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 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 MXller, symbolic-module at steffen-mueller dot net
         Stray Toaster, mwk at users dot sourceforge dot net
         Oliver EbenhXh

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