NAME
Math::Matrix - multiply and invert matrices
SYNOPSIS
use Math::Matrix;
# Generate a random 3-by-3 matrix.
srand(time);
$A = Math::Matrix -> new([rand, rand, rand],
[rand, rand, rand],
[rand, rand, rand]);
$A -> print("A\n");
# Append a fourth column to $A.
$x = Math::Matrix -> new([rand, rand, rand]);
$E = $A -> concat($x -> transpose);
$E -> print("Equation system\n");
# Compute the solution.
$s = $E -> solve;
$s -> print("Solutions s\n");
# Verify that the solution equals $x.
$A -> multiply($s) -> print("A*s\n");
DESCRIPTION
This module implements various constructors and methods for creating and
manipulating matrices.
All methods return new objects, so, for example, "$X->add($Y)" does not
modify $X.
$X -> add($Y); # $X not modified; output is lost
$X = $X -> add($Y); # this works
Some operators are overloaded (see "OVERLOADING") and allow the operand
to be modified directly.
$X = $X + $Y; # this works
$X += $Y; # so does this
METHODS
Constructors
new Constructor arguments are a list of references to arrays of the same
length. The arrays are copied. The method returns undef in case of
error.
$a = Math::Matrix->new([rand,rand,rand],
[rand,rand,rand],
[rand,rand,rand]);
If you call "new" with no input arguments, a zero filled matrix with
identical dimensions is returned:
$b = $a->new(); # $b is a zero matrix with the size of $a
new_identity
Returns a new identity matrix.
$a = Math::Matrix -> new(3); # $a is a 3-by-3 identity matrix
eye This is an alias for "new_identity".
clone
Clones a matrix and returns the clone.
$b = $a->clone;
diagonal
A constructor method that creates a diagonal matrix from a single
list or array of numbers.
$p = Math::Matrix->diagonal(1, 4, 4, 8);
$q = Math::Matrix->diagonal([1, 4, 4, 8]);
The matrix is zero filled except for the diagonal members, which
take the values of the vector.
The method returns undef in case of error.
tridiagonal
A constructor method that creates a matrix from vectors of numbers.
$p = Math::Matrix->tridiagonal([1, 4, 4, 8]);
$q = Math::Matrix->tridiagonal([1, 4, 4, 8], [9, 12, 15]);
$r = Math::Matrix->tridiagonal([1, 4, 4, 8], [9, 12, 15], [4, 3, 2]);
In the first case, the main diagonal takes the values of the vector,
while both of the upper and lower diagonals's values are all set to
one.
In the second case, the main diagonal takes the values of the first
vector, while the upper and lower diagonals are each set to the
values of the second vector.
In the third case, the main diagonal takes the values of the first
vector, while the upper diagonal is set to the values of the second
vector, and the lower diagonal is set to the values of the third
vector.
The method returns undef in case of error.
Other methods
size
You can determine the dimensions of a matrix by calling:
($m, $n) = $a->size;
concat
Concatenate matrices horizontally. The matrices must have the same
number or rows. The result is a new matrix or undef in case of
error.
$x = Math::Matrix -> new([1, 2], [4, 5]); # 2-by-2 matrix
$y = Math::Matrix -> new([3], [6]); # 2-by-1 matrix
$z = $x -> concat($y); # 2-by-3 matrix
transpose
Returns the transposed matrix. This is the matrix where colums and
rows of the argument matrix are swapped.
negative
Negate a matrix and return it.
$a = Math::Matrix -> new([-2, 3]);
$b = $a -> negative(); # $b = [[2, -3]]
multiply
Multiplies two matrices where the length of the rows in the first
matrix is the same as the length of the columns in the second
matrix. Returns the product or undef in case of error.
solve
Solves a equation system given by the matrix. The number of colums
must be greater than the number of rows. If variables are dependent
from each other, the second and all further of the dependent
coefficients are 0. This means the method can handle such systems.
The method returns a matrix containing the solutions in its columns
or undef in case of error.
invert
Invert a Matrix using "solve".
pinvert
Compute the pseudo-inverse of the matrix: ((A'A)^-1)A'
multiply_scalar
Multiplies a matrix and a scalar resulting in a matrix of the same
dimensions with each element scaled with the scalar.
$a->multiply_scalar(2); scale matrix by factor 2
add Add two matrices of the same dimensions.
subtract
Shorthand for "add($other->negative)"
equal
Decide if two matrices are equal. The criterion is, that each pair
of elements differs less than $Math::Matrix::eps.
slice
Extract columns:
a->slice(1,3,5);
diagonal_vector
Extract the diagonal as an array:
$diag = $a->diagonal_vector;
tridiagonal_vector
Extract the diagonals that make up a tridiagonal matrix:
($main_d, $upper_d, $lower_d) = $a->tridiagonal_vector;
determinant
Compute the determinant of a matrix.
$a = Math::Matrix->new([3, 1],
[4, 2]);
$d = $a->determinant; # $d = 2
dot_product
Compute the dot product of two vectors. The second operand does not
have to be an object.
# $x and $y are both objects
$x = Math::Matrix -> new([1, 2, 3]);
$y = Math::Matrix -> new([4, 5, 6]);
$p = $x -> dot_product($y); # $p = 32
# Only $x is an object.
$p = $x -> dot_product([4, 5, 6]); # $p = 32
absolute
Compute the absolute value (i.e., length) of a vector.
$v = Math::Matrix -> new([3, 4]);
$a = $v -> absolute(); # $v = 5
normalize
Normalize a vector, i.e., scale a vector so its length becomes 1.
$v = Math::Matrix -> new([3, 4]);
$u = $v -> normalize(); # $u = [ 0.6, 0.8 ]
cross_product
Compute the cross-product of vectors.
$x = Math::Matrix -> new([1,3,2],
[5,4,2]);
$p = $x -> cross_product(); # $p = [ -2, 8, -11 ]
as_string
Creates a string representation of the matrix and returns it.
$x = Math::Matrix -> new([1, 2], [3, 4]);
$s = $x -> as_string();
print
Prints the matrix on STDOUT. If the method has additional
parameters, these are printed before the matrix is printed.
OVERLOADING
The following operators are overloaded.
"+" and "+="
Matrix addition. The two operands must have the same size.
$C = $A + $B; # assign $A + $B to $C
$A += $B; # assign $A + $B to $A
"-" and "-="
Matrix subtraction. The two operands must have the same size.
$C = $A + $B; # assign $A - $B to $C
$A += $B; # assign $A - $B to $A
"*" and "*="
Matrix multiplication. The number of columns in the first operand
must be equal to the number of rows in the second operand.
$C = $A * $B; # assign $A * $B to $C
$A *= $B; # assign $A * $B to $A
"~" Transpose.
$B = ~$A; # $B is the transpose of $A
BUGS
Please report any bugs through the web interface at
<https://rt.cpan.org/Ticket/Create.html?Queue=Math-Matrix> (requires
login). We will be notified, and then you'll automatically be notified
of progress on your bug as I make changes.
SUPPORT
You can find documentation for this module with the perldoc command.
perldoc Math::Matrix
You can also look for information at:
* GitHub Source Repository
<https://github.com/pjacklam/p5-Math-Matrix>
* RT: CPAN's request tracker
<https://rt.cpan.org/Public/Dist/Display.html?Name=Math-Matrix>
* CPAN Ratings
<https://cpanratings.perl.org/dist/Math-Matrix>
* MetaCPAN
<https://metacpan.org/release/Math-Matrix>
* CPAN Testers Matrix
<http://matrix.cpantesters.org/?dist=Math-Matrix>
LICENSE AND COPYRIGHT
Copyright (c) 2020, Peter John Acklam.
Copyright (C) 2013, John M. Gamble <jgamble@ripco.com>, all rights
reserved.
Copyright (C) 2009, oshalla
https://rt.cpan.org/Public/Bug/Display.html?id=42919
Copyright (C) 2002, Bill Denney <gte273i@prism.gatech.edu>, all rights
reserved.
Copyright (C) 2001, Brian J. Watson <bjbrew@power.net>, all rights
reserved.
Copyright (C) 2001, Ulrich Pfeifer <pfeifer@wait.de>, all rights
reserved. Copyright (C) 1995, Universität Dortmund, all rights reserved.
Copyright (C) 2001, Matthew Brett <matthew.brett@mrc-cbu.cam.ac.uk>
This program is free software; you may redistribute it and/or modify it
under the same terms as Perl itself.
AUTHORS
Peter John Acklam <pjacklam@gmail.com> (2020)
Ulrich Pfeifer <pfeifer@ls6.informatik.uni-dortmund.de> (1995-2013)
Brian J. Watson <bjbrew@power.net>
Matthew Brett <matthew.brett@mrc-cbu.cam.ac.uk>