[manpage_begin math::calculus::symdiff n 1.0]
[see_also math::calculus]
[see_also math::interpolate]
[copyright "2010 by Kevin B. Kenny <kennykb@acm.org>
Redistribution permitted under the terms of the Open\
Publication License <http://www.opencontent.org/openpub/>"]
[moddesc "Symbolic differentiation for Tcl"]
[titledesc "Symbolic differentiation for Tcl"]
[require Tcl 8.5]
[require grammar::aycock 1.0]
[require math::calculus::symdiff 1.0]
[description]
[para]
The [cmd 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.
[section "Procedures"]
The [cmd math::calculus::symdiff] package exports the two procedures:
[list_begin definitions]
[call [cmd math::calculus::symdiff::symdiff] [arg expression] [arg variable]]
Differentiates the given [arg expression] with respect to the specified
[arg variable]. (See [sectref "Expressions"] below for a discussion of the
subset of Tcl math expressions that are acceptable to
[cmd math::calculus::symdiff].)
The result is a Tcl expression that evaluates the derivative. Returns an
error if [arg expression] is not a well-formed expression or is not
differentiable.
[call [cmd math::calculus::jacobian] [arg variableDict]]
Computes the Jacobian of a system of equations.
The system is given by the dictionary [arg variableDict], whose keys
are the names of variables in the system, and whose values are Tcl expressions
giving the values of those variables. (See [sectref "Expressions"] below
for a discussion of the subset of Tcl math expressions that are acceptable
to [cmd 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 [arg variableDict]
is not a well-formed dictionary.
[list_end]
[section "Expressions"]
The [cmd math::calculus::symdiff] package accepts only a small subset of the expressions
that are acceptable to Tcl commands such as [cmd expr] or [cmd if].
Specifically, the only constructs accepted are:
[list_begin itemized]
[item]Floating-point constants such as [const 5] or [const 3.14159e+00].
[item]References to Tcl variable using $-substitution. The variable names
must consist of alphanumerics and underscores: the [const \$\{...\}] notation
is not accepted.
[item]Parentheses.
[item]The [const +], [const -], [const *], [const /]. and [const **]
operators.
[item]Calls to the functions [cmd acos], [cmd asin], [cmd atan],
[cmd atan2], [cmd cos], [cmd cosh], [cmd exp], [cmd hypot], [cmd log],
[cmd log10], [cmd pow], [cmd sin], [cmd sinh]. [cmd sqrt], [cmd tan],
and [cmd tanh].
[list_end]
Command substitution, backslash substitution, and argument expansion are
not accepted.
[section "Examples"]
[example {
math::calculus::symdiff::symdiff {($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}}
}]

[vset CATEGORY {math :: calculus}]
[include ../doctools2base/include/feedback.inc]
[manpage_end]
