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math::calculus::symdiff - Symbolic differentiation for Tcl

- package require
**Tcl 8.5** - package require
**grammar::aycock 1.0** - package require
**math::calculus::symdiff 1.0**

The **math::calculus::symdiff** 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.

The **math::calculus::symdiff** package exports the two procedures:

**math::calculus::symdiff::symdiff***expression**variable*Differentiates the given

*expression*with respect to the specified*variable*. (See Expressions below for a discussion of the subset of Tcl math expressions that are acceptable to**math::calculus::symdiff**.) The result is a Tcl expression that evaluates the derivative. Returns an error if*expression*is not a well-formed expression or is not differentiable.**math::calculus::jacobian***variableDict*Computes the Jacobian of a system of equations. The system is given by the dictionary

*variableDict*, whose keys are the names of variables in the system, and whose values are Tcl expressions giving the values of those variables. (See Expressions below for a discussion of the subset of Tcl math expressions that are acceptable to**math::calculus::symdiff**. 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*variableDict*is not a well-formed dictionary.

The **math::calculus::symdiff** package accepts only a small subset of the expressions
that are acceptable to Tcl commands such as **expr** or **if**.
Specifically, the only constructs accepted are:

Floating-point constants such as

**5**or**3.14159e+00**.References to Tcl variable using $-substitution. The variable names must consist of alphanumerics and underscores: the

**${...}**notation is not accepted.Parentheses.

The

**+**,**-**,*****,**/**. and******operators.Calls to the functions

**acos**,**asin**,**atan**,**atan2**,**cos**,**cosh**,**exp**,**hypot**,**log**,**log10**,**pow**,**sin**,**sinh**.**sqrt**,**tan**, and**tanh**.

Command substitution, backslash substitution, and argument expansion are not accepted.

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}}

This document, and the package it describes, will undoubtedly contain
bugs and other problems.
Please report such in the category *math :: calculus* of the
Tcllib Trackers.
Please also report any ideas for enhancements you may have for either
package and/or documentation.

Copyright © 2010 by Kevin B. Kenny <kennykb@acm.org>

Redistribution permitted under the terms of the Open Publication License <http://www.opencontent.org/openpub/>