Math::Symbolic::MiscAlgebra(3pm)

1Math::Symbolic::MiscAlgUesberra(C3o)ntributed Perl DocumMeanttha:t:iSoynmbolic::MiscAlgebra(3)
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NAME

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

9         use Math::Symbolic qw/:all/;
10         use Math::Symbolic::MiscAlgebra qw/:all/; # not loaded by Math::Symbolic
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12         @matrix = (['x*y', 'z*x', 'y*z'],['x', 'z', 'z'],['x', 'x', 'y']);
13         $det = det @matrix;
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15         @vector = ('x', 'y', 'z');
16         $solution = solve_linear(\@matrix, \@vector);
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DESCRIPTION

19       This module provides several subroutines related to algebra such as
20       computing the determinant of quadratic matrices, solving linear
21       equation systems and computation of Bell Polynomials.
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23       Please note that the code herein may or may not be refactored into the
24       OO-interface of the Math::Symbolic module in the future.
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26   EXPORT
27       None by default.
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29       You may choose to have any of the following routines exported to the
30       calling namespace. ':all' tag exports all of the following:
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32         det
33         linear_solve
34         bell_polynomial
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SUBROUTINES

37   det
38       det() computes the determinant of a matrix of Math::Symbolic trees (or
39       strings that can be parsed as such). First argument must be a literal
40       array: "det @matrix", where @matrix is an n x n matrix.
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42       Please note that calculating determinants of matrices using the
43       straightforward Laplace algorithm is a slow (O(n!))  operation. This
44       implementation cannot make use of the various optimizations resulting
45       from the determinant properties since we are dealing with symbolic
46       matrix elements. If you have a matrix of reals, it is strongly
47       suggested that you use Math::MatrixReal or Math::Pari to get the
48       determinant which can be calculated using LR decomposition much faster.
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50       On a related note: Calculating the determinant of a 20x20 matrix would
51       take over 77146 years if your Perl could do 1 million calculations per
52       second.  Given that we're talking about several method calls per
53       calculation, that's much more than todays computers could do. On the
54       other hand, if you'd be using this straightforward algorithm with
55       numbers only and in C, you might be done in 26 years alright, so please
56       go for the smarter route (better algorithm) instead if you have numbers
57       only.
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59   linear_solve
60       Calculates the solutions x (vector) of a linear equation system of the
61       form "Ax = b" with "A" being a matrix, "b" a vector and the solution
62       "x" a vector. Due to implementation limitations, "A" must be a
63       quadratic matrix and "b" must have a dimension that is equivalent to
64       that of "A". Furthermore, the determinant of "A" must be non-zero. The
65       algorithm used is devised from Cramer's Rule and thus inefficient. The
66       preferred algorithm for this task is Gaussian Elimination. If you have
67       a matrix and a vector of real numbers, please consider using either
68       Math::MatrixReal or Math::Pari instead.
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70       First argument must be a reference to a matrix (array of arrays) of
71       symbolic terms, second argument must be a reference to a vector (array)
72       of symbolic terms. Strings will be automatically converted to
73       Math::Symbolic trees.  Returns a reference to the solution vector.
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75   bell_polynomial
76       This functions returns the nth Bell Polynomial. It uses memoization for
77       speed increase.
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79       First argument is the n. Second (optional) argument is the variable or
80       variable name to use in the polynomial. Defaults to 'x'.
81
82       The Bell Polynomial is defined as follows:
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84         phi_0  (x) = 1
85         phi_n+1(x) = x * ( phi_n(x) + partial_derivative( phi_n(x), x ) )
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87       Bell Polynomials are Exponential Polynimals with phi_n(1) = the nth
88       bell number. Please refer to the bell_number() function in the
89       Math::Symbolic::AuxFunctions module for a method of generating these
90       numbers.
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AUTHOR

93       Please send feedback, bug reports, and support requests to the
94       Math::Symbolic support mailing list: math-symbolic-support at lists dot
95       sourceforge dot net. Please consider letting us know how you use
96       Math::Symbolic. Thank you.
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98       If you're interested in helping with the development or extending the
99       module's functionality, please contact the developers' mailing list:
100       math-symbolic-develop at lists dot sourceforge dot net.
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102       List of contributors:
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104         Steffen Müller, symbolic-module at steffen-mueller dot net
105         Stray Toaster, mwk at users dot sourceforge dot net
106         Oliver Ebenhöh
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NAME

SYNOPSIS

DESCRIPTION

SUBROUTINES

AUTHOR

SEE ALSO