{booktabs@\texttt{booktabs}} package which +\texttt{numerica-tables} uses is most emphatic that one should `1. +Never, ever use vertical rules.\index{rules@\texttt{rules}!edicts} +2. Never use double rules.' Most of the tables proper in \emph{HMF} +lack rules of any kind although closer inspection shows smaller tables +within the text generally \emph{are} delimited by horizontal rules +(often also with vertical rules). In the various examples in the present +document I have used horizontal rules because these too are tables +within text. Some form of delineation seems necessary although I think +I have `over ruled'. The question should always be: is a rule necessary +at all? Usually, less is more. (Although many of \emph{HMF}'s tables +are inelegantly typeset, I have used it as a valuable resource for +the variety of structures needed to present a multitude of different +kinds of numerical data.) + +The \verb`rules` key allows one to specify precisely which rules +are used. The content of the key is a `word' \textendash{} a sequence +of letters \textendash{} where the characters have the significance +and default thicknesses \textendash{} from the \texttt{booktabs}\index{booktabs@\texttt{booktabs}} +package \textendash{} shown in Table~\ref{tab:Rules}. (To adjust +the thickness of rules used, see §\ref{subsec:Trim-and-thickness} +below.) No particular order is required in the \verb`rules` specification; +\verb`rules=hBT` would work as well as \verb`rules=ThB` but the +latter, reflecting the order of the rules in the table, is simpler +to grasp at a glance. + +In previous versions of \texttt{numerica-tables} the initial value +of this key was \verb`rules=ThB` so that if the \verb`rules` key +was not explicitly set then a rule was automatically drawn at the +top of the table, at the bottom of the table, and beneath the header +row. From version 3.2, no initial value is assigned to \verb`rules`. +Instead, a package option of the same name\index{rules@\texttt{rules}!package option}\index{package options} +can be set (see §\ref{chap:Introduction}) which is initialized to +\verb`rules=ThB` to maintain previous behaviour. What is gained is +that by assigning a different value to the package option it is now +possible for a user to alter the default behaviour. A \verb`norules` +package option is also available if you want your tables free of rules +by default. The default set by the package option can always be overridden +for an individual table by means of the \verb`rules` \emph{setting}. + +\subsubsection{Header and footer rules} + +The header (resp. footer) rule specified by including \verb`h` (resp. +\verb`f`) in the \verb`rules` specification spans the whole table, +from the left of the first column to the right of the last column. +It can give a better visual appearance sometimes to trim (shorten) +these rules both left and right. This is specified by priming the +\verb`h` (resp. \verb`f`) in the specification. Thus \verb`rules=hf` +will produce header and footer rules spanning the whole table; \verb`rules=h'f'` +will produce slightly shorter header and footer rules trimmed by the +default $0.5\text{\,em}$ at both ends \textendash{} unless some other +trim value has been specified; see §\ref{subsec:Trim-and-thickness} +below. + +\subsubsection{Title and subtitle rules} + +\begin{table}[t] +\centering +\centering{}\caption{\label{tab:Rules}Rules. {\small (In the \textquoteleft span\textquoteright{} +column, \textquoteleft fv\textquoteright =function-value; \textquoteleft rv\textquoteright =row +variable.)}} \begin{center} -\begin{tabular}{lll>{\raggedright}p{3cm}l} +\begin{tabular}{lll>{\raggedright}p{2.2cm}V{\linewidth}V{\linewidth}} \toprule -{\small char} & {\small rule} & {\small position} & {\small span} & {\small default rule thickness}\tabularnewline +{\footnotesize char} & {\footnotesize rule} & {\footnotesize position} & {\footnotesize span} & {\footnotesize trim} & {\footnotesize default rule thickness}\tabularnewline \midrule -{\small\texttt{T}} & {\small top} & {\small above table} & {\small table} & {\small\texttt{\textbackslash heavyrulewidth=.08em}}\tabularnewline -{\small\texttt{t}} & {\small title} & {\small below title} & {\small f-v cols} & {\small\texttt{\textbackslash cmidrulewidth =.03em}}\tabularnewline -{\small\texttt{s}} & {\small subtitle} & {\small below subtitle} & {\small f-v cols (if $1$ r-v col.)}{\small\par} - -{\small$<$ table (if $2$ r-v cols)} & {\small\texttt{\textbackslash cmidrulewidth =.03em}}\tabularnewline -{\small\texttt{h}} & {\small header} & {\small below header} & {\small table} & {\small\texttt{\textbackslash lightrulewidth=.05em}}\tabularnewline -{\small\texttt{f}} & {\small footer} & {\small above footer} & {\small < table} & {\small\texttt{\textbackslash cmidrulewidth =.03em}}\tabularnewline -{\small\texttt{B}} & {\small bottom} & {\small below table} & {\small table} & {\small\texttt{\textbackslash heavyrulewidth=.08em}}\tabularnewline +{\small\texttt{T}} & {\footnotesize top} & {\footnotesize above table} & {\footnotesize table} & & {\footnotesize\texttt{\textbackslash heavyrulewidth=.08em}}\tabularnewline +{\small\texttt{t}} & {\footnotesize title} & {\footnotesize below title} & {\footnotesize fv cols} & {\small\texttt{.5em}} & {\footnotesize\texttt{\textbackslash cmidrulewidth=.03em}}\tabularnewline +{\small\texttt{s}} & {\footnotesize subtitle} & {\footnotesize below subtitle} & {\footnotesize fv cols ($1$ rv col.) table ($2$ rv cols)} & {\small\texttt{.5em}}{\small\par} + +{\small\texttt{.5em}} & {\footnotesize\texttt{\textbackslash cmidrulewidth=.03em}}{\footnotesize\par} + +{\footnotesize\texttt{\textbackslash lightrulewidth=.05em}}\tabularnewline +{\small\texttt{h}} & {\footnotesize header} & {\footnotesize below header} & {\footnotesize table} & & {\footnotesize\texttt{\textbackslash lightrulewidth=.05em}}\tabularnewline +{\small\texttt{h'}} & & & & {\small\texttt{.5em}} & {\footnotesize\texttt{\textbackslash cmidrulewidth=.03em}}\tabularnewline +{\small\texttt{f}} & {\footnotesize footer} & {\footnotesize above footer} & {\footnotesize table} & & {\footnotesize\texttt{\textbackslash lightrulewidth=.05em}}\tabularnewline +{\small\texttt{f}}' & & & & {\small\texttt{.5em}} & {\footnotesize\texttt{\textbackslash cmidrulewidth=.03em}}\tabularnewline +{\small\texttt{B}} & {\footnotesize bottom} & {\footnotesize below table} & {\footnotesize table} & & {\footnotesize\texttt{\textbackslash heavyrulewidth=.08em}}\tabularnewline \bottomrule \end{tabular} \par\end{center} \end{table} -The \texttt{booktabs} \index{booktabs@\texttt{booktabs}} package -which \texttt{numerica} uses is most emphatic that one should `1. -Never, ever use vertical rules.\index{rules!edicts} 2. Never use -double rules.' Most of the tables proper in \emph{HMF} lack rules -of any kind although closer inspection shows smaller tables within -the text generally \emph{are} delimited by horizontal rules (often -also with vertical rules). In the various examples in the present -document I have used horizontal rules because these too are tables -within text. Some form of delineation seems necessary. (Although many -of \emph{HMF}'s tables are inelegantly typeset, I have used it as -a valuable source for the variety of structures that one might need -for presenting a multitude of different kinds of numerical data.) - -The \verb`rules` key allows one to specify precisely which rules -are used. The content of the key is a `word' \textendash{} a sequence -of letters \textendash{} where the characters have the significance -and default thicknesses (from \texttt{booktabs}\index{booktabs@\texttt{booktabs}}) -shown in Table~\ref{tab:Rules}. The default setting is \verb`rules=ThB`. -To insert a rule beneath the title, for example, change this to \verb`rules=TthB`. - -If you are using a subtitle row between header and title rows and -want a rule beneath that too, then the spec. is \verb`rules=TtshB`. -(For legacy reasons, \verb`m` \textendash{} from `midrow' \textendash{} -can also be used instead of \verb`s`, as in version 2 of \texttt{numerica-tables}.) -To my eye rules beneath both title and subtitle don't work; a rule -beneath the subtitle alone gives a better result. The subtitle rule -changes its behaviour depending on whether there are two row variable -columns \textendash{} on the left and right of the table \textendash{} -or not. If there is only such column then, like the title rule, the -subtitle rule spans only the function-value columns. If there are -two row variable columns then the subtitle rule spans the table but -is trimmed by 0.5 em at each end. That degree of trim is the \texttt{booktabs}\index{booktabs@\texttt{booktabs}} -default but can be changed by giving a different value to \verb`\cmidrulekern`\index{cmidrulekern@\texttt{\textbackslash cmidrulekern}} -in the preamble, e.g. \verb`\cmidrulekern=1em`. Note that the changed -trim will also apply to the title rule. - -If you are using a footer row\index{footer row} and want a rule above -it, then add \verb`f` to the specification, e.g. \verb`rules=TthfB`. -In version 3 of \texttt{numerica-tables} the rule is trimmed at each -end. Visually, having two table-spanning rules close together, the -\verb`f` and \verb`B` rules, doesn't work. The trimming makes a -difference. (For the \verb`T` and \verb`h` rules, the occurrence -of the table body beneath the \verb`h` rule seems to make a difference -to the visual impact of the rules.) But the question should always -be: is a rule necessary at all? Usually, less is more. - -If you wish to change the thickness of a rule from its default, then -enter new values for any or all of \verb`\heavyrulewidth`\index{heavyrulewidth@\texttt{\textbackslash heavyrulewidth}}, -\verb`\lightrulewidth`\index{lightrulewidth@\texttt{\textbackslash lightrulewidth}}, -\verb`\cmidrulewidth`\index{cmidrulewidth@\texttt{\textbackslash cmidrulewidth}} -in the preamble. The values listed in Table~\ref{tab:Rules} are -the default values in the \texttt{booktabs}\index{booktabs@\texttt{booktabs}} -package (except for the midrow and footer rules, which \texttt{booktabs}\index{booktabs@\texttt{booktabs}} -does not cover; in \texttt{numerica-tables} these rules are assigned -a thickness of \verb`\cmidrulewidth`). - -The order in which rules are placed in the specification doesn't matter. -I have entered them in their `natural' order simply because it feels -natural to do so, but it is their occurrence in the spec., not their -position, that matters. +If \verb`t` is included in the \verb`rules` specification, then +a rule will be drawn beneath the title. This will span the function-value +columns only and be trimmed at each end (by default by 0.5 em). If +you are also using a subtitle row, between title and header rows, +and want a rule beneath that too, then include \verb`s` in the setting +\textendash{} for instance \verb`rules=TtshB`. (For legacy reasons, +\verb`m` \textendash{} from `midrow' \textendash{} can also be +used instead of \verb`s`.) To my eye rules beneath \emph{both} title +and subtitle don't work; a rule beneath the subtitle alone gives a +better result \textendash{} if a rule is needed at all. + +The span of the subtitle rule depends on whether the row variable +column is placed only on one side of the table or on both (\verb`rpos<2` +or \verb`rpos>2`). If on only one then the rule spans only the function-value +columns \textendash{} like the title rule. If on both then the subtitle +rule spans the table but is trimmed by 0.5~em at each end. To the +author's eye, the extra span in this second case gives a better sense +of the row variable columns as part of the table as a whole rather +than as optional `clip ons' at the edges. In the example table below, a rule for the column title has been specified (the \verb`t` in the setting \verb`rules=TthB`). Also note the use -of \verb`ctitle=**`\index{ctitle@\texttt{ctitle}!** setting@\texttt{{*}{*}} setting}. +of \verb`ctitle=**`\index{ctitle@\texttt{ctitle}!{*},{*}{*} settings}. The formula contains an extra parameter $a$, assigned a value in -the vv-list. It now makes sense to display the vv-list in the column -title (but note the braces around \verb`k` and \verb`x` in the vv-list -so that \emph{they} don't display). +the vv-list, so that it now makes sense to display the vv-list in +the column title (the braces around \verb`k` and \verb`x` in the +vv-list ensure they don't display). \begin{verbatim} - \tabulate - [rspec={x,0.25,5},rround=2,rhnudge=9, - cspec={k,0.25,3},chstyle=2,chround=2, - ctitle=**,rules=TthB] - { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] + \tabulate[rspec={x,0.25,5,2},cspec={k,0.25,3,2}, + rhnudge=9,chstyle=2,ctitle=**,rules=TthB] + { a\sin kx }[a=2/\pi,{k}=3,{x}=0.25][*] \end{verbatim} -$\Longrightarrow$ \tabulate - [rspec={x,0.25,5},rround=2,rhnudge=9, - cspec={k,0.25,3},chstyle=2,chround=2, - ctitle=**,rules=TthB] - { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*]\index{rules|)}\index{rules@\texttt{rules}|)}\index{formatting!rules} - -\subsection{Second row variable column: \texttt{rpos=4}} - -\label{subsec:Second-row-var-col}In\index{row variable column!position in table} -\index{rpos@\texttt{rpos}|(}§\ref{subsec:Row-var-col-pos} I discussed -the settings \verb`rpos=0,1,2` and in §\ref{sec:Multiple-function-tables} -gave an example of using {\ttfamily\verb`rpos=3`} where -repeating the row variable column on the right is helpful. There is -another value available for this key, \verb`rpos=4`. Like {\ttfamily\verb`rpos=3`} -this adds the row variable column to both left and right sides of -the table, but for the right column the values are a function of those -in the left column ({\ttfamily\verb`rpos=3`} corresponds -to the function being the identity). The value given to the key \verb`rvar'`\index{rvar@\texttt{rvar'}|(} -determines the function used and the value given to the key \verb`rhead'`\index{rhead@\texttt{rhead'}|(} -determines the header for the right-hand row variable column. If \verb`rhead'` -is omitted it defaults to a blank header, unless the \verb`rvar'` -setting is also omitted, when \verb`rpos=4` behaves like \verb`rpos=3`. - -For example, the sine and cosine are complementary functions\index{complementary functions|(}; -when working in degrees, $\cos\theta=\sin(90-\theta)$. We can exploit -this fact to halve the table size needed to tabulate the two functions. -In the table, $\theta'=90-\theta$ and \verb`rhead'=\theta'`. Simply -to illustrate the use of \verb`rhnudge'`\index{rhnudge@\texttt{rhnudge'}|(} -I have nudged the header in the second (right) row variable column -to sit above the tens digits of the row variable values. The example -also gives an illustration of the use of an expression in the third -element of \verb`rspec`.\index{foot@\texttt{foot}} -\begin{verbatim} - \tabulate[ff=;,o,rpos=4, - rspec={\theta,5,1+45/5},rround=0, - chnudge=14,rvar'=90-\theta,rhnudge'=4, - rhead'=\theta',rules=ThfB,foot=*] - { \sin\theta;\cos\theta }[\theta=0][*] -\end{verbatim} -$\Longrightarrow$ \tabulate[ff=;,o,rpos=4, - rspec={\theta,5,1+45/5},rround=0, - chnudge=14,rvar'=90-\theta,rhnudge'=4, - rhead'=\theta',rules=ThfB,foot=*] - { \sin\theta;\cos\theta }[\theta=0][*]\index{footer row!complementary functions}\index{rhnudge@\texttt{rhnudge'}|)}\index{rhead@\texttt{rhead'}|)}\index{ff@\texttt{ff}}\medskip{} +$\Longrightarrow$ \tabulate[rspec={x,0.25,5,2},cspec={k,0.25,3,2}, + rhnudge=9,chstyle=2,ctitle=**,rules=TthB] + { a\sin kx }[a=2/\pi,{k}=3,{x}=0.25][*] + +\subsubsection{Trim and thickness of rules} + +\label{subsec:Trim-and-thickness}The trim applied to rules \textendash{} +the amount cut from the ends \textendash{} is the \texttt{booktabs}'\index{booktabs@\texttt{booktabs}} +default value, $0.5$~em. To change it, enter the command +\begin{verbatim} + \setlength\cmidrulekern{<trim>} +\end{verbatim} +in the preamble where \verb`<trim>` is a length, something like \verb`0.5em`, +\verb`2.5mm`, etc. (Or, one could write, e.g. \verb`\cmidrulekern=2.5mm`.) +The new default will apply to the settings \verb`h'`, \verb`f'`, +\verb`t` and sometimes \verb`s`, since these all use \texttt{booktabs}' +\verb`\cmidrule`. + +To change the thickness of a rule from its default value, enter new +values in the preamble for any or all of \verb`\heavyrulewidth`\index{heavyrulewidth@\texttt{\textbackslash heavyrulewidth}}, +\verb`\lightrulewidth`\index{lightrulewidth@\texttt{\textbackslash lightrulewidth}}, +\verb`\cmidrulewidth`\index{cmidrulewidth@\texttt{\textbackslash cmidrulewidth}}, +e.g. +\begin{verbatim} + \setlength\heavyrulewidth{<width>} +\end{verbatim} +where \verb`<width>` is a length. The values listed in Table~\ref{tab:Rules} +are the default values, from the \texttt{booktabs}\index{booktabs@\texttt{booktabs}} +package, used in \texttt{numerica-tables}.\index{rules@\texttt{rules}|)}\index{formatting!rules} + +\subsection{Second row variable column: \texttt{rpos=3,4}} + +\label{subsec:Second-row-var-col}\index{row variable column!position in table}\index{rpos@\texttt{rpos}|(}The +settings \verb`rpos=0,1,2` have been discussed earlier; {\ttfamily\verb`rpos=3`} +duplicates the row variable column on the right of the table, with +the proviso that if the header of the row variable column on the left +is automatically generated and like $x\!\:\backslash\!\:k$, the header +on the right is changed to a form like $k\!\:/\!\:x$. Repeating the +first example of §\ref{subsec:Headerstyle:-multi-column-case} but +with the setting \verb`rpos=3`, +\begin{verbatim} + \tabulate[rpos=3,rspec={x,0.2,5}, + ccspec={k,2,cstop=7,chnudge=18] + { \sin kx }[k=3,x=0.2][4*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rpos=3,rspec={x,0.2,5}, + cvar=k,cstep=2,cstop=7,chnudge=18] + { \sin kx }[k=3,x=0.2][4*]\medskip{} + +\noindent An example of using \verb`rpos=3` in a multi-function table +is provided near the end of §\ref{sec:Multiple-function-tables}. + +The \verb`rpos` key can also take the value $4$. \verb`rpos=4` +adds the row variable column to both left and right sides of the table, +but the values displayed in the right column are a \emph{function +of those in the left column} ({\ttfamily\verb`rpos=3`} +corresponds to the function being the identity). The value of the +key \verb`rvar'`\index{rvar@\texttt{rvar'}|(} specifies the function +used. +\begin{verbatim} + \tabulate[rpos=4,rspec={x,0.5,5,2},rhnudge=9, + cspec={k,2,3},chstyle=2,ctitle=*, + rvar'=x^2,rhnudge'=4] + { \sin(kx^2) }[k=3,x=1][4*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rpos=4,rspec={x,0.5,5,2},rhnudge=9, + chstyle=2,cspec={k,2,3},ctitle=*, + rvar'=x^2,rhnudge'=4] + { \sin(kx^2) }[k=3,x=1][4*]\medskip{} + +\noindent If no value is given to \verb`rvar'` (or if \verb`rvar'=*`) +then the row variable column on the right of the table is the left +column reversed (with the possible exception of its header). +\begin{itemize} +\item Note:\index{transpose@\texttt{transpose}!and rpos@and \texttt{rpos}} +\verb`rpos=4` is incompatible with the \verb`transpose` setting. +If both settings are used, the table will be transposed as if \verb`rpos=3`. +\end{itemize} +When \verb`chstyle` is not zero, as in the example above, and \verb`rhead'` +is omitted or empty, the header for the right-hand row variable column +is automatically generated, displaying the value of the \verb`rvar'` +key. The positioning of the right-hand row variable header can be +adjusted with the use of \verb`rhnudge'`\index{rhnudge@\texttt{rhnudge}'} +as in the example. If \verb`chstyle=0`, the header is up to the user +to supply by assigning a value to the key \verb`rhead'`\index{rhead@\texttt{rhead'}}. +In all cases, \verb`rhead'` will override any automatically generated +header. + +The sine and cosine are complementary functions\index{complementary functions|(}; +when working in degrees (the \verb`o` setting in the next example), +$\cos\theta=\sin(90-\theta)$. We can exploit this fact to halve the +table size needed to tabulate the two functions. The following `toy' +example (see\emph{ HMF} Tables 4.10\textendash 4.12 for the real thing) +of a multi-function table (hence the \verb`ff`\index{ff@\texttt{ff}} +setting) also gives an example of the use of the \verb`foot=*`\index{foot@\texttt{foot}} +setting, and the primed versions of header and footer rules\index{rules@\texttt{rules}}. +Note also the use of an expression in the third element of \verb`rspec`. +The \verb`rhead'` setting is of no significance; it simply avoids +a large block of white space. +\begin{verbatim} + \tabulate[ff,o,rpos=4,rules=Th'f'B,foot=*,calign=c, + rspec={\theta,9,1+45/9,0}, + rvar'=90-\theta,rhead'=-] + { \sin\theta,\cos\theta }[\theta=0][*] +\end{verbatim} +$\Longrightarrow$ \tabulate[ff,o,rpos=4,rules=Th'f'B,foot=*,calign=c, + rspec={\theta,9,1+45/9,0}, + rvar'=90-\theta,rhead'=-] + { \sin\theta,\cos\theta }[\theta=0][*]\medskip{} \noindent The values of sines from $0$ to $45$ degrees are read -downwards from the first column of function values, and from 45 to -90 degrees are read upwards from the second column of function values. +downwards from the first column of function-values, and from 45 to +90 degrees are read upwards from the second column of function-values. For cosines it is downwards from the second column and upwards from the first column. The reversed footer line indicates the change of -columns to use. In the example note -\begin{itemize} -\item the setting of \verb`rvar'`\index{rvar@\texttt{rvar'}|)} to a function -\verb`90-\theta` of the row variable; -\item the footer setting \verb`foot=*` to obtain the reversed header in -the footer; -\item the rule \emph{above} the footer row specified by the \verb`f` added -to the \verb`rules` setting, \verb`rules=ThfB`. -\end{itemize} -Note also the degree setting \verb`o` in the settings option. +columns to use. -Although there is a significant space saving with tables like this -(see \emph{HMF} Tables 4.10, 4.11, 4.12), they are not `kind to the -reader'. They require a certain concentration to read and in my view -should be avoided unless space is seriously constrained. \emph{HMF} -Tables 6.1 and 6.2 are tables of the gamma function and its relatives -where $y=x-1$ is used in the row variable column on the right (stemming -from $y!=\Gamma(x-1)$); \emph{HMF }Table 6.5 in effect uses $\langle1/x\rangle$ -(the nearest integer to $1/x$) for the row variable on the right.\index{complementary functions|)} +Although there is a significant space saving with tables like this, +they are not `kind to the reader'. They require a certain concentration +to read and should be avoided unless space is seriously constrained. +\emph{HMF} Tables 6.1 and 6.2 are tables of the gamma function and +its relatives where $y=x-1$ is used in the row variable column on +the right (stemming from $x!=\Gamma(x-1)$); \emph{HMF }Table 6.5 +in effect uses $\langle1/x\rangle$ (the nearest integer to $1/x$) +for the row variable on the right.\index{complementary functions|)} \subsection{Separating blocks of rows: \texttt{rbloc}} Readability\index{formatting!grouping rows|(}\index{row grouping|(} -of long columns of figures can be aided by breaking the columns into -blocks with extra white space between blocks of rows. This is achieved -with the \verb`rbloc`\index{rbloc@\texttt{rbloc}|(} key: +of long columns of figures can be aided by adding extra white space +between blocks of rows. This is achieved with the \verb`rbloc`\index{rbloc@\texttt{rbloc}|(} +key: \begin{verbatim} rbloc = <comma list of positive integers> \end{verbatim} @@ -1565,14 +2030,7 @@ say, the row variable runs from $0$ to $1$ in increments of $0.1$ \textendash{} there are $11$ rows of which the first (when the row variable is zero) may have distinctive values. -\noindent{}% -\noindent\begin{minipage}[t]{1\columnwidth}% -\begin{shaded}% - -\subsubsection*{The pull of the nice round number} - -However, this is not how \emph{HMF} sets out its tables.\emph{ }The -dominant practice in \emph{HMF} is division into blocks of (generally) +The dominant practice in \emph{HMF} is division into blocks of (generally) $5$ rows, many of which start with a zero value for the row variable. Rather than isolate this initial value, they include it in the first block of $5$, then continue with blocks of $5$ until a single isolated @@ -1583,70 +2041,115 @@ than $0$ to $9$, from $0$ to $1$ rather than $0$ to $0.9$, and even from $0$ to $30$ or $0$ to $2$ rather than $0$ to $29$ or $0$ to $1.9$. Using blocks of $5$ the consequence is that there is always an isolated line at the end \textendash{} a kind of punctuation -mark to signal the end of the page or the table.\end{shaded}% -\end{minipage} +mark to signal the end of the page or the table. -\medskip{} -In the next example I have divided the rows into blocks of $5$ by -means of the setting \verb`rbloc=5`. +In the next example I have divided the rows into blocks of $4$ but +separated the special values of the first and last rows by setting +\verb`rbloc={1,4}`. \begin{verbatim} - \tabulate[ff=;,o,rspec={\theta,10,1+90/10}, - rround=0,rbloc=5] - { \sin\theta; \cos\theta }[\theta=0][*] + \tabulate[ff,o,rspec={\theta,10,1+90/10,0},rbloc={1,4}] + { \sin\theta, \cos\theta }[\theta=0][*] \end{verbatim} -$\Longrightarrow$ \tabulate[ff=;,o,rspec={\theta,10,1+90/10}, - rround=0,rbloc=5] - { \sin \theta; \cos \theta}[\theta=0][*]\index{rbloc@\texttt{rbloc}|)} +$\Longrightarrow$ \tabulate[ff,o,rspec={\theta,10,1+90/10,0},rbloc={1,4}] + { \sin\theta, \cos\theta }[\theta=0][*]\index{rbloc@\texttt{rbloc}|)} \subsubsection{Adjusting the extra space\texttt{ rblocsep} } -By default \texttt{numerica} sets the extra space between blocks of -rows at \verb`1 ex`. This value can easily by changed with the setting -\verb`rblocsep=<length>`\index{rblocsep@\texttt{rblocsep}}. The -units need to be included in the specification. \index{formatting!grouping rows|)}\index{row grouping|)} +By default \texttt{numerica-tables} sets the extra space between blocks +of rows at \verb`1 ex`. This value can easily by changed with the +setting \verb`rblocsep=<length>`\index{rblocsep@\texttt{rblocsep}}. +The units need to be included in the specification. \index{formatting!grouping rows|)}\index{row grouping|)} + +\subsection{\textquoteleft Horizontal\textquoteright{} tables: \texttt{transpose}} + +\label{subsec:Transpose}Traditionally\index{tables!`horizontal'/`vertical'|(}\index{transpose@\texttt{transpose}|(} +in a book of mathematical tables the function values are arranged +vertically in columns. This is why the row variable has been given +primacy in \texttt{numerica-tables}. But that may not always be what +is wanted nor make the best use of available space. From version 3.2.0 +of \texttt{numerica-tables} there is a setting, \verb`transpose`, +that will convert a `vertical' table (function values in columns) +to a `horizontal' table (function values in rows). The table is +\emph{constructed} as if the function values will be listed vertically +in columns, but with \verb`transpose` entered in the settings option +the table is displayed with the function values listed in rows. + +(Strictly, \verb`transpose` `is really' \verb`transpose=1`, but +\verb`transpose` alone suffices; \verb`transpose=0` corresponds +to the default situation of function values in columns.) + +With transposition, the row variable column and header row are interchanged. +The building of the row variable column and header row occurs \emph{before} +transposition. Hence \verb`rround` and \verb`chround` are applied +to these elements \emph{before} transposition. It is the rounded values +of the row and column variables that are used to calculate function +values. But `cosmetic effects' like placement of the (new) row variable +column or (new) header row, as well as header styling (\verb`chstyle`), +alignment (\verb`ralign`, \verb`calign`), nudging (\verb`rhnudge`, +\verb`chnudge`), fonts (\verb`rfont`, \verb`chfont`), and choice +of display-, text- or scriptstyle (\verb`rmath`, \verb`chmath`) +are applied \emph{after} transposition. A footer row is created \emph{after} +transposition. +\begin{itemize} +\item Transposition\index{transpose@\texttt{transpose}!and rpos@and \texttt{rpos}} +does not work for tables using a second row variable column with \verb`rpos=4`. +Such a table \emph{is} transposed, but treated as if \verb`rpos=3`. +\end{itemize} +In a `shallow' transposed table \textendash{} like one with a single +row of function values \textendash{} rules should be used sparingly +if at all. See the example at §\ref{sec:Row-variable-settings} where +the \verb`norules` setting is used. In the following example, with +three function-value rows resulting from the transposition, a single +header rule suffices. +\begin{verbatim} + \tabulate[rspec={x,0.2,5},ralign=c,cspec={k,0.25,3,2}, + chstyle=1,rules=h',transpose] + { \sin kx }[k=3,x=0.2][*] +\end{verbatim} +$\Longrightarrow$ \tabulate[rspec={x,0.2,5},ralign=c,cspec={k,0.25,3,2}, + chstyle=1,rules=h',transpose] + { \sin kx }[k=3,x=0.2][*]\index{tables!`horizontal'/`vertical'|)}\index{transpose@\texttt{transpose}|)} \subsection{Table placement} -\label{sec:Table-placement}Tables\index{table placement!on the page} +\label{sec:Table-placement}Tables\index{tables!position on line/page} can be nudged vertically with the \LaTeX{} commands \verb`\bigskip`\index{bigskip@\texttt{\textbackslash bigskip}}, -\verb`\medskip`\verb`\medskip`, \verb`\smallskip`\index{smallskip@\texttt{\textbackslash smallskip}}, +\verb`\medskip`, \verb`\smallskip`\index{smallskip@\texttt{\textbackslash smallskip}}, usually about $1$, $\sfrac12$ and $\sfrac14$ line spaces (with stretch and shrink). \texttt{booktabs}\index{booktabs@\texttt{booktabs}} provides \verb`\abovetopsep`\index{abovetopsep@\texttt{\textbackslash abovetopsep}} and \verb`\belowbottomsep`\index{belowbottomsep@\texttt{\textbackslash belowbottomsep}}, both set by default to \verb`0ex` and easily changed by writing, -e.g., \verb`\abovetopsep=1.25ex` if you want to insert \verb`1.25ex` -of space above the table (perhaps to fit captions). +e.g. \verb`\setlength\abovetopsep{1.25ex}` (or \verb`\abovetopsep=1.25ex`) +if you want to insert \verb`1.25ex` of space above the table (perhaps +to fit captions). \subsubsection{Vertical~alignment} -\label{subsec:Verticalalignment}By\index{table placement!on the line} +\label{subsec:Verticalalignment}By\index{tables!position on line/page} writing\texttt{ }{\ttfamily\verb`valign=<char>`}\index{valign@\texttt{valign}} where \verb`<char>` is one of \verb`t`, \verb`m` or \verb`b` the -vertical alignment of the table can be set relative to the text baseline.\texttt{ }\verb`valign=t`\texttt{ -}aligns the top of the table with the text baseline, \verb`valign=b` -the bottom of the table with the text baseline, \verb`valign=m` aligns -the middle of the table with the text baseline. By default \verb`valign=m` -is set. Repeating an example from earlier (§\ref{sec:Row-variable-settings}) -I have added letters A, B, C to show where the baseline is. In the -first table the top of the table aligns with the baseline; in the -second table (default case) the middle of the table aligns with the -baseline; in the third table, the bottom of the table aligns with -the baseline. -\begin{verbatim} - A \tabulate[valign=t,rvar=x,rstep=0.2,rows=6] +vertical alignment of the table can be set relative to the text baseline:\texttt{ }\verb`t`\texttt{ +}aligns the top of the table, \verb`b` the bottom of the table, and +\verb`m` the middle of the table with the text baseline. By default +\verb`valign=m` is set. Repeating an example from earlier (§\ref{sec:Row-variable-settings}) +I have added letters A, B, C to show where the baseline is. Clearly +the top, middle and bottom of the respective tables aligns with the +baseline. +\begin{verbatim} + A \tabulate[valign=t,rvar=x,rstep=0.2,rows=5] { \sin x/\cos x }[x=0][*] \quad - B \tabulate[rspec={x,0.2,1+(1/0.2)}] + B \tabulate[rspec={x,0.2,1/0.2}] { \tan x }[x=0][*] \quad - C \tabulate[valign=b,rspec={x,0.2,(6)}] + C \tabulate[valign=b,rspec={x,0.2,5}] { \sqrt{\sec^2 x - 1} }[x=0][*] \end{verbatim} -$\Longrightarrow$ A \tabulate[valign=t,rvar=x,rstep=0.2,rows=6,rstop=2] - { \sin x/\cos x }[x=0][*] \quad - B \tabulate[rspec={x,0.2,1+(1/0.2)}] - { \tan x }[x=0][*] \quad - C \tabulate[valign=b,rspec={x,0.2,6}] - { \sqrt{\sec^2 x - 1} }[x=0][*]\medskip{} +$\Longrightarrow$ A \tabulate[valign=t,rvar=x,rstep=0.2,rows=5,rstop=2] + { \sin x/\cos x }[x=0.2][*] \quad + B \tabulate[rspec={x,0.2,1/0.2}] + { \tan x }[x=0.2][*] \quad + C \tabulate[valign=b,rspec={x,0.2,5}] + { \sqrt{\sec^2 x - 1} }[x=0.2][*]\medskip{} As explained in §\ref{subsec:Adjoining-tables}, tables can be adjoined to give the appearance of a single larger table. If tables with different @@ -1654,34 +2157,17 @@ numbers of rows are adjoined in this manner, then a middle alignment fails and a top alignment is necessary (so that the header rows of the tables align). -\section{Function value formatting} +\section{Formatting function values} -\label{sec:Function-value-formatting}\index{formatting!function values|(}\index{function value formatting|(} -\begin{table} -\centering{}\caption{\protect\label{tab:Function-value-formatting}Formatting function -values} -\begin{center} -\begin{tabular}{cc>{\raggedright}p{5cm}c} -\toprule -{\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline -\midrule -{\small\texttt{(pad)}} & {\small int} & {\small t-notation phantom padding} & \tabularnewline -{\small\texttt{signs}} & {\small int} & {\small sign handling for function-values} & {\small\texttt{0}}\tabularnewline -{\small\texttt{diffs}} & {\small int} & {\small insert differences \& pre-pad with zeros} & {\small\texttt{0}}\tabularnewline -{\small\texttt{round}} & {\small tokens} & {\small row or col. dependent rounding value} & \tabularnewline -{\small\texttt{Q?}} & {\small tokens} & {\small special cell conditional} & \tabularnewline -{\small\texttt{A!}} & {\small tokens} & {\small special cell formatting} & \tabularnewline -\bottomrule -\end{tabular} -\par\end{center} -\end{table} -In previous tables in this document, function values have generally -been limited to a fairly narrow range of values. What happens when -function values span orders of magnitude? Can we accommodate scientific -notation, expressly designed to cope with such orders of magnitude, -in a natural way? Can we round rows or columns to different rounding -values? Or, in a different direction, can we form tables of function -values in fraction form? +\label{sec:Function-value-formatting}\index{formatting!function values|(}\index{function-value formatting|(}In +previous tables in this document, function values have generally been +limited to a fairly narrow range of values. What happens when they +span orders of magnitude? Can scientific notation, expressly designed +to cope with such differing orders of magnitude, be accommodated in +a table in a natural way? Can rows or columns \textendash{} or individual +cells \textendash{} be rounded to different rounding values? Can we +indicate differences? Or form tables of function values in fraction +form? These and similar questions are our concern here. \subsection{Trailing optional argument} @@ -1694,26 +2180,39 @@ specified. \subsubsection{Fraction-form output} -\label{subsec:Fraction-form-output}In\index{fraction form!function values|(}\index{function value formatting!fraction form|(} -§\ref{subsec:rfracFraction-form-rvar} we saw how to display the row -variable in fraction form. Function values can also be presented in -that form. The problem is that such output requires far more computation -than other output since finding denominators at the specified accuracy -is an iterative process that needs to be performed for every function -value. However, it is feasible for small tables. In the tables below, -approximations to small positive and inverse powers of $\pi$ are -listed to $2$ and $4$ decimal places of accuracy. It is interesting -that all the powers listed can be approximated to $4$-place accuracy -by $3$-figure denominators (and $\pi^{2}$ by a $2$-figure denominator).\index{row variable!verbatim values|(}\index{sfrac@\texttt{\textbackslash sfrac}} -\begin{verbatim} - \def\mydataiii{\pi,\pi^2,\pi^3, - \pi^{\sfrac12},\pi^{\sfrac13}} - \tabulate[rdata=\mydataiii,rverb=1,rpos=1, - rvar=k,ralign=l,chead={\small $2$ places}] - { k }[2/s] \qquad - \tabulate[rdata=\mydataiii,rverb=1,rpos=1, - rvar=k,ralign=l,chead={\small $4$ places}] - { k }[4/s] +\begin{table}[t] +\centering +\centering{}\caption{\label{tab:Function-value-formatting}Formatting function values} +\begin{center} +\begin{tabular}{cc>{\raggedright}p{5cm}c} +\toprule +{\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline +\midrule +{\small\texttt{(pad)}} & {\small int} & {\small t-notation phantom padding} & \tabularnewline +{\small\texttt{signs}} & {\small int} & {\small sign handling for function values} & {\small\texttt{0}}\tabularnewline +{\small\texttt{diffs}} & {\small int} & {\small insert differences, pre-pad with $0$s} & {\small\texttt{0}}\tabularnewline +{\small\texttt{round}} & {\small tokens} & {\small row/col. dependent rounding value} & \tabularnewline +{\small\texttt{Q?}} & {\small tokens} & {\small special cell conditional} & \tabularnewline +{\small\texttt{A!}} & {\small tokens} & {\small special cell formatting} & \tabularnewline +\bottomrule +\end{tabular} +\par\end{center} +\end{table} +\label{subsec:Fraction-form-output}\index{fraction form!function values|(}\index{function-value formatting!fraction form|(}Function +values in a table can be presented in fraction form, but such output +requires far more computation than other forms since finding denominators +at the specified accuracy is an iterative process that needs doing +for every function value. But for small tables it is feasible. In +the tables below, approximations to small positive and inverse powers +of $\pi$ are listed to $2$ and $4$ decimal places of accuracy. +All the powers listed can be approximated to $4$-place accuracy by +$3$-figure denominators (and $\pi^{2}$ by a $2$-figure denominator).\index{row variable!verbatim values|(}\index{sfrac@\texttt{\textbackslash sfrac}} +\begin{verbatim} + \def\mypi{\pi,\pi^2,\pi^3,\pi^{\sfrac12},\pi^{\sfrac13}} + \tabulate[rdata=\mypi,rverb=1,rpos=1,rvar=k,ralign=l, + chead={\small $2$ places}]{ k }[2/s] \qquad + \tabulate[rdata=\mypi,rverb=1,rpos=1,rvar=k,ralign=l, + chead={\small $4$ places}]{ k }[4/s] \end{verbatim} $\Longrightarrow$ \def\mydataiii{\pi,\pi^2,\pi^3, \pi^{\sfrac12},\pi^{\sfrac13}} @@ -1724,71 +2223,73 @@ $\Longrightarrow$ \def\mydataiii{\pi,\pi^2,\pi^3, rvar=k,ralign=l,chead={\small $4$ places}] { k }[4/s]\medskip{} -A second example shows that all four built-in constants to \texttt{numerica} -and their first few inverse powers can be approximated to $5$ decimal -places with $3$-figure denominators:\index{sfrac@\texttt{\textbackslash sfrac}} +A second example shows that all four of \texttt{numerica}'s built-in +constants and their first few inverse powers can be approximated to +$5$ decimal places with $3$-figure denominators:\index{sfrac@\texttt{\textbackslash sfrac}} \begin{verbatim} - \tabulate[rdata={\pi ,e,\phi,\gamma},rverb=1,rvar=k, - cspec={n,1,4},chstyle=3,chnudge=9,rules=TthB,/max=1000, - ctitle=\lvert k^{\sfrac1n}-p/q \rvert<0.5\times10^{-5}] + \tabulate[rdata={\pi,e,\phi,\gamma},rverb=1,/max=1000, + rvar=k,cspec={n,1,4},chstyle=3,chnudge=9,rules=TthB, + ctitle=\abs{k^{\sfrac1n}-p/q}<0.5\times10^{-5}] { k^{\sfrac1n} }[n=1,k=1][/s5] \end{verbatim} $\Longrightarrow$ \tabulate[rdata={\pi ,e,\phi,\gamma},rverb=1,rvar=k, cspec={n,1,4},chstyle=3,chnudge=9,rules=TthB,/max=1000, ctitle=\lvert k^{\sfrac1n}-p/q \rvert<0.5\times10^{-5}] - { k^{\sfrac1n} }[n=1,k=1][/s5]\index{function value formatting!fraction form|)}\index{fraction form!function values|)}\index{row variable!verbatim values|)} + { k^{\sfrac1n} }[n=1,k=1][/s5]\index{function-value formatting!fraction form|)}\index{fraction form!function values|)}\index{row variable!verbatim values|)} \subsubsection{Scientific notation} -Elegant\index{function value formatting!scientific notations|(} scientific -notation\index{scientific notation},\index{scientific notation|seealso{t-notation}} -set with an \verb`x` in the trailing optional argument, is generally -not appropriate for use in tables; see the first table below. Repeating -the \verb`x` \textendash{} \verb`xx` \textendash{} in the trailing -optional argument (the second table) so that scientific notation extends -to numbers in the range $[1,10)$ helps, particularly with the \emph{left} -alignment chosen for the function-value column, but the result is -wasteful of space and the repetition of the `$\times10$' is distracting -and would be more so for a larger table. The \verb`x` specification -should be used in tables, if at all, only for \emph{small} tables -and special cases. The \verb`t` option is much preferred; see §\ref{subsec:t-option} -below. +\index{function-value formatting!scientific notations|(}Scientific +notation\index{scientific notation} \textendash \index{scientific notation|seealso{t-notation}} +put \verb`x` in the trailing optional argument \textendash{} is generally +inappropriate for use in tables; see the first table below. Entering +\verb`xx` (second table) so that the notation extends to numbers +in the range $[1,10)$ helps, particularly with the \emph{left} alignment +of the function values, but the result is wasteful of space and the +repetition of the `$\times10$' is distracting and would be more +so for a larger table. The \verb`x` specification should be used +in tables, if at all, only for \emph{small} tables and special cases. +The \verb`t` option is much preferred; see §\ref{subsec:t-option} +following. \begin{verbatim} - \tabulate[rspec={x,1,2*3+1},rround=0] - { e^x}[x=-5][*x]\qquad - \tabulate[rspec={x,1,2*3+1},rround=0,calign=l] - { e^x}[x=-3][*xx] + \tabulate[rspec={x,1,6,0},chnudge=58] + { e^x}[x=-2][*x]\qquad + \tabulate[rspec={x,1,6,0},calign=l,chnudge=45] + { e^x}[x=-2][*xx] \end{verbatim} -$\Longrightarrow$ \tabulate[rspec={x,1,2*3+1},rround=0] - { e^x}[x=-3][*x]\qquad - \tabulate[rspec={x,1,2*3+1},rround=0,calign=l] - { e^x}[x=-3][*xx] +$\Longrightarrow$ \tabulate[rspec={x,1,6,0},chnudge=58] + { e^x}[x=-2][*x]\qquad + \tabulate[rspec={x,1,6,0},calign=l,chnudge=45] + { e^x}[x=-2][*xx] \subsection{The \texttt{t} option} \label{subsec:t-option}\emph{HMF}\index{t-notation|(} uses a special -notation\index{function value formatting|(seealso{t-notation}} for -coping with function values spanning orders of magnitude. This notation -can be invoked by inserting \verb`t` in the trailing optional argument. -Repeating the previous two tables, and adding a \verb`chnudge` value, -gives a more compact and visually appealing result: +notation\index{function-value formatting|(seealso{t-notation}} for +coping with function values spanning different orders of magnitude. +This notation can be invoked by inserting \verb`t` in the trailing +optional argument. For the previous two tables the notation gives +a more compact and visually appealing result: \begin{verbatim} - \tabulate[rspec={x,1,2*3+1},rround=0,chnudge=24] + \tabulate[rspec={x,1,2*3+1,0},chnudge=24] { e^x}[x=-3][*t]\qquad - \tabulate[rspec={x,1,2*3+1},rround=0,chnudge=24] + \tabulate[rspec={x,1,2*3+1,0},chnudge=24] { e^x}[x=-3][*tt] \end{verbatim} -$\Longrightarrow$ \tabulate[rspec={x,1,2*3+1},rround=0,chnudge=24] +$\Longrightarrow$ \tabulate[rspec={x,1,2*3+1,0},chnudge=24] { e^x}[x=-3][*t] \qquad - \tabulate[rspec={x,1,2*3+1},rround=0,chnudge=24] - { e^x}[x=-3][*tt] + \tabulate[rspec={x,1,2*3+1,0},chnudge=24] + { e^x}[x=-3][*tt]\medskip{} + +Doubling the \verb`t` in the trailing optional argument (second table) +extends the notation to numbers in the range $[1,10)$ \textendash{} +as doubling \verb`x` extends normal scientific notation to this range. \subsubsection{Padding the exponent: \texttt{(pad)}} -In\index{padding with phantoms|(} the second table of the last example -some might quibble at the lack of alignment of the left parentheses. -\emph{HMF} tends to align these and \verb`numerica-tables` offers -the setting +In\index{padding with phantoms|(} the second table one might quibble +at the lack of alignment of the left parentheses. \emph{HMF} tends +to align these and \verb`numerica-tables` offers the setting \begin{verbatim} (pad) = <integer> \end{verbatim} @@ -1797,11 +2298,9 @@ a reminder of the \verb`t`-form of scientific notation.) \verb`<integer>` is the number of digits/characters to pad to. Repeating the last two tables with the setting \verb`(pad)=2` produces the following results: \begin{verbatim} - \tabulate[rspec={x,1,2*3+1},rround=0, - chnudge=24,(pad)=2] + \tabulate[rspec={x,1,2*3+1,0},chnudge=24,(pad)=2] { e^x}[x=-3][*t]\qquad - \tabulate[rspec={x,1,2*3+1},rround=0, - chnudge=24,(pad)=2] + \tabulate[rspec={x,1,2*3+1,0},chnudge=24,(pad)=2] { e^x}[x=-3][*tt] \end{verbatim} $\Longrightarrow$ \tabulate[rspec={x,1,2*3+1},rround=0, @@ -1811,15 +2310,19 @@ $\Longrightarrow$ \tabulate[rspec={x,1,2*3+1},rround=0, chnudge=24,(pad)=2] { e^x}[x=-3][*tt]\medskip{} -Note that this setting is relevant only when the \texttt{t} option -is used in the trailing number-formatting argument of the \verb`\tabulate` -command. Examples in \emph{HMF} of the style exemplified by the first +\noindent Examples in \emph{HMF} of the style exemplified by the first table are, among others, Tables 8.6, 9.2, 20.1, and of the style exemplified by the second table, among many, Tables 9.9, 10.5, 13.1, 14.1, 19.1.\index{padding with phantoms|)} +But note: +\begin{itemize} +\item the \verb`(pad)` setting is relevant only when the \verb`t` option +is used in the trailing number-format argument of the \verb`\tabulate` +command and should have no effect otherwise. +\end{itemize} -\paragraph{Accommodating signs: \texttt{signs}} +\subsubsection{Accommodating signs in the \texttt{t}-notation} -\label{subsec:Signs}Instead\index{t-notation!accommodating signs}\index{function value formatting!sign handling|(} +\label{subsec:Signs}Instead\index{t-notation!accommodating signs}\index{function-value formatting!sign handling|(} of $e^{x}$ as the test function, use $e^{x}-1.$ Now there are positive, zero and negative function values to contend with. Recall that in the \verb`t`-notation the \emph{exponent} is the parenthesized integer @@ -1829,8 +2332,8 @@ key to align (or not) the exponents. The setting is \begin{verbatim} signs = <integer> \end{verbatim} -Besides the do-nothing default \verb`(signs=0)`, there are four effective -values for \verb`<integer>`: +Besides the do-nothing default (\verb`signs=0` or omitted), there +are four effective values for \verb`<integer>`: \begin{itemize} \item \verb`signs=2`\texttt{ }inserts a $+$ sign between exponent and significand of every non-negative number; @@ -1847,14 +2350,11 @@ exponent and significand of every non-negative number; In the following examples, \verb`signs=-2`, \verb`signs=-1` and \verb`signs=2`, all give acceptable results. \begin{verbatim} - \tabulate[rspec={x,1,2*3+1},rround=0, - (pad)=2,signs=-2] - { e^x-1}[x=-3][4*tt]\qquad - \tabulate[rspec={x,1,2*3+1},rround=0, - (pad)=2,signs=-1] - { e^x-1}[x=-3][4*tt]\qquad - \tabulate[rspec={x,1,2*3+1},rround=0, - (pad)=2,signs=2] + \tabulate[rspec={x,1,2*3+1,0},(pad)=2,signs=-2] + { e^x-1}[x=-3][4*tt] \qquad + \tabulate[rspec={x,1,2*3+1,0},(pad)=2,signs=-1] + { e^x-1}[x=-3][4*tt] \qquad + \tabulate[rspec={x,1,2*3+1,0},(pad)=2,signs=2] { e^x-1}[x=-3][4*tt] \end{verbatim} $\Longrightarrow$ \tabulate[rspec={x,1,2*3+1},rround=0, @@ -1871,16 +2371,7 @@ $\Longrightarrow$ \tabulate[rspec={x,1,2*3+1},rround=0, 10.1, 13.1, 14.1, 19.1 among many others illustrate \verb`signs=-1`; and Tables 9.4, 10.6, 20.2, 22.11 among others illustrate \verb`signs=2`. -\verb`signs=1`, however, is an inappropriate setting for these function -values in the \verb`t`-notation: -\begin{verbatim} - \tabulate[rspec={x,1,2*3+1},rround=0, - (pad)=2,signs=1] - { e^x-1}[x=-3][4*tt] \qquad -\end{verbatim} -$\Longrightarrow$ \tabulate[rspec={x,1,2*3+1},rround=0, - (pad)=2,signs=1] - { e^x-1}[x=-3][4*tt]\index{t-notation|)}\index{function value formatting!scientific notations|)}\index{padding with phantoms|)}\medskip{} +For \verb`signs=1`, see the second table in the next example.\index{t-notation|)}\index{function-value formatting!scientific notations|)}\index{padding with phantoms|)}\medskip{} \subsection{Indicating signs outside the t-notation} @@ -1888,69 +2379,69 @@ $\Longrightarrow$ \tabulate[rspec={x,1,2*3+1},rround=0, The \verb`signs` key is not limited to the \verb`t`-notation. In the following tables where the notation is not used, positive values for the key, including \verb`signs=1`\index{signs@\texttt{signs}}, -give good results (I've included also the default setting \textendash{} -the third table): -\begin{verbatim} - \tabulate[rspec={x,0.1,9},(pad)=2,signs=2] - { 10\sin 5x}[x=-0.4][*4]\qquad - \tabulate[rspec={x,0.1,9},(pad)=2,signs=1] - { 10\sin 5x}[x=-0.4][*4]\qquad - \tabulate[rspec={x,0.1,9},(pad)=2] - { 10\sin 5x}[x=-0.4][*4] -\end{verbatim} -$\Longrightarrow$ \tabulate[rspec={x,0.1,9},(pad)=2,signs=2] - { 10\sin 5x}[x=-0.4][*4]\qquad - \tabulate[rspec={x,0.1,9},(pad)=2,signs=1] - { 10\sin 5x}[x=-0.4][*4]\qquad - \tabulate[rspec={x,0.1,9},(pad)=2] - { 10\sin 5x}[x=-0.4][*4]\medskip{} +give good results. The third table, corresponding to \verb`signs=0`, +shows that the comment extends to \emph{non-negative} values of the +\verb`signs` key. +\begin{verbatim} + \tabulate[rspec={x,0.1,7},(pad)=2,signs=2] + { 10\sin 5x}[x=-0.3][*4]\qquad + \tabulate[rspec={x,0.1,7},(pad)=2,signs=1] + { 10\sin 5x}[x=-0.3][*4]\qquad + \tabulate[rspec={x,0.1,7},(pad)=2] + { 10\sin 5x}[x=-0.3][*4] +\end{verbatim} +$\Longrightarrow$ \tabulate[rspec={x,0.1,7},(pad)=2,signs=2] + { 10\sin 5x}[x=-0.3][*4]\qquad + \tabulate[rspec={x,0.1,7},(pad)=2,signs=1] + { 10\sin 5x}[x=-0.3][*4]\qquad + \tabulate[rspec={x,0.1,7},(pad)=2] + { 10\sin 5x}[x=-0.3][*4]\medskip{} \noindent\emph{HMF} seems to use \verb`signs=2` when the sign of the function values changes every few entries and \verb`signs=1` when there are runs of entries of the same sign. Over the range tabulated here for $10\sin5x$, they would use the middle table of the three, -\verb`signs=1`.\index{function value formatting!sign handling|)}\index{signs@\texttt{signs}|)} +\verb`signs=1`.\index{function-value formatting!sign handling|)}\index{signs@\texttt{signs}|)} -\subsection{Rounding to varying values} +\subsection{Cell-, row-, column-dependent rounding} -\label{subsec:Rounding-varying} Above\index{rounding value!row/col-dependent|(}, -in §\ref{subsec:Fraction-form-output}, we created two tables of fraction-form -approximations to simple power functions of $\pi$, one accurate to -two places of decimals, one to four places. Version 3.1 of \texttt{numerica-tables} -(as distinct from version 3.0) offers the means of producing tables +\label{subsec:Rounding-varying} \index{rounding value!row/col.dependent@row/col.\ dependent|(}In +§\ref{subsec:Fraction-form-output}, we created two tables of fraction-form +approximations to simple powers of $\pi$, one accurate to two places +of decimals, one to four. From version 3.1 (as distinct from version +3.0) \texttt{numerica-tables} offers the means of producing tables with rounding values depending on position in the table. This is effected -through the key \index{round@\texttt{round}|(} -\begin{verbatim} - round = f(rvar,cvar) -\end{verbatim} -where \texttt{f(rvar,cvar)} denotes a function of row and column variables. -Usually this will mean dependence either on row variable or column -variable rather than both. In the present instance we form a multi-function -table and with the \verb`round` key let the rounding value equal -the row variable value \verb`r` (\verb`round=r`) to obtain fractional -approximations to simple powers of $\pi$ at rounding values from -$1$ to $5$ (and discover that all these values can be approximated -to $5$ decimal places with $3$ figure denominators \textendash{} -$\pi^{2}$ only just).\index{multi-function tables}\index{fraction form!row/col-dependent rounding} +through the key \verb`round`\index{round@\texttt{round}|(} which +sets the rounding value as a function of row and column variables. +In practice this usually means dependence on row or column variable +alone rather than both. In the example below, the rounding equals +the row variable value, \verb`round=r`, producing fractional approximations +to simple powers of $\pi$ at rounding values from $1$ to $5$, and +learn that all these powers can be approximated to $5$ decimal places +with $3$ figure denominators \textendash{} $\pi^{2}$ only just. +(The command \verb`\abs` used in \verb`ctitle` is defined in \texttt{numerica}.)\index{multi-function tables}\index{fraction form!row/col.-dependent rounding} \begin{verbatim} \tabulate[ff,rspec={r,1,5},round=r,/max=999,chstyle=2, - ctitle=\lvert\pi^k-\sfrac mn\rvert<0.5\times10^{-r}] + ctitle=\abs{\pi^k-\sfrac mn}<0.5\times10^{-r}] { \pi,\pi^2,\pi^3,\pi^{1/2},\pi^{1/3}}[r=1][/s] \end{verbatim} $\Longrightarrow$ \tabulate[ff,rspec={r,1,5},round=r,/max=999,chstyle=2, - ctitle=\lvert\pi^k-\sfrac mn\rvert<0.5\times10^{-r}] - { \pi ,\pi^2,\pi^3,\pi^{1/2},\pi^{1/3}}[r=1][/s]\index{fraction form!function values}\index{function value formatting!fraction form}\medskip{} + ctitle=\abs{\pi^k-\sfrac mn} < 0.5 + \times10^{-r}] + { \pi ,\pi^2,\pi^3,\pi^{1/2},\pi^{1/3}}[r=1][/s]\index{fraction form!function values}\index{function-value formatting!fraction form}\medskip{} Another place where a variable rounding value can be of value is when a function being tabulated changes slowly for each step in the row -variable value; the value of the cosine for instance changes from +variable value. The value of the cosine for instance changes from $1.0000$ to $\eval{\cos10\degree}[4]$ between $0\degree$ and $10\degree$. \emph{Part} of a table of the cosine might be something like the following, where values in the initial rows of the table are rounded to a higher -value than in later rows. \verb`round` is set to an expression involving -the row variable in boolean elements \verb`\theta<11` and \verb`\theta>10` -which evaluate to $0$ or $1$ so that \verb`round` takes the value -$6$ for the initial rows of the table and the value $4$ thereafter. +value than in later rows. In the example \verb`round` is set to an +expression in the row variable \verb`\theta` involving the boolean +expressions \verb`\theta<11` and \verb`\theta>10` which evaluate +to $0$ or $1$ depending on the value of \verb`\theta`. Thus \verb`round` +takes the value $6$ for the initial rows of the table and the value +$4$ thereafter. \begin{verbatim} \tabulate[o,rspec={\theta,1,6},calign=l,chnudge=15, round=6(\theta<11)+4(\theta>10)] @@ -1958,7 +2449,7 @@ $6$ for the initial rows of the table and the value $4$ thereafter. \end{verbatim} $\Longrightarrow$ \tabulate[o,rspec={\theta,1,6},calign=l,chnudge=15, round=6(\theta<11)+4(\theta>10)] - { \cos\theta }[\theta=8]\index{rounding value!row/col-dependent|)}\index{round@\texttt{round}|)} + { \cos\theta }[\theta=8]\index{rounding value!row/col.dependent@row/col.\ dependent|)}\index{round@\texttt{round}|)} \subsection{Differences: \texttt{diffs}} @@ -1973,59 +2464,58 @@ the \verb`\tabulate` command will include differences in a table. The \texttt{<non-negative integer>} is the maximum number of digits in a difference. \begin{verbatim} - \tabulate[rspec={x,0.01,1+(1.05-1)/0.01},rround=2, - rhnudge=9,chnudge=21,diffs=3] + \tabulate[rspec={x,0.01,6,2},diffs=3, + rhnudge=9,chnudge=21] + { \sinh x }[x=1,k=1][*4] \qquad + \tabulate[rspec={x,0.01,6,2},diffs=2, + rhnudge=9,chnudge=25] + { \sinh x }[x=1][*4]\qquad + \tabulate[rspec={x,0.01,6,2},diffs=4, + rhnudge=9,chnudge=30] { \sinh x }[x=1][*4] \end{verbatim} -$\Longrightarrow$ \tabulate[rspec={x,0.01,1+(1.05-1)/0.01},rround=2, - rhnudge=9,chnudge=21,diffs=3] +$\Longrightarrow$ \tabulate[rspec={x,0.01,6,2},diffs=3, + rhnudge=9,chnudge=21] + { \sinh x }[x=1,k=1][*4] \qquad + \tabulate[rspec={x,0.01,6,2},diffs=2, + rhnudge=9,chnudge=25] + { \sinh x }[x=1][*4]\qquad + \tabulate[rspec={x,0.01,6,2},diffs=4, + rhnudge=9,chnudge=30] { \sinh x }[x=1][*4]\medskip{} -I have deliberately chosen the function and settings here \textendash{} -particularly \texttt{diffs=3} \textendash{} to give a good result. -With the default right alignment of the function-value columns, it -is easy to get this wrong. The evidence will be either in the misalignment -of the first row of function values or unnecessary padding of differences -with leading zeros. It is a good idea to create your table first, -see how function values change between successive rows and judge how -many digits there will be in a difference. In the following examples -I have deliberately put \texttt{diffs=2} and \texttt{diffs=4} to show -the effect of a misjudgement. In the first table the first row of -function values is misaligned by one character. (\texttt{diffs=1} -would have produced a two-character misalignment.) In the second table -the unnecessary fourth digit for the differences results in pre-padding -with $0$. - -In the second table the function $-\sinh x$ is \emph{decreasing}, -showing how it is the \emph{absolute value of the difference} between -successive function values that is tabulated. A difference is always -a non-negative value. -\begin{verbatim} - \tabulate[rspec={x,0.01,1+(1.05-1)/0.01},rround=2, - rhnudge=9,chnudge=21,diffs=2] - { \sinh x }[x=1][*4]\qquad - \tabulate[rspec={x,0.01,1.05},rround=2, - rhnudge=9,chnudge=30,diffs=4] - { -\sinh x }[x=1][*4] +I have deliberately chosen the settings in the first table \textendash{} +\texttt{diffs=3} \textendash{} to give a good result. With the default +right alignment of the function-value columns, it is easy to get this +wrong. The evidence will be either in the misalignment of the first +row of function values (second table) or unnecessary padding of differences +with leading zeros (third table). It is a good idea to create your +table first, see how function values change between successive rows +and judge how many digits there will be in a difference. When the +\texttt{diffs} setting is too small, function values in the first +row are misaligned, the amount depending on how much too small. (A +left alignment of the function-value column is another way of tackling +this issue.) When the \texttt{diffs} setting is too big, alignment +is fine but differences are padded with unnecessary leading zeros, +meaning the column header will need a bigger nudge to bring \emph{it} +into alignment. + +In the next example the function is \emph{decreasing}, showing how +it is the \emph{absolute value of the difference} between successive +function values that is tabulated. A difference is always a non-negative +value. +\begin{verbatim} + \tabulate[rspec={x,0.01,6,2},diffs=2, + rhnudge=9,chnudge=21] + { e^{-x^2} }[x=0.05][*4] \end{verbatim} -$\Longrightarrow$ \tabulate[rspec={x,0.01,1+(1.05-1)/0.01},rround=2, - rhnudge=9,chnudge=21,diffs=2] - { \sinh x }[x=1][*4]\qquad - \tabulate[rspec={x,0.01,1+(1.05-1)/0.01},rround=2, - rhnudge=9,chnudge=21,diffs=4] - { -\sinh x }[x=1][*4]\medskip{} - -\noindent When the \texttt{diffs} setting is too small, function values -in the first row are misaligned, the amount depending on how much -too small. (A left alignment of the function value column is another -way of tackling this issue.) When the \texttt{diffs} setting is too -big, alignment is fine but differences are padded with unnecessary -leading zeros, meaning the column header will need a bigger nudge -to bring \emph{it} into alignment. \index{differences|)} +$\Longrightarrow$ \tabulate[rspec={x,0.01,6,2},diffs=2, + rhnudge=9,chnudge=21] + { e^{-x^2} }[x=0.05][*4] \index{differences|)} \subsection{Formatting special values: \texttt{Q?} and \texttt{A!}} -You\index{function value formatting!highlighting special values|(}\index{Q?@\texttt{Q?}|(}\index{A@\texttt{A"!}|(} +\label{subsec:QA-formatting}You\index{function-value formatting!highlighting special values|(}\index{Q?@\texttt{Q?}|(}\index{A@\texttt{A"!}|(} may wish to highlight or display in some special way a particular function value or values. \verb`\nmcTabulate` has two related settings that enable this: \verb`Q?=<tokens>` and \verb`A!=<tokens>`. As @@ -2033,53 +2523,55 @@ the names suggest: Question? and Answer! The question should be an expression that \texttt{l3fp} can digest and produce a boolean answer to (1 for `true', 0 for `false'). -\emph{This is not a \LaTeX{} expression};\emph{ this is an }\verb`l3fp` -\emph{expression}.\footnote{Documentation about \texttt{l3fp }can be found in \texttt{interface3.pdf}, -which is part of the \texttt{l3kernel} bundle.} For the user it should be enough to know that an expression formed -from decimal numbers (but only with the dot decimal point), parentheses -\verb`( )`, the familiar arithmetic symbols, \verb`+`,\texttt{ }\verb`-`,\texttt{ -}\verb`*`,\texttt{ }\verb`/` and \verb`^`, relation symbols \verb`<`,\texttt{ -}\verb`>`,\texttt{ }\verb`=` and combinations like \verb`!=` (for -$\ne$),\texttt{ }\verb`>=` (for $\ge$), and\texttt{ }\verb`<=` -(for $\le$) will be digested by \verb`l3fp`. In addition there are -\verb`||` for logical Or, \verb`&&` for logical And, and \verb`!` -for logical Not; \verb`exp(1)` for $e$ and \verb`pi` (no backslash) -for $\pi$. \texttt{numerica-tables} provides \verb`MAX`\index{MAX@\texttt{MAX}} -and \verb`MIN`\index{MIN@\texttt{MIN}} for the maximum and minimum -function values tabulated, and uses \verb`@` to denote the current -function value. +\emph{This is not a \LaTeX{} expression but an }\verb`l3fp` \emph{expression}.\footnote{Documentation about \texttt{l3fp }can be found in \texttt{interface3.pdf}, +which is part of the \texttt{l3kernel} bundle.} For the user it should suffice to know that an expression formed +from decimal numbers (but \verb`l3fp` knows only the decimal dot), +parentheses \verb`( )`, the familiar arithmetic symbols, \verb`+`,\texttt{ +}\verb`-`,\texttt{ }\verb`*`,\texttt{ }\verb`/` and \verb`^`, +relation symbols \verb`<`,\texttt{ }\verb`>`,\texttt{ }\verb`=` +and combinations like \verb`!=` (for $\ne$),\texttt{ }\verb`>=` +(for $\ge$), and\texttt{ }\verb`<=` (for $\le$) will be digested +by \verb`l3fp`. In addition there are \verb`||` for logical Or, +\verb`&&` for logical And, and \verb`!` for logical Not; \verb`exp(1)` +for $e$ and \verb`pi` (no backslash) for $\pi$. \texttt{numerica-tables} +provides \verb`MAX`\index{MAX@\texttt{MAX}} and \verb`MIN`\index{MIN@\texttt{MIN}} +for the maximum and minimum function values tabulated, and uses \verb`@` +to denote the function value of the current cell. So, a query might be \verb`Q?=@<0`, \emph{Is the current function value negative?}, or \verb`Q?={@>=pi}`, \emph{Is the current function value greater than or equal to $\pi$?} (The braces hide the equality -sign in the \emph{key=value} settings option.) \verb`Q?={@=MIN}` -(again note the braces) is the question: \emph{Is the current function -value equal to the minimum function value for the whole table?} +sign.) \verb`Q?={@=MIN}` (again note the braces) is the question: +\emph{Is the current function value equal to the minimum function +value for the whole table?} The answer must be in the form of a \LaTeXe{} formatting statement, -again using \verb`@` to denote the current function value. Thus \verb`A!=\mathbf{@}` -is a valid answer; so is \verb`A!=\color{red}{@}` (provided you have -\verb`\usepackage{color}` in the preamble); and so is \verb`A!=(@)`. -Another valid answer is \verb`A!= ` , meaning that function values -satisfying the \verb`Q?` question are omitted from the output. - -This can be useful to suppress `irrelevant' values in a particular -context. For example, suppose we wish to focus on the values of $\cos(m\pi/n)$ -lying between $0$ and $\tfrac{1}{2}$ inclusive for certain values -of $m$ and $n$. Rather than cluttering the table with values outside -that interval, we suppress them (the two occurrences of `\verb`1e-14`' -in the query are there to prevent rounding errors confusing the result): -\begin{verbatim} - \tabulate[rspec={n,1,1+(15-4)},rules=Tth,rround=0, - rpos=2,cspec={m,1,1+(5-2)},chstyle=2, - ctitle=*,Q?={@<-1e-14||@>0.5+1e-14},A!=] - { \cos(m\pi/n) }[n=4,m=2][*4] +again using \verb`@` to denote the function value of the current +cell. Thus \verb`A!=\mathbf{@}` is a valid answer; so is \verb`A!=\color{red}{@}` +(provided you have \verb`\usepackage{color}` in the preamble); and +so is \verb`A!=(@)`. Another valid answer is \verb`A!= ` , meaning +that function values satisfying the \verb`Q?` question are omitted +from the output. + +This can be useful to suppress `irrelevant' values in particular +contexts. For example, the \emph{non-zero} values of \verb`\binom{n}{m}`, +displaying as $\binom{n}{m}$, are the ones of interest. Rather than +cluttering the table with $0$s, suppress them, first by finding them +(\verb`Q?={@=0}`), and then by excising them (\verb`A!= ` ). The +table suppresses display of the header row with the \verb`headless`\index{headless@\texttt{headless}} +setting and instead displays it in the footer with the \verb`foot=''`\index{foot@\texttt{foot}}\index{footer!copies header} +setting, automatically changing what would have been $n\:\!\backslash\:\!m$ +in the header to $n\:\!/\:\!m$ in the footer. +\begin{verbatim} + \tabulate[rspec={n,1,5},rhnudge=-18,rfont=bf,chfont=bf, + cspec={m,1,5},ctitle=\binom nm\qquad, + headless,rules=f',foot='',Q?={@=0},A!=] + { \binom{n}{m} }[n=1,m=1] \end{verbatim} -$\Longrightarrow$ \tabulate - [rspec={n,1,1+(15-4)},rround=0,rpos=2,rules=Tth, - cspec={m,1,1+(5-2)},ctitle=*,chstyle=2, - Q?={@<-1e-14||@>0.5+1e-14},A!=] - { \cos(m\pi/n) }[n=4,m=2][*4] +$\Longrightarrow$ \tabulate[rspec={n,1,5},rhnudge=-18,rfont=bf,chfont=bf, + cspec={m,1,5},ctitle=\binom nm\qquad, + headless,rules=f',foot='',Q?={@=0},A!=] + { \binom{n}{m} }[n=1,m=1] \subsubsection{Star option: \texttt{\textbackslash nmcTabulate{*}}} @@ -2093,79 +2585,88 @@ display the first such instance. Like other starred commands in the you do not need an answering \verb`A!` to the query \verb`Q?` since no formatting of table values is involved. \begin{verbatim} - \tabulate*[rspec={n,1,1+(15-4)},cspec={m,1,1+(5-2)}, + \tabulate*[rspec={n,1,12},cspec={m,1,4)}, Q?={@<-1e-14||@>0.5+1e-14}] { \cos(m\pi/n) }[n=4,m=2][*4] \end{verbatim} $\Longrightarrow$ \tabulate* - [rspec={n,1,1+(15-4)},cspec={m,1,1+(5-2)}, + [rspec={n,1,12},cspec={m,1,4}, Q?={@<-1e-14||@>0.5+1e-14}] - { \cos(m\pi/n) }[n=4,m=2][*4]. Indeed, if you omit the \verb`Q?` and \verb`A!` settings from -the previous table so that all function values are visible then this -is the value that follows $0.5000$ in the \verb`m=2` column \textendash{} -the first function value encountered outside the interval $[0,0.5]$. + { \cos(m\pi/n) }[n=4,m=2][*4]. Indeed, if you omit the \verb`Q?` setting from the previous table +so that all function values are visible then this is the value that +follows $0.5000$ in the \verb`m=2` column, the first function value +encountered outside the interval $[0,0.5]$. (The \verb`1e-14` is +to ensure rounding errors don't give a spurious result.) If you want the \emph{maximum} value that has been tabulated then, from version 3 of \texttt{numerica-tables}, you do not even need the query: when \verb`\tabulate` is starred, \verb`Q?` is initialized -behind the scenes to \verb`@=MAX`\index{MAX@\texttt{MAX}}.\footnote{\noindent In the unlikely event that someone \emph{consistently} wanted -some other query to be asked \textendash{} the minimum value perhaps, -or first negative value or \ldots{} \textendash{} please let the author -know. It would be straightforward to use a package option to give -a choice in this matter.} Thus, repeating the example from §\ref{sec:Column-variable-settings}, -\begin{verbatim} - \tabulate[rspec={x,0.2,6},rhead=x\backslash k, - cvar=k,cstep=2,cstop=9] - { \sin kx }[k=3,x=0] - \tabulate*[rspec={x,0.2,6},rhead=x\backslash k, - cvar=k,cstep=2,cstop=9] - { \sin kx }[k=3,x=0] -\end{verbatim} -$\Longrightarrow$ \tabulate - [rspec={x,0.2,6},rhead=x\backslash k, - cvar=k,cstep=2,cstop=9] - { \sin kx }[k=3,x=0][*] -\tabulate* - [rspec={x,0.2,6},rhead=x\backslash k, - cvar=k,cstep=2,cstop=9] - { \sin kx }[k=3,x=0] [*] +behind the scenes to \verb`@=MAX`\index{MAX@\texttt{MAX}}. Thus, +repeating the example from §\ref{sec:Column-variable-settings}, a +visual check of the table shows that the starred command has, indeed, +isolated the maximum value displayed: +\begin{verbatim} + \tabulate[rspec={x,0.2,6},cspec={k,2,4},chnudge=27] + { \sin kx }[k=3,x=0] $\longrightarrow + \tabulate*[rspec={x,0.2,6},cspec={k,2,4},chnudge=27] + { \sin kx }[k=3,x=0] $ +\end{verbatim} +$\Longrightarrow$ \tabulate[rspec={x,0.2,6},cspec={k,2,4},chnudge=27] + { \sin kx }[k=3,x=0.2][*] $\longrightarrow + \tabulate*[rspec={x,0.2,6},cspec={k,2,4},chnudge=27] + { \sin kx }[k=3,x=0.2]$ + +\paragraph{Package option} + +There\index{package options} is a package option \verb`Q?*`\index{Q?*@\texttt{Q?{*}}} +available which can be set to give some other default query than \verb`@=MAX` +if you so wish. For instance +\begin{verbatim} + \usepackage[Q?*={@=MIN}]{numerica-tables} +\end{verbatim} +makes `what is the minimum value tabulated?' the default query for +a \verb`\tabulate*` command. Or, \verb`Q?*=@>0` makes `what is +the first positive value encountered?' the default query for a \verb`\tabulate*` +command, and so on. \paragraph{Errors} If\index{error messages} \emph{no} function value satisfies a query then a message is generated: \begin{verbatim} - \tabulate*[rspec={n,1,1+(15-4)}, - cspec={m,1,1+(5-2)},Q?=@>1] + \tabulate*[rspec={n,1,12)},cspec={m,1,4},Q?=@>1] { \cos(m\pi/n) }[n=4,m=2][*4] \end{verbatim} $\Longrightarrow$ \tabulate* [rspec={n,1,1+(15-4)},cspec={m,1,1+(5-2)}, Q?=@>1] - { \cos(m\pi/n) }[n=4,m=2][*4]\index{Q?@\texttt{Q?}|)}\index{A@\texttt{A"!}|)} + { \cos(m\pi/n) }[n=4,m=2][*4] + +\noindent Note that the same message is presented if the error arises +from the package option query, which may well be confusing (for a +moment) since there is no explicit query present in the settings option.\index{Q?@\texttt{Q?}|)}\index{A@\texttt{A"!}|)} \paragraph{Scientific notation} If you want the number output in scientific notation when the star option is chosen, then enter the exponent mark in the trailing number-format -option. This is straightforward for a letter like the commonly used -\verb`e`, but remember that if you enter the \verb`x` option you -will need to place the \verb`\tabulate*` command between math delimiters, -otherwise the \verb`\times` symbol resulting from the \verb`x` option -will generate a \LaTeX{} error (`Missing \$ inserted'): -\begin{verbatim} - $ - \tabulate*[rspec={n,1,1+(15-4)},cspec={m,1,1+(5-2)}, - Q?={@<-1e-14||@>0.5+1e-14},A!=] - { \cos(m\pi/n) }[n=4,m=2][*4x] - $ +option of the \verb`\tabulate*` command. This is straightforward +for a mark like the commonly used letter \verb`e`, but remember that +if you enter \verb`x` you will need to place the \verb`\tabulate*` +command between math delimiters, otherwise the \verb`\times` symbol +resulting from the \verb`x` option will generate a \LaTeX{} error +(`Missing \$ inserted'). Remembering the \verb`$` signs gives, +e.g., +\begin{verbatim} + $ \tabulate*[rspec={n,1,12},cspec={m,1,4}, + Q?={@<-1e-14||@>0.5+1e-14}] + { \cos(m\pi/n) }[n=4,m=2][*4x] $ \end{verbatim} $\Longrightarrow$ $ - \tabulate* - [rspec={n,1,1+(15-4)},cspec={m,1,1+(5-2)}, - Q?={@<-1e-14||@>0.5+1e-14},A!=] + \tabulate*[rspec={n,1,12},cspec={m,1,4}, + Q?={@<-1e-14||@>0.5+1e-14}] { \cos(m\pi/n) }[n=4,m=2][*4x] - $.\index{formatting!function values|)}\index{function value formatting|)}\index{function value formatting!highlighting special values|)} + $.\index{formatting!function values|)}\index{function-value formatting|)}\index{function-value formatting!highlighting special values|)} \section{Other matters} @@ -2188,36 +2689,32 @@ one tabulated function value. \eval[env=$]{(\tabulate [rspec={n,1,15},cspec={m,1,5},Q?={@=MAX}] { \cos(m\pi/n) }[n=4,m=2][*4])\sinh t + - (\tabulate[rspec={n,1,15},cspec={m,1,5}, - Q?={@=MIN}] - { \sin(m\pi/n) }[n=4,m=2][*4])\cosh t - }[t=2][4] + (\tabulate[rspec={n,1,15},cspec={m,1,5},Q?={@=MIN}] + { \sin(m\pi/n) }[n=4,m=2][*4])\cosh t } + [t=2][4] \end{verbatim} -$\Longrightarrow$ \eval{$ - (\tabulate - [rspec={n,1,15},cspec={m,1,5}, - Q?={@=MAX}] - { \cos(m\pi/n) }[n=4,m=2][*4])\sinh t + - (\tabulate - [rspec={n,1,15},cspec={m,1,5}, - Q?={@=MIN}] - { \sin(m\pi/n) }[n=4,m=2][*4])\cosh t - $}[t=2][4]. +$\Longrightarrow$ \eval[env=$]{(\tabulate + [rspec={n,1,15},cspec={m,1,5},Q?={@=MAX}] + { \cos(m\pi/n) }[n=4,m=2][*4])\sinh t + + (\tabulate[rspec={n,1,15},cspec={m,1,5},Q?={@=MIN}] + { \sin(m\pi/n) }[n=4,m=2][*4])\cosh t } + [t=2][4]. Forming the table \begin{verbatim} - \tabulate[rspec={n,1,15},rround=0,rpos=2,rules=Tth, + \tabulate[rspec={n,1,15,0},rpos=2,rules=Tth, cspec={m,1,5},ctitle=*,chstyle=2] { \cos(m\pi/n) }[n=4,m=2][*4] \end{verbatim} -for the cosine and the table +for the cosine (not presented here) and the table \begin{verbatim} - \tabulate[rspec={n,1,15},rround=0,rpos=2,rules=Tth, - cspec={m,1,5},ctitle=*,chstyle=2] + \tabulate[rspec={n,1,15,0},rpos=2,rules=Tth, + cspec={m,1,5},chstyle=2,ctitle=*] { \sin(m\pi/n) }[n=4,m=2][*4] \end{verbatim} -for the sine and checking the entries shows that indeed the maximum -and minimum values are $0.9397$ and $-1.0000$ respectively. +for the sine (also not presented here) and checking the entries shows +that indeed the maximum and minimum values are $0.9397$ and $-1.0000$ +respectively. If the \verb`Q?` setting is not satisfied by any function value a familiar error message is shown \textendash{} with a tweak: @@ -2237,10 +2734,10 @@ $\Longrightarrow$ \eval{$ \noindent Here, the \verb`(2)` tells us that the message refers to a command at the second level, a \emph{nested} command. -Perhaps a more likely situation is to want to nest other commands -within a \verb`\tabulate` command. I give an example in the documentation -to the associated package \verb`numerica-plus` around the timing -of signals between points fixed on a rotating disk. +A more likely situation is to want to nest other commands within a +\verb`\tabulate` command. I give an example in the documentation +to the associated package \verb`numerica-plus` about timing of signals +between points fixed on a rotating disk. \subsection{Saving tables to file} @@ -2257,19 +2754,15 @@ In the following example, a table is created and then saved to file and to the macro \verb`\mytable` by the subsequent \verb`\reuse` command: \begin{verbatim} - \tabulate - [rspec={x,0.25,5},rround=2,rhead=x, - ralign=r,rhnudge=9, - cspec={k,0.25,3},chstyle=2, - chround=2,calign=r,ctitle=**, - rules=TthB] - { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] - \reuse{mytable} + \tabulate[rspec={x,0.25,5,2},rhnudge=9,rules=TthB, + cspec={k,0.25,3,2},chstyle=2,ctitle=**] + { a\sin kx }[a=2/\pi,{k}=3,{x}=0.25][*] + \reuse[renew]{mytable} \end{verbatim} $\Longrightarrow$ \tabulate - [rspec={x,0.25,(5)},rround=2, + [rspec={x,0.25,5},rround=2, rhead=x,ralign=r,rhnudge=9, - cspec={k,0.25,(3)},chstyle=2, + cspec={k,0.25,3},chstyle=2, chround=2,calign=r,ctitle=**,rules=TthB] { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] \reuse[renew]{mytable}\medskip{} @@ -2278,37 +2771,29 @@ $\Longrightarrow$ \tabulate \noindent\verb`\mytable` $\Longrightarrow$ \mytable \medskip{} -\noindent Certainly \verb`\mytable` contains the table. - -If we use the \verb`view` setting with \verb`\reuse` we can see -that \verb`\mytable` and its contents have also been saved to file: -\begin{verbatim} - \reuse[view]{} -\end{verbatim} -$\Longrightarrow$ \reuse[view]{} - -The file that \verb`\mytable` is saved to is the \texttt{.nmc} file +\noindent Yes, \verb`\mytable` contains the table. The macro and +its contents have also been saved to file, the \texttt{.nmc} file of the current document, hence \texttt{numerica-tables.nmc} in the present instance. The contents of this file can be edited in a text -editor, or some limited file operations can be effected with the \verb`\reuse` +editor, or some limited file operations are possible with the \verb`\reuse` command. These have been described in the associated document \texttt{numerica.pdf}. \subsection{Viewing the \protect\LaTeX{} form} -In\index{tables!viewing LaTeX form@viewing \LaTeX{} form} previous -versions of \texttt{numerica-tables} the \verb`dbg` and \verb`view`\index{view@\texttt{view}} -settings were disabled. In version 3, they have been enabled to the -extent that the \LaTeX{} form of a table can be viewed by entering -either \verb`dbg=11` or, less nerdishly, \verb`view` into the settings -option of \verb`\nmcTabulate`. In the example I first create the -table and then use the \verb`view` setting: +\label{subsec:Viewing-latex-form}In\index{tables!viewing LaTeX form@viewing \LaTeX{} form} +previous versions of \texttt{numerica-tables} the \verb`dbg` and +\verb`view`\index{view@\texttt{view}} settings were disabled. In +version 3, they have been enabled to the extent that the \LaTeX{} form +of a table can be viewed by entering either \verb`dbg=11` or, less +nerdishly, \verb`view` into the settings option of \verb`\nmcTabulate`. +In the example a familiar table is specified and built in \LaTeX{} +and its form displayed with the \verb`view` setting: \begin{verbatim} - - \tabulate[view,rvar=x,rstep=0.2,rstop=1] - { \sin x }[x=0] + \tabulate[view,rspec={x,0.2,5}] + { \sin x }[x=0.2] \end{verbatim} -$\Longrightarrow$ \tabulate [view,rvar=x,rstep=0.2,rstop=1] - { \sin x }[x=0] +$\Longrightarrow$ \tabulate[view,rspec={x,0.2,5}] + { \sin x }[x=0.2] \chapter{Reference summary} @@ -2316,134 +2801,166 @@ $\Longrightarrow$ \tabulate [view,rvar=x,rstep=0.2,rstop=1] \texttt{\textbackslash nmcTabulate, \textbackslash tabulate} +\subsubsection*{Package options} +\begin{itemize} +\item default rule configuration: \verb`rules`, \verb`norules`; see §§\ref{sec:Package-options}, +\ref{subsec:Rules:-rules-setting}. +\item default query for \verb`\tabulate*`: \verb`Q?*`; see §\ref{sec:Star-option}. +\end{itemize} + \section{Settings for \texttt{\textbackslash nmcTabulate}} -\subsubsection*{Row-variable specification: uniform case §\ref{subsec:Row-var-spec-uniform}} +\subsubsection*{Row variable specification: uniform case } + +See §\ref{subsec:Row-var-spec-uniform}. \begin{center} \begin{center} -\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{3cm}} +\begin{tabular}{l>{\raggedright}p{2cm}>{\raggedright}p{4cm}>{\raggedright}p{3cm}} \toprule -{\small key} & {\small type} & {\small meaning} & {\small comment}\tabularnewline +{\small key} & {\small type} & {\small meaning } & {\small comment}\tabularnewline \midrule {\small\texttt{rvar}} & {\small token(s)} & {\small row variable} & \tabularnewline -{\small\texttt{rstep}} & {\small real num} & {\small step size} & \tabularnewline -{\small\texttt{rstop}} & {\small real num} & {\small stop value} & \multirow{2}{3cm}{either {\small\texttt{rstop}}{\small{} or }{\small\texttt{rows}}{\small , -not both}}\tabularnewline -{\small\texttt{rows}} & {\small int} & {\small number of rows} & \tabularnewline -{\small\texttt{rspec}} & {\small comma list} & {\small\texttt{\{start}}{\small , }{\small\texttt{step}}{\small , }{\small\texttt{rows\}}} & {\small short form spec.}\tabularnewline +{\small\texttt{rstep}} & {\small real num.} & {\small step size} & \tabularnewline +{\small\texttt{rstop}} & {\small real num.} & {\small stop value} & {\small excludes }{\small\texttt{rows}}\tabularnewline +{\small\texttt{rows}} & {\small int} & {\small number of rows} & {\small excludes }{\small\texttt{rstop}}\tabularnewline +{\small\texttt{rspec}} & {\small comma list} & {\small\texttt{\{rvar}}{\small , }{\small\texttt{rstep}}{\small , }{\small\texttt{rows\}}} & {\small short form spec.}\tabularnewline +{\small\texttt{rround}} & {\small int} & {\small rounding} & {\small default: }{\small\texttt{1}}\tabularnewline \bottomrule \end{tabular} -\par\end{center} +\par\end{center}\pagebreak{} \par\end{center} -\subsubsection*{Row-variable specification: non-uniform case §\ref{subsec:Row-var-spec-nonuniform}} -\begin{center} +\subsubsection*{Row variable specification: non-uniform case} + +See §\ref{subsec:Row-var-spec-nonuniform}. + \begin{center} -\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{3cm}} +\begin{tabular}{l>{\raggedright}p{2cm}>{\raggedright}p{4cm}>{\raggedright}p{3cm}} \toprule {\small key} & {\small type} & {\small meaning} & {\small comment}\tabularnewline \midrule -{\small\texttt{rdata}} & {\small comma list} & {\small list of row-var. values} & {\small may be stored in a macro}\tabularnewline -{\small\texttt{rfile}} & {\small chars} & {\small file of row-var. values} & {\small file path/name}\tabularnewline -{\small\texttt{rverb}} & {\small int (}{\small\texttt{0}}{\small /}{\small\texttt{1}}{\small )} & {\small display }{\small\texttt{rdata}}{\small{} or }{\small\texttt{rfile}}{\small{} -values verbatim} & {\small default }{\small\texttt{0}}\tabularnewline -{\small\texttt{rfunc}} & {\small token(s)} & {\small step function specifying row-var. values} & \tabularnewline +{\small\texttt{rfunc}} & {\small token(s)} & {\small formula for row var. values} & \tabularnewline +{\small\texttt{rdata}} & {\small comma list or macro} & {\small list or macro (containing list) of row var. values} & \tabularnewline +{\small\texttt{rfile}} & {\small chars} & {\small filepath/filename} & {\small file contains list of row var. values}\tabularnewline +{\small\texttt{rdelim}} & {\small char} & {\small item separator for }{\small\texttt{rdata}}{\small{} or }{\small\texttt{rfile}}{\small{} +lists} & {\small defaults to }{\small\texttt{,}}{\small{} or }{\small\texttt{;}}{\small{} +depending as }{\small\texttt{.}}{\small{} or }{\small\texttt{,}}{\small{} +is decimal mark}\tabularnewline +{\small\texttt{rverb}} & {\small fp (}{\small\texttt{0/0.5/1}}{\small )} & {\small display }{\small\texttt{rdata}}{\small , }{\small\texttt{rfile}}{\small{} +values verbatim (}{\small\texttt{1}}{\small ), or slash fractions formatted +(}{\small\texttt{0.5}}{\small )} & {\small initialized to }{\small\texttt{0}}\tabularnewline \bottomrule \end{tabular} \par\end{center} -\par\end{center} -\subsubsection*{Row-variable column formatting §\ref{subsec:Row-var-col-formatting}} +\subsubsection*{Row variable column formatting} + +See §\ref{subsec:Row-var-col-formatting}. \begin{center} \begin{center} -\begin{tabular}{ll>{\raggedright}p{4cm}l} +\begin{tabular}{l>{\raggedright}p{2cm}>{\raggedright}p{4cm}>{\raggedright}p{2cm}} \toprule {\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline \midrule -{\small\texttt{rround}} & {\small int} & {\small rounding} & {\small\texttt{1}}\tabularnewline +{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small column placement} & {\small\texttt{1}}\tabularnewline {\small\texttt{ralign}} & {\small char (}{\small\texttt{r/c/l}}{\small )} & {\small horizontal alignment} & {\small\texttt{r}}\tabularnewline {\small\texttt{rfont}} & {\small chars} & {\small font (}{\small\verb`\math<chars>`}{\small )} & \tabularnewline -{\small\texttt{rhead}} & {\small tokens} & {\small header} & {\small\texttt{rvar}}\tabularnewline -{\small\texttt{rhnudge}} & int & {\small nudge header }{\small{\small\verb`rhnudge`}}{\small{} -mu} & {\small\texttt{0}}\tabularnewline -{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & & {\small\texttt{1}}\tabularnewline -{\small\texttt{rvar'}} & {\small token(s)} & {\small 2nd row variable col. spec.} & {\small\texttt{rvar}}\tabularnewline -{\small\texttt{rhead'}} & {\small token(s)} & {\small header of 2nd r-v col. (if it exists)} & {\small\texttt{rvar'}}\tabularnewline -{\small\texttt{rhnudge'}} & int & {\small nudge 2nd r-v col. header }{\small{\small\verb`rhnudge'`}}{\small{} +{\small\texttt{rhead}} & {\small tokens} & {\small header} & \tabularnewline +{\small\texttt{rhnudge}} & {\small fp} & {\small nudge header }{\small{\small\verb`<fp>`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline +{\small\texttt{rhsize}} & {\small int (}{\small\texttt{-4}}{\small\ldots}{\small\texttt{5}}{\small )} & {\small font size relative to }{\small\verb`0=\normalsize`} & {\small\texttt{0}}\tabularnewline +{\small\texttt{rmath}} & {\small char (}{\small\texttt{s/t/d}}{\small )} & {\small script-, text-, displaystyle} & {\small\texttt{t}}\tabularnewline +{\small\texttt{rvar'}} & {\small tokens} & {\small 2nd row variable col. spec.} & \tabularnewline +{\small\texttt{rhead'}} & {\small tokens} & {\small header of 2nd row var. col. (if it exists)} & \tabularnewline +{\small\texttt{rhnudge'}} & {\small fp} & {\small nudge 2nd row var. col. header }{\small{\small\verb`<fp>`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline -{\small\texttt{rfrac}} & {\small int (}{\small\texttt{0}}{\small\ldots 5)} & {\small fraction form} & {\small\texttt{0}}\tabularnewline \bottomrule \end{tabular} +\par\end{center}\pagebreak{} \par\end{center} -\par\end{center} -\subsubsection*{Column-variable specification §\ref{sec:Column-variable-settings}.} +\subsubsection*{Column-variable specification } + +See §\ref{sec:Column-variable-settings}. \begin{center} \begin{center} -\begin{tabular}{ll>{\raggedright}p{4cm}l} +\begin{tabular}{ll>{\raggedright}p{3cm}l} \toprule -{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline +{\small key} & {\small type} & {\small meaning} & {\small comment}\tabularnewline \midrule {\small\texttt{cvar}} & {\small token(s)} & {\small column variable} & \tabularnewline -{\small\texttt{cstep}} & {\small real num} & {\small step size} & \tabularnewline -{\small\texttt{cstop}} & {\small real num} & {\small stop value} & {\small either }{\small\texttt{cstop}}\tabularnewline -{\small\texttt{cols}} & {\small int} & {\small number of columns} & {\small or }{\small\texttt{cols}}{\small , not both}\tabularnewline +{\small\texttt{cstep}} & {\small real num.} & {\small step size} & \tabularnewline +{\small\texttt{cstop}} & {\small real num.} & {\small stop value} & {\small either }{\small\texttt{cstop}}\tabularnewline +{\small\texttt{cols}} & {\small int} & {\small number of columns} & {\small or }{\small\texttt{cols}}\tabularnewline {\small\texttt{cspec}} & {\small comma list} & {\small\texttt{\{cvar,cstep}}{\small ,}{\small\texttt{cols\}}} & {\small short form spec.}\tabularnewline +{\small\texttt{chround}} & {\small int} & {\small col. var. rounding} & {\small default: }{\small\texttt{0}}\tabularnewline \bottomrule \end{tabular} \par\end{center} \par\end{center} -\subsubsection*{Column-variable header formatting §\ref{subsec:Column-header-formatting}.} +\subsubsection*{Column-variable header formatting} + +See §\ref{subsec:Column-header-formatting}. \begin{center} \begin{center} +{\small{}% \begin{tabular}{ll>{\raggedright}p{4cm}l} \toprule {\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline \midrule {\small\texttt{chstyle}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small header style} & {\small\texttt{0}}\tabularnewline -{\small\texttt{ctitle}} & {\small token(s)} & {\small single col. alternative header} & \tabularnewline -{\small\texttt{chead}} & {\small token(s)} & {\small user-defined header} & \tabularnewline -{\small\texttt{calign}} & {\small char (r/c/l)} & {\small column alignment} & {\small\texttt{r}}\tabularnewline -{\small\texttt{chnudge}} & {\small int} & {\small nudge header }{\small{\small\verb`chnudge`}}{\small{} +{\small\texttt{chead}} & {\small token(s)} & {\small user-defined col. var. header} & \tabularnewline +{\small\texttt{calign}} & {\small char (}{\small\texttt{r}}{\small /}{\small\texttt{c}}{\small /}{\small\texttt{l}}{\small )} & {\small column alignment} & {\small\texttt{r}}\tabularnewline +{\small\texttt{chnudge}} & {\small fp} & {\small nudge header }{\small{\small\verb`chnudge`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline -{\small\texttt{chround}} & {\small int} & {\small{} rounding} & {\small\texttt{0}}\tabularnewline +{\small\texttt{chfont}} & {\small chars} & {\small font (}{\small\verb`\math<chars>`}{\small )} & \tabularnewline +{\small\texttt{chmath}} & {\small char (}{\small\texttt{s}}{\small /}{\small\texttt{t}}{\small /}{\small\texttt{d}}{\small )} & {\small script-, text- or displaystyle} & {\small\texttt{t}}\tabularnewline +{\small\texttt{chsize}} & {\small int (}{\small\texttt{-4}}{\small\ldots}{\small\texttt{5}}{\small )} & {\small font size relative to }{\small\verb`0 => \normalsize`} & {\small\texttt{0}}\tabularnewline \bottomrule -\end{tabular} -\par\end{center}\newpage{} +\end{tabular}} +\par\end{center} \par\end{center} -\subsubsection*{Function-value formatting §\ref{sec:Function-value-formatting}.} +\subsubsection*{Function-value formatting} + +See §\ref{sec:Function-value-formatting}. \begin{center} \begin{center} -\begin{tabular}{cc>{\raggedright}p{4cm}c} +\begin{tabular}{cc>{\raggedright}p{5cm}c} \toprule -{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline +{\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline \midrule {\small\texttt{(pad)}} & {\small int} & {\small t-notation phantom padding} & \tabularnewline -{\small\texttt{signs}} & {\small int} & {\small sign handling for function-values} & {\small\texttt{0}}\tabularnewline -{\small\texttt{diffs}} & {\small int} & {\small insert differences \& pre-pad with zeros} & {\small\texttt{0}}\tabularnewline +{\small\texttt{signs}} & {\small int} & {\small sign handling for function values} & {\small\texttt{0}}\tabularnewline +{\small\texttt{diffs}} & {\small int} & {\small insert differences, pre-pad with $0$s} & {\small\texttt{0}}\tabularnewline +{\small\texttt{round}} & {\small tokens} & {\small row/col. dependent rounding value} & \tabularnewline {\small\texttt{Q?}} & {\small tokens} & {\small special cell conditional} & \tabularnewline -{\small\texttt{A!}} & {\small token(s)} & {\small special cell formatting} & \tabularnewline +{\small\texttt{A!}} & {\small tokens} & {\small special cell formatting} & \tabularnewline \bottomrule \end{tabular} +\par\end{center}\pagebreak{} \par\end{center} -\par\end{center} -\subsubsection*{Whole-of-table formatting §\ref{sec:Whole-of-table-formatting}.} +\subsubsection*{Whole-of-table formatting } + +See §\ref{sec:Whole-of-table-formatting}. \begin{center} \begin{center} \begin{tabular}{ll>{\raggedright}p{4cm}l} \toprule -{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline +{\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline \midrule -{\small\texttt{ctitle}} & {\small token(s)} & {\small collective title for function-value columns} & \tabularnewline - & {\small token(s)} & {\small inter-header/title row for} {\small function-value columns} & \tabularnewline -{\small\texttt{header}} & {\small int (}{\small\texttt{0/1}}{\small )} & {\small suppress/show header row} & {\small\texttt{1}}\tabularnewline -{\small\texttt{rules}} & {\small char(s)} & {\small horizontal rules template} & {\small\texttt{ThB}}\tabularnewline -{\small\texttt{foot}} & {\small token(s)} & {\small content of footer line} & \tabularnewline -{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small row variable } & {\small\texttt{1}}\tabularnewline -{\small\texttt{rbloc}} & {\small comma list} & {\small division of rows into blocks} & \tabularnewline +{\small\texttt{ctitle}} & {\small token(s)} & {\small collective title for columns} & \tabularnewline +{\small\texttt{csubttl}} & {\small token(s)} & {\small subtitle row for} {\small function- value cols} & \tabularnewline +{\small\texttt{calign}} & {\small char(}{\small\texttt{r}}{\small /}{\small\texttt{c}}{\small /}{\small\texttt{l}}{\small )} & {\small column alignment} & {\small\texttt{r}}\tabularnewline +{\small\texttt{headless}} & & {\small suppress header row} & \tabularnewline +{\small\texttt{foot}} & {\small token(s)} & {\small table-wide footer row} & \tabularnewline +{\small\texttt{rules}} & {\small char(s)} & {\small horizontal rule spec.} & {\small\texttt{ThB}}\tabularnewline +{\small\texttt{norules}} & & {\small cancel all rules} & \tabularnewline +{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small row variable col. position(s)} & {\small\texttt{1}}\tabularnewline +{\small\texttt{rbloc}} & {\small integer comma list} & {\small row block specification} & \tabularnewline +{\small\texttt{rblocsep}} & {\small length} & {\small extra space between row blocks} & {\small\texttt{1ex}}\tabularnewline +{\small\texttt{transpose}} & & {\small show funct. vals in rows} & \tabularnewline {\small\texttt{valign}} & {\small char (}{\small\texttt{t}}/{\small\texttt{m}}/{\small\texttt{b}}{\small )} & {\small vertical alignment of table relative to text baseline} & {\small\texttt{m}}\tabularnewline \bottomrule \end{tabular} @@ -2451,9 +2968,9 @@ mu} & {\small\texttt{0}}\tabularnewline \par\end{center} \subsubsection*{Miscellaneous settings} - -\verb`view`, equivalent to \verb`dbg=11`: show the \LaTeX{} expression -for the table. - +\begin{itemize} +\item \verb`view`, equivalent to \verb`dbg=11`: show the \LaTeX{} expression +for the table; see §\ref{subsec:Viewing-latex-form}. +\end{itemize} {\small\printindex}{\small\par} \end{document} |