# FreeBSD Manual Pages

```Math::Symbolic::MiscAlUseraContributed Perl DocuMath::Symbolic::MiscAlgebra(3)

NAME
Math::Symbolic::MiscAlgebra - Miscellaneous algebra routines like det()

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

@matrix = (['x*y', 'z*x', 'y*z'],['x',	'z', 'z'],['x',	'x', 'y']);
\$det =	det @matrix;

@vector = ('x', 'y', 'z');
\$solution = solve_linear(\@matrix, \@vector);

DESCRIPTION
This module provides several subroutines	related	to algebra such	as
computing the determinant of quadratic matrices,	solving	linear
equation	systems	and computation	of Bell	Polynomials.

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

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:

det
linear_solve
bell_polynomial

SUBROUTINES
det
det() computes the determinant of a matrix of Math::Symbolic trees (or
strings that can	be parsed as such). First argument must	be a literal
array: "det @matrix", where @matrix is an n x n matrix.

Please note that	calculating determinants of matrices using the
straightforward Laplace algorithm is a slow (O(n!))  operation. This
implementation cannot make use of the various optimizations resulting
from the	determinant properties since we	are dealing with symbolic
matrix elements.	If you have a matrix of	reals, it is strongly
suggested that you use Math::MatrixReal or Math::Pari to	get the
determinant which can be	calculated using LR decomposition much faster.

On a related note: Calculating the determinant of a 20x20 matrix	would
take over 77146 years if	your Perl could	do 1 million calculations per
second.	Given that we're talking about several method calls per
calculation, that's much	more than todays computers could do. On	the
other hand, if you'd be using this straightforward algorithm with
numbers only and	in C, you might	be done	in 26 years alright, so	please
go for the smarter route	(better	algorithm) instead if you have numbers
only.

linear_solve
Calculates the solutions	x (vector) of a	linear equation	system of the
form "Ax	= b" with "A" being a matrix, "b" a vector and the solution
"x" a vector. Due to implementation limitations,	"A" must be a
quadratic matrix	and "b"	must have a dimension that is equivalent to
that of "A". Furthermore, the determinant of "A"	must be	non-zero. The
algorithm used is devised from Cramer's Rule and	thus inefficient. The
preferred algorithm for this task is Gaussian Elimination. If you have
a matrix	and a vector of	real numbers, please consider using either

First argument must be a	reference to a matrix (array of	arrays)	of
symbolic	terms, second argument must be a reference to a	vector (array)
of symbolic terms. Strings will be automatically	converted to
Math::Symbolic trees.  Returns a	reference to the solution vector.

bell_polynomial
This functions returns the nth Bell Polynomial. It uses memoization for
speed increase.

First argument is the n.	Second (optional) argument is the variable or
variable	name to	use in the polynomial. Defaults	to 'x'.

The Bell	Polynomial is defined as follows:

phi_0	(x) = 1
phi_n+1(x) = x	* ( phi_n(x) + partial_derivative( phi_n(x), x ) )

Bell Polynomials	are Exponential	Polynimals with	phi_n(1) = the nth
bell number. Please refer to the	bell_number() function in the
Math::Symbolic::AuxFunctions module for a method	of generating these
numbers.

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
math-symbolic-develop at	lists dot sourceforge dot net.

List of contributors:

Steffen Mi?1/2ller, symbolic-module at	steffen-mueller	dot net
Stray Toaster,	mwk at users dot sourceforge dot net
Oliver	Ebenhi?1/2h