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

       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

       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

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

       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 v5.40.0                                       2024-09-16                   Math::Symbolic::MiscAlgebra(3pm)

detdet() 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.

         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);