 Math::Symbolic::MiscAlgebra(3) 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
```

## 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
```