/ 
Math-Symbolic-0.118
River stage two • 32 direct dependents • 34 total dependents
/ Math::Symbolic::VectorCalculus

NAME

Math::Symbolic::VectorCalculus - Symbolically comp. grad, Jacobi matrices etc.

SYNOPSIS

use Math::Symbolic qw/:all/;
use Math::Symbolic::VectorCalculus; # not loaded by Math::Symbolic

@gradient = grad 'x+y*z';
# or:
$function = parse_from_string('a*b^c');
@gradient = grad $function;
# or:
@signature = qw(x y z);
@gradient = grad 'a*x+b*y+c*z', @signature; # Gradient only for x, y, z
# or:
@gradient = grad $function, @signature;

# Similar syntax variations as with the gradient:
$divergence = div @functions;
$divergence = div @functions, @signature;

# Again, similar DWIM syntax variations as with grad:
@rotation = rot @functions;
@rotation = rot @functions, @signature;

# Signatures always inferred from the functions here:
@matrix = Jacobi @functions;
# $matrix is now array of array references. These hold
# Math::Symbolic trees.

# Similar to Jacobi:
@matrix = Hesse $function;

DESCRIPTION

This module provides several subroutines related to vector calculus such as computing gradients, divergence, rotation, and Jacobi matrices of Math::Symbolic trees.

Please note that the code herein may or may not be refactored into the OO-interface of the Math::Symbolic module.

EXPORT

None by default.

You may choose to have any of the following routines exported to the calling namespace. ':all' tag exports all of the following:

grad
div
rot
Jacobi

SUBROUTINES

grad

This subroutine computes the gradient of a Math::Symbolic tree representing a function.

The gradient of a function f(x1, x2, ..., xn) is defined as the vector:

( df(x1, x2, ..., xn) / d(x1),
  df(x1, x2, ..., xn) / d(x2),
  ...,
  df(x1, x2, ..., xn) / d(xn) )

(These are all partial derivatives.) Any good book on calculus will have more details on this.

grad uses prototypes to allow for a variety of usages. In its most basic form, it accepts only one argument which may either be a Math::Symbolic tree or a string both of which will be interpreted as the function to compute the gradient for. Optionally, you may specify a second argument which must be a (literal) array of Math::Symbolic::Variable objects or valid Math::Symbolic variable names (strings). These variables will the be used for the gradient instead of the x1, ..., xn inferred from the function signature.

div

This subroutine computes the divergence of a set of Math::Symbolic trees representing a vectorial function.

The divergence of a vectorial function F = (f1(x1, ..., xn), ..., fn(x1, ..., xn)) is defined like follows:

sum_from_i=1_to_n( dfi(x1, ..., xn) / dxi )

That is, the sum of all partial derivatives of the i-th component function to the i-th coordinate. See your favourite book on calculus for details. Obviously, it is important to keep in mind that the number of function components must be equal to the number of variables/coordinates.

Similar to grad, div uses prototypes to offer a comfortable interface. First argument must be a (literal) array of strings and Math::Symbolic trees which represent the vectorial function's components. If no second argument is passed, the variables used for computing the divergence will be inferred from the functions. That means the function signatures will be joined to form a signature for the vectorial function.

If the optional second argument is specified, it has to be a (literal) array of Math::Symbolic::Variable objects and valid variable names (strings). These will then be interpreted as the list of variables for computing the divergence.

rot

This subroutine computes the rotation of a set of three Math::Symbolic trees representing a vectorial function.

The rotation of a vectorial function F = (f1(x1, x2, x3), f2(x1, x2, x3), f3(x1, x2, x3)) is defined as the following vector:

( ( df3/dx2 - df2/dx3 ),
  ( df1/dx3 - df3/dx1 ),
  ( df2/dx1 - df1/dx2 ) )

Or "nabla x F" for short. Again, I have to refer to the literature for the details on what rotation is. Please note that there have to be exactly three function components and three coordinates because the cross product and hence rotation is only defined in three dimensions.

As with the previously introduced subroutines div and grad, rot offers a prototyped interface. First argument must be a (literal) array of strings and Math::Symbolic trees which represent the vectorial function's components. If no second argument is passed, the variables used for computing the rotation will be inferred from the functions. That means the function signatures will be joined to form a signature for the vectorial function.

If the optional second argument is specified, it has to be a (literal) array of Math::Symbolic::Variable objects and valid variable names (strings). These will then be interpreted as the list of variables for computing the rotation. (And please excuse my copying the last two paragraphs from above.)

Jacobi

Jacobi() returns the Jacobi matrix of a given vectorial function. It expects any number of arguments (strings and/or Math::Symbolic trees) which will be interpreted as the vectorial function's components. Variables used for computing the matrix are inferred from the combined signature of the components. There is currently no way to override this behaviour.

The Jacobi matrix is the vector of gradient vectors of the vectorial function's components.

Hesse

Hesse() returns the Hesse matrix of a given scalar function. First and only argument must be a string (to be parsed as a Math::Symbolic tree) or a Math::Symbolic tree. As with Jacobi(), Hesse() always infers the variables used for computing the matrix.

The Hesse matrix is the Jacobi matrix of the gradient of a scalar function.

AUTHOR

Please send feedback, bug reports, and support requests to the Math::Symbolic support mailing list: math-symbolic-support at lists dot sourceforge dot net. Please consider letting us know how you use Math::Symbolic. Thank you.

If you're interested in helping with the development or extending the module's functionality, please contact the developers' mailing list: math-symbolic-develop at lists dot sourceforge dot net.

List of contributors:

Steffen Müller, symbolic-module at steffen-mueller dot net
Stray Toaster, mwk at users dot sourceforge dot net

SEE ALSO

New versions of this module can be found on http://steffen-mueller.net or CPAN. The module development takes place on Sourceforge at http://sourceforge.net/projects/math-symbolic/

Math::Symbolic

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