'\" '\" Generated from file 'symdiff\&.man' by tcllib/doctools with format 'nroff' '\" Copyright (c) 2010 by Kevin B\&. Kenny '\" Redistribution permitted under the terms of the Open Publication License '\" .TH "math::calculus::symdiff" 3tcl 1\&.0\&.1 tcllib "Symbolic differentiation for Tcl" .\" The -*- nroff -*- definitions below are for supplemental macros used .\" in Tcl/Tk manual entries. .\" .\" .AP type name in/out ?indent? .\" Start paragraph describing an argument to a library procedure. .\" type is type of argument (int, etc.), in/out is either "in", "out", .\" or "in/out" to describe whether procedure reads or modifies arg, .\" and indent is equivalent to second arg of .IP (shouldn't ever be .\" needed; use .AS below instead) .\" .\" .AS ?type? ?name? .\" Give maximum sizes of arguments for setting tab stops. Type and .\" name are examples of largest possible arguments that will be passed .\" to .AP later. If args are omitted, default tab stops are used. .\" .\" .BS .\" Start box enclosure. From here until next .BE, everything will be .\" enclosed in one large box. .\" .\" .BE .\" End of box enclosure. .\" .\" .CS .\" Begin code excerpt. .\" .\" .CE .\" End code excerpt. .\" .\" .VS ?version? ?br? .\" Begin vertical sidebar, for use in marking newly-changed parts .\" of man pages. The first argument is ignored and used for recording .\" the version when the .VS was added, so that the sidebars can be .\" found and removed when they reach a certain age. If another argument .\" is present, then a line break is forced before starting the sidebar. .\" .\" .VE .\" End of vertical sidebar. .\" .\" .DS .\" Begin an indented unfilled display. .\" .\" .DE .\" End of indented unfilled display. .\" .\" .SO ?manpage? .\" Start of list of standard options for a Tk widget. The manpage .\" argument defines where to look up the standard options; if .\" omitted, defaults to "options". The options follow on successive .\" lines, in three columns separated by tabs. .\" .\" .SE .\" End of list of standard options for a Tk widget. .\" .\" .OP cmdName dbName dbClass .\" Start of description of a specific option. cmdName gives the .\" option's name as specified in the class command, dbName gives .\" the option's name in the option database, and dbClass gives .\" the option's class in the option database. .\" .\" .UL arg1 arg2 .\" Print arg1 underlined, then print arg2 normally. .\" .\" .QW arg1 ?arg2? .\" Print arg1 in quotes, then arg2 normally (for trailing punctuation). .\" .\" .PQ arg1 ?arg2? .\" Print an open parenthesis, arg1 in quotes, then arg2 normally .\" (for trailing punctuation) and then a closing parenthesis. .\" .\" # Set up traps and other miscellaneous stuff for Tcl/Tk man pages. .if t .wh -1.3i ^B .nr ^l \n(.l .ad b .\" # Start an argument description .de AP .ie !"\\$4"" .TP \\$4 .el \{\ . ie !"\\$2"" .TP \\n()Cu . el .TP 15 .\} .ta \\n()Au \\n()Bu .ie !"\\$3"" \{\ \&\\$1 \\fI\\$2\\fP (\\$3) .\".b .\} .el \{\ .br .ie !"\\$2"" \{\ \&\\$1 \\fI\\$2\\fP .\} .el \{\ \&\\fI\\$1\\fP .\} .\} .. .\" # define tabbing values for .AP .de AS .nr )A 10n .if !"\\$1"" .nr )A \\w'\\$1'u+3n .nr )B \\n()Au+15n .\" .if !"\\$2"" .nr )B \\w'\\$2'u+\\n()Au+3n .nr )C \\n()Bu+\\w'(in/out)'u+2n .. .AS Tcl_Interp Tcl_CreateInterp in/out .\" # BS - start boxed text .\" # ^y = starting y location .\" # ^b = 1 .de BS .br .mk ^y .nr ^b 1u .if n .nf .if n .ti 0 .if n \l'\\n(.lu\(ul' .if n .fi .. .\" # BE - end boxed text (draw box now) .de BE .nf .ti 0 .mk ^t .ie n \l'\\n(^lu\(ul' .el \{\ .\" Draw four-sided box normally, but don't draw top of .\" box if the box started on an earlier page. .ie !\\n(^b-1 \{\ \h'-1.5n'\L'|\\n(^yu-1v'\l'\\n(^lu+3n\(ul'\L'\\n(^tu+1v-\\n(^yu'\l'|0u-1.5n\(ul' .\} .el \}\ \h'-1.5n'\L'|\\n(^yu-1v'\h'\\n(^lu+3n'\L'\\n(^tu+1v-\\n(^yu'\l'|0u-1.5n\(ul' .\} .\} .fi .br .nr ^b 0 .. .\" # VS - start vertical sidebar .\" # ^Y = starting y location .\" # ^v = 1 (for troff; for nroff this doesn't matter) .de VS .if !"\\$2"" .br .mk ^Y .ie n 'mc \s12\(br\s0 .el .nr ^v 1u .. .\" # VE - end of vertical sidebar .de VE .ie n 'mc .el \{\ .ev 2 .nf .ti 0 .mk ^t \h'|\\n(^lu+3n'\L'|\\n(^Yu-1v\(bv'\v'\\n(^tu+1v-\\n(^Yu'\h'-|\\n(^lu+3n' .sp -1 .fi .ev .\} .nr ^v 0 .. .\" # Special macro to handle page bottom: finish off current .\" # box/sidebar if in box/sidebar mode, then invoked standard .\" # page bottom macro. .de ^B .ev 2 'ti 0 'nf .mk ^t .if \\n(^b \{\ .\" Draw three-sided box if this is the box's first page, .\" draw two sides but no top otherwise. .ie !\\n(^b-1 \h'-1.5n'\L'|\\n(^yu-1v'\l'\\n(^lu+3n\(ul'\L'\\n(^tu+1v-\\n(^yu'\h'|0u'\c .el \h'-1.5n'\L'|\\n(^yu-1v'\h'\\n(^lu+3n'\L'\\n(^tu+1v-\\n(^yu'\h'|0u'\c .\} .if \\n(^v \{\ .nr ^x \\n(^tu+1v-\\n(^Yu \kx\h'-\\nxu'\h'|\\n(^lu+3n'\ky\L'-\\n(^xu'\v'\\n(^xu'\h'|0u'\c .\} .bp 'fi .ev .if \\n(^b \{\ .mk ^y .nr ^b 2 .\} .if \\n(^v \{\ .mk ^Y .\} .. .\" # DS - begin display .de DS .RS .nf .sp .. .\" # DE - end display .de DE .fi .RE .sp .. .\" # SO - start of list of standard options .de SO 'ie '\\$1'' .ds So \\fBoptions\\fR 'el .ds So \\fB\\$1\\fR .SH "STANDARD OPTIONS" .LP .nf .ta 5.5c 11c .ft B .. .\" # SE - end of list of standard options .de SE .fi .ft R .LP See the \\*(So manual entry for details on the standard options. .. .\" # OP - start of full description for a single option .de OP .LP .nf .ta 4c Command-Line Name: \\fB\\$1\\fR Database Name: \\fB\\$2\\fR Database Class: \\fB\\$3\\fR .fi .IP .. .\" # CS - begin code excerpt .de CS .RS .nf .ta .25i .5i .75i 1i .. .\" # CE - end code excerpt .de CE .fi .RE .. .\" # UL - underline word .de UL \\$1\l'|0\(ul'\\$2 .. .\" # QW - apply quotation marks to word .de QW .ie '\\*(lq'"' ``\\$1''\\$2 .\"" fix emacs highlighting .el \\*(lq\\$1\\*(rq\\$2 .. .\" # PQ - apply parens and quotation marks to word .de PQ .ie '\\*(lq'"' (``\\$1''\\$2)\\$3 .\"" fix emacs highlighting .el (\\*(lq\\$1\\*(rq\\$2)\\$3 .. .\" # QR - quoted range .de QR .ie '\\*(lq'"' ``\\$1''\\-``\\$2''\\$3 .\"" fix emacs highlighting .el \\*(lq\\$1\\*(rq\\-\\*(lq\\$2\\*(rq\\$3 .. .\" # MT - "empty" string .de MT .QW "" .. .BS .SH NAME math::calculus::symdiff \- Symbolic differentiation for Tcl .SH SYNOPSIS package require \fBTcl 8\&.5\fR .sp package require \fBgrammar::aycock 1\&.0\fR .sp package require \fBmath::calculus::symdiff 1\&.0\&.1\fR .sp \fBmath::calculus::symdiff::symdiff\fR \fIexpression\fR \fIvariable\fR .sp \fBmath::calculus::jacobian\fR \fIvariableDict\fR .sp .BE .SH DESCRIPTION .PP The \fBmath::calculus::symdiff\fR package provides a symbolic differentiation facility for Tcl math expressions\&. It is useful for providing derivatives to packages that either require the Jacobian of a set of functions or else are more efficient or stable when the Jacobian is provided\&. .SH PROCEDURES The \fBmath::calculus::symdiff\fR package exports the two procedures: .TP \fBmath::calculus::symdiff::symdiff\fR \fIexpression\fR \fIvariable\fR Differentiates the given \fIexpression\fR with respect to the specified \fIvariable\fR\&. (See \fBExpressions\fR below for a discussion of the subset of Tcl math expressions that are acceptable to \fBmath::calculus::symdiff\fR\&.) The result is a Tcl expression that evaluates the derivative\&. Returns an error if \fIexpression\fR is not a well-formed expression or is not differentiable\&. .TP \fBmath::calculus::jacobian\fR \fIvariableDict\fR Computes the Jacobian of a system of equations\&. The system is given by the dictionary \fIvariableDict\fR, whose keys are the names of variables in the system, and whose values are Tcl expressions giving the values of those variables\&. (See \fBExpressions\fR below for a discussion of the subset of Tcl math expressions that are acceptable to \fBmath::calculus::symdiff\fR\&. The result is a list of lists: the i'th element of the j'th sublist is the partial derivative of the i'th variable with respect to the j'th variable\&. Returns an error if any of the expressions cannot be differentiated, or if \fIvariableDict\fR is not a well-formed dictionary\&. .PP .SH EXPRESSIONS The \fBmath::calculus::symdiff\fR package accepts only a small subset of the expressions that are acceptable to Tcl commands such as \fBexpr\fR or \fBif\fR\&. Specifically, the only constructs accepted are: .IP \(bu Floating-point constants such as \fB5\fR or \fB3\&.14159e+00\fR\&. .IP \(bu References to Tcl variable using $-substitution\&. The variable names must consist of alphanumerics and underscores: the \fB${\&.\&.\&.}\fR notation is not accepted\&. .IP \(bu Parentheses\&. .IP \(bu The \fB+\fR, \fB-\fR, \fB*\fR, \fB/\fR\&. and \fB**\fR operators\&. .IP \(bu Calls to the functions \fBacos\fR, \fBasin\fR, \fBatan\fR, \fBatan2\fR, \fBcos\fR, \fBcosh\fR, \fBexp\fR, \fBhypot\fR, \fBlog\fR, \fBlog10\fR, \fBpow\fR, \fBsin\fR, \fBsinh\fR\&. \fBsqrt\fR, \fBtan\fR, and \fBtanh\fR\&. .PP Command substitution, backslash substitution, and argument expansion are not accepted\&. .SH EXAMPLES .CS math::calculus::symdiff::symdiff {($a*$x+$b)*($c*$x+$d)} x ==> (($c * (($a * $x) + $b)) + ($a * (($c * $x) + $d))) math::calculus::symdiff::jacobian {x {$a * $x + $b * $y} y {$c * $x + $d * $y}} ==> {{$a} {$b}} {{$c} {$d}} .CE .SH "BUGS, IDEAS, FEEDBACK" This document, and the package it describes, will undoubtedly contain bugs and other problems\&. Please report such in the category \fImath :: calculus\fR of the \fITcllib Trackers\fR [http://core\&.tcl\&.tk/tcllib/reportlist]\&. Please also report any ideas for enhancements you may have for either package and/or documentation\&. .PP When proposing code changes, please provide \fIunified diffs\fR, i\&.e the output of \fBdiff -u\fR\&. .PP Note further that \fIattachments\fR are strongly preferred over inlined patches\&. Attachments can be made by going to the \fBEdit\fR form of the ticket immediately after its creation, and then using the left-most button in the secondary navigation bar\&. .SH "SEE ALSO" math::calculus, math::interpolate .SH COPYRIGHT .nf Copyright (c) 2010 by Kevin B\&. Kenny Redistribution permitted under the terms of the Open Publication License .fi