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# math::calculus::symdiff(n) 1.0.1 tcllib "Symbolic differentiation for Tcl"

## Name

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

## Synopsis

- package require Tcl 8.5
- package require grammar::aycock 1.0
- package require math::calculus::symdiff 1.0.1

- math::calculus::symdiff::symdiff expression variable
- math::calculus::jacobian variableDict

## Description

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.

## Procedures

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.

## Expressions

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.

## Examples

    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}}

## Bugs, Ideas, Feedback

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.

## See Also

math::calculus, math::interpolate

## Copyright

Copyright © 2010 by Kevin B. Kenny <kennykb@acm.org>
Redistribution permitted under the terms of the Open Publication License <http://www.opencontent.org/openpub/>