Math::Symbolic::MiscAlgebra - Miscellaneous algebra routines like det()
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);
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.
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
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.
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.
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.
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 M�ller, symbolic-module at steffen-mueller dot net
Stray Toaster, mwk at users dot sourceforge dot net
Oliver Ebenh�h
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