NAME

Math::Orthonormalize - Gram-Schmidt Orthonormalization of vectors

SYNOPSIS

  use Math::Orthonormalize qw(:all);
  
  my @base_of_r_2 = (
    [2, 1],
    [1, 3]
  );
  my $vector = [1, 2, 3];
  
  my @orthonormalized = orthonormalize(@base_of_r_2);
  my @orthogonalized  = orthogonalize(@base_of_r_2);
  
  my $normalized      = normalize($vector);
  my $scaled          = scale(2, $vector);
  my $scalar          = scalar_product($vector1, $vector2);

DESCRIPTION

Math::Orthonormalize offers subroutines to compute normalized or non-normalized orthogonal bases of Euclidean vector spaces. That means: Given a vector base of R^n, it computes a new base of R^n whose individual vectors are all orthogonal. If those new base vectors all have a length of 1, the base is orthonormalized.

The module uses the Gram-Schmidt Algorithm.

EXPORT

No subroutines are exported by default, but the standart Exporter semantics are in place, including the ':all' tag that imports all of the exportable subroutines which are listed below.

SUBROUTINES

orthonormalize

Takes any number (>1) of vectors (array refs of vector components) as argument which form a base (that is, they are linearly independent) and returns an orthogonalized and normalized base of the same vector space (that is, n new array references).

orthogonalize

Takes any number (>1) of vectors (array refs of vector components) as argument which form a base (that is, they are linearly independent) and returns an orthogonalized base of the same vector space (that is, n new array references).

normalize

Normalizes a vector. That is, it changes the vector length to 1 without changing the vector's direction.

Takes an array reference with the vector components as argument and returns a new array reference containing the normalized vector components.

scale

Takes a scalar and a vector (array reference of vector components) as arguments. Multiplies every component of the vector by the specified scalar and returns a new array reference containing the scaled vector components.

scalar_product

Computes the scalar product of two vectors. Expects two array references with vector components (same number of components) as argument and returns their scalar product.

AUTHOR

Steffen Mueller, orthonormalize-module at steffen-mueller dot net

SEE ALSO

(German) Merziger, Wirth: "Repetitorium der Höheren Mathematik" (Binomi, 1999)

You may find the current versions of this module at http://steffen-mueller.net/ or on CPAN.

COPYRIGHT AND LICENSE

Copyright (C) 2004-2005 by Steffen Mueller

This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself, either Perl version 5.6.1 or, at your option, any later version of Perl 5 you may have available.