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# NAME

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

# Table Of Contents

# 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.

When proposing code changes, please provide *unified diffs*, i.e the output of
**diff -u**.

Note further that *attachments* are strongly preferred over inlined patches.
Attachments can be made by going to the **Edit** form of the ticket
immediately after its creation, and then using the left-most button in the
secondary navigation bar.

# SEE ALSO

math::calculus, math::interpolate

# COPYRIGHT

Copyright © 2010 by Kevin B. Kenny