Math::Symbolic::MiscAlUseraContributed Perl DocuMath::Symbolic::MiscAlgebra(3)NAMEMath::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
Math::MatrixReal or Math::Pari instead.
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
module's functionality, please contact the developers' mailing list:
math-symbolic-develop at lists dot sourceforge dot net.
List of contributors:
Steffen MA~Xller, symbolic-module at steffen-mueller dot net
Stray Toaster, mwk at users dot sourceforge dot net
Oliver EbenhA~Xh
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
perl v5.14.1 2011-07-26 Math::Symbolic::MiscAlgebra(3)
