# FreeBSD Manual Pages

```Math::Matrix(3)	      User Contributed Perl Documentation      Math::Matrix(3)

NAME
Math::Matrix - multiply and invert matrices

SYNOPSIS
use Math::Matrix;

# Generate a	random 3-by-3 matrix.
srand(time);
my \$A = Math::Matrix	-> new([rand, rand, rand],
[rand, rand, rand],
[rand, rand, rand]);
\$A -> print("A\n");

# Append a fourth column to \$A.
my \$x = Math::Matrix	-> new([rand, rand, rand]);
my \$E = \$A -> concat(\$x -> transpose);
\$E -> print("Equation system\n");

# Compute the solution.
my \$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

to be modified directly.

\$X =	\$X + \$Y;	  # this works
\$X += \$Y;		  # so does this

METHODS
Constructors
new()
Creates a new object	from the input arguments and returns it.

If a	single input argument is given,	and that argument is a
reference to	array whose first element is itself a reference	to an
array, it is	assumed	that the argument contains the whole matrix,
like	this:

\$x = Math::Matrix->new([[1, 2, 3], [4, 5, 6]]); # 2-by-3	matrix
\$x = Math::Matrix->new([[1, 2, 3]]);	       # 1-by-3	matrix
\$x = Math::Matrix->new([, , ]);	       # 3-by-1	matrix

If a	single input argument is given,	and that argument is not a
reference to	an array, a 1-by-1 matrix is returned.

\$x = Math::Matrix->new(1);		       # 1-by-1	matrix

Note	that all the folling cases result in an	empty matrix:

\$x = Math::Matrix->new([[], [], []]);
\$x = Math::Matrix->new([[]]);
\$x = Math::Matrix->new([]);

If "new()" is called	as an instance method 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

Each	row must contain the same number of elements.

new_from_sub()
Creates a new matrix	object by doing	a subroutine call to create
each	element.

\$sub = sub { ...	};
\$x = Math::Matrix -> new_from_sub(\$sub);		 # 1-by-1
\$x = Math::Matrix -> new_from_sub(\$sub, \$m);	 # \$m-by-\$m
\$x = Math::Matrix -> new_from_sub(\$sub, \$m, \$n);	 # \$m-by-\$n

The subroutine is called in scalar context with two input
arguments, the row and column indices of the	element	to be created.
Note	that no	checks are performed on	the output of the subroutine.

Example 1, a	4-by-4 identity	matrix can be created with

\$sub = sub { \$_ == \$_ ? 1 : 0 };
\$x = Math::Matrix -> new_from_sub(\$sub, 4);

Example 2, the code

\$x = Math::Matrix -> new_from_sub(sub { 2**\$_	}, 1, 11);

creates the following 1-by-11 vector	with powers of two

[ 1, 2, 4, 8, 16, 32, 64, 128, 256, 512,	1024 ]

Example 3, the code,	using \$i and \$j	for increased readability

\$sub = sub {
(\$i,	\$j) = @_;
\$d =	\$j - \$i;
return \$d ==	-1 ? 5
: \$d ==	 0 ? 6
: \$d ==	 1 ? 7
: 0;
};
\$x = Math::Matrix -> new_from_sub(\$sub, 5);

creates the tridiagonal matrix

[ 6 7 0 0 0 ]
[ 5 6 7 0 0 ]
[ 0 5 6 7 0 ]
[ 0 0 5 6 7 ]
[ 0 0 0 5 6 ]

new_from_rows()
Creates a new matrix	by assuming each argument is a row vector.

\$x = Math::Matrix -> new_from_rows(\$y, \$z, ...);

For example

\$x = Math::Matrix -> new_from_rows([1, 2, 3],[4,	5, 6]);

returns the matrix

[ 1 2 3 ]
[ 4 5 6 ]

new_from_cols()
Creates a matrix by assuming	each argument is a column vector.

\$x = Math::Matrix -> new_from_cols(\$y, \$z, ...);

For example,

\$x = Math::Matrix -> new_from_cols([1, 2, 3],[4,	5, 6]);

returns the matrix

[ 1 4 ]
[ 2 5 ]
[ 3 6 ]

id()
Returns a new identity matrix.

\$I = Math::Matrix -> id(\$n);    # \$n-by-\$n identity matrix
\$I = \$x -> id(\$n);	       # \$n-by-\$n identity matrix
\$I = \$x -> id();		       # identity matrix with size of \$x

new_identity()
This	is an alias for	"id()".

eye()
This	is an alias for	"id()".

exchg()
Exchange matrix.

\$x = Math::Matrix -> exchg(\$n);	   # \$n-by-\$n exchange matrix

scalar()
Returns a scalar matrix, i.e., a diagonal matrix with all the
diagonal elements set to the	same value.

# Create	an \$m-by-\$m scalar matrix where	each element is	\$c.
\$x = Math::Matrix -> scalar(\$c, \$m);

# Create	an \$m-by-\$n scalar matrix where	each element is	\$c.
\$x = Math::Matrix -> scalar(\$c, \$m, \$n);

Multiplying a matrix	A by a scalar matrix B is effectively the same
as multiply each element in A by the	constant on the	diagonal of B.

zeros()
Create a zero matrix.

# Create	an \$m-by-\$m matrix where each element is 0.
\$x = Math::Matrix -> zeros(\$m);

# Create	an \$m-by-\$n matrix where each element is 0.
\$x = Math::Matrix -> zeros(\$m, \$n);

ones()
Create a matrix of ones.

# Create	an \$m-by-\$m matrix where each element is 1.
\$x = Math::Matrix -> ones(\$m);

# Create	an \$m-by-\$n matrix where each element is 1.
\$x = Math::Matrix -> ones(\$m, \$n);

inf()
Create a matrix of positive infinities.

# Create	an \$m-by-\$m matrix where each element is Inf.
\$x = Math::Matrix -> inf(\$m);

# Create	an \$m-by-\$n matrix where each element is Inf.
\$x = Math::Matrix -> inf(\$m, \$n);

nan()
Create a matrix of NaNs (Not-a-Number).

# Create	an \$m-by-\$m matrix where each element is NaN.
\$x = Math::Matrix -> nan(\$m);

# Create	an \$m-by-\$n matrix where each element is NaN.
\$x = Math::Matrix -> nan(\$m, \$n);

constant()
Returns a constant matrix, i.e., a matrix whose elements all	have
the same value.

# Create	an \$m-by-\$m matrix where each element is \$c.
\$x = Math::Matrix -> constant(\$c, \$m);

# Create	an \$m-by-\$n matrix where each element is \$c.
\$x = Math::Matrix -> constant(\$c, \$m, \$n);

rand()
Returns a matrix of uniformly distributed random numbers in the
range [0,1).

\$x = Math::Matrix -> rand(\$m);	       # \$m-by-\$m matrix
\$x = Math::Matrix -> rand(\$m, \$n);      # \$m-by-\$n matrix

To generate an \$m-by-\$n matrix of uniformly distributed random
numbers in the range	[0,\$a),	use

\$x = \$a * Math::Matrix -> rand(\$m, \$n);

To generate an \$m-by-\$n matrix of uniformly distributed random
numbers in the range	[\$a,\$b), use

\$x = \$a + (\$b - \$a) * Math::Matrix -> rand(\$m, \$n);

randi()
Returns a matrix of uniformly distributed random integers.

\$x = Math::Matrix -> randi(\$max);		 # 1-by-1 matrix
\$x = Math::Matrix -> randi(\$max,	\$n);		 # \$n-by-\$n matrix
\$x = Math::Matrix -> randi(\$max,	\$m, \$n);	 # \$m-by-\$n matrix

\$x = Math::Matrix -> randi([\$min, \$max]);	 # 1-by-1 matrix
\$x = Math::Matrix -> randi([\$min, \$max],	\$n);	 # \$n-by-\$n matrix
\$x = Math::Matrix -> randi([\$min, \$max],	\$m, \$n); # \$m-by-\$n matrix

randn()
Returns a matrix of random numbers from the standard	normal
distribution.

\$x = Math::Matrix -> randn(\$m);	       # \$m-by-\$m matrix
\$x = Math::Matrix -> randn(\$m, \$n);     # \$m-by-\$n matrix

To generate an \$m-by-\$n matrix with mean \$mu	and standard deviation
\$sigma, use

\$x = \$mu	+ \$sigma * Math::Matrix	-> randn(\$m, \$n);

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.

blkdiag()
Create block	diagonal matrix. Returns a block diagonal matrix given
a list of matrices.

\$z = Math::Matrix -> blkdiag(\$x,	\$y, ...);

Identify matrices
is_empty()
Returns 1 is	the invocand is	empty, i.e., it	has no elements.

\$bool = \$x -> is_empty();

is_scalar()
Returns 1 is	the invocand is	a scalar, i.e.,	it has one element.

\$bool = \$x -> is_scalar();

is_vector()
Returns 1 is	the invocand is	a vector, i.e.,	a row vector or	a
column vector.

\$bool = \$x -> is_vector();

is_row()
Returns 1 if	the invocand has exactly one row, and 0	otherwise.

\$bool = \$x -> is_row();

is_col()
Returns 1 if	the invocand has exactly one column, and 0 otherwise.

\$bool = \$x -> is_col();

is_square()
Returns 1 is	the invocand is	square,	and 0 otherwise.

\$bool = \$x -> is_square();

is_symmetric()
Returns 1 is	the invocand is	symmetric, and 0 otherwise.

\$bool = \$x -> is_symmetric();

An symmetric	matrix satisfies x(i,j)	= x(j,i) for all i and j, for
example

[  1  2 -3 ]
[  2 -4	5 ]
[ -3  5	6 ]

is_antisymmetric()
Returns 1 is	the invocand is	antisymmetric a.k.a. skew-symmetric,
and 0 otherwise.

\$bool = \$x -> is_antisymmetric();

An antisymmetric matrix satisfies x(i,j) = -x(j,i) for all i	and j,
for example

[  0  2 -3 ]
[ -2  0	4 ]
[  3 -4	0 ]

is_persymmetric()
Returns 1 is	the invocand is	persymmetric, and 0 otherwise.

\$bool = \$x -> is_persymmetric();

A persymmetric matrix is a square matrix which is symmetric with
respect to the anti-diagonal, e.g.:

[ f  h  j  k ]
[ c  g  i  j ]
[ b  d  g  h ]
[ a  b  c  f ]

is_hankel()
Returns 1 is	the invocand is	a Hankel matric	a.k.a. a catalecticant
matrix, and 0 otherwise.

\$bool = \$x -> is_hankel();

A Hankel matrix is a	square matrix in which each ascending skew-
diagonal from left to right is constant, e.g.:

[ e f g h i ]
[ d e f g h ]
[ c d e f g ]
[ b c d e f ]
[ a b c d e ]

is_zero()
Returns 1 is	the invocand is	a zero matrix, and 0 otherwise.	A zero
matrix contains no element whose value is different from zero.

\$bool = \$x -> is_zero();

is_one()
Returns 1 is	the invocand is	a matrix of ones, and 0	otherwise. A
matrix of ones contains no element whose value is different from
one.

\$bool = \$x -> is_one();

is_constant()
Returns 1 is	the invocand is	a constant matrix, and 0 otherwise. A
constant matrix is a	matrix where no	two elements have different
values.

\$bool = \$x -> is_constant();

is_identity()
Returns 1 is	the invocand is	an identity matrix, and	0 otherwise.
An identity matrix contains ones on the main	diagonal and zeros
elsewhere.

\$bool = \$x -> is_identity();

is_exchg()
Returns 1 is	the invocand is	an exchange matrix, and	0 otherwise.

\$bool = \$x -> is_exchg();

An exchange matrix contains ones on the main	anti-diagonal and
zeros elsewhere, for	example

[ 0 0 1 ]
[ 0 1 0 ]
[ 1 0 0 ]

is_bool()
Returns 1 is	the invocand is	a boolean matrix, and 0	otherwise.

\$bool = \$x -> is_bool();

A boolean matrix is a matrix	is a matrix whose entries are either 0
or 1, for example

[ 0 1 1 ]
[ 1 0 0 ]
[ 0 1 0 ]

is_perm()
Returns 1 is	the invocand is	an permutation matrix, and 0
otherwise.

\$bool = \$x -> is_perm();

A permutation matrix	is a square matrix with	exactly	one 1 in each
row and column, and all other elements 0, for example

[ 0 1 0 ]
[ 1 0 0 ]
[ 0 0 1 ]

is_int()
Returns 1 is	the invocand is	an integer matrix, i.e., a matrix of
integers, and 0 otherwise.

\$bool = \$x -> is_int();

is_diag()
Returns 1 is	the invocand is	diagonal, and 0	otherwise.

\$bool = \$x -> is_diag();

A diagonal matrix is	a square matrix	where all non-zero elements,
if any, are on the main diagonal. It	has the	following pattern,
where only the elements marked as "x" can be	non-zero,

[ x 0 0 0 0 ]
[ 0 x 0 0 0 ]
[ 0 0 x 0 0 ]
[ 0 0 0 x 0 ]
[ 0 0 0 0 x ]

Returns 1 is	the invocand is	anti-diagonal, and 0 otherwise.

A diagonal matrix is	a square matrix	where all non-zero elements,
if any, are on the main antidiagonal. It has	the following pattern,
where only the elements marked as "x" can be	non-zero,

[ 0 0 0 0 x ]
[ 0 0 0 x 0 ]
[ 0 0 x 0 0 ]
[ 0 x 0 0 0 ]
[ x 0 0 0 0 ]

is_tridiag()
Returns 1 is	the invocand is	tridiagonal, and 0 otherwise.

\$bool = \$x -> is_tridiag();

A tridiagonal matrix	is a square matrix with	nonzero	elements only
on the diagonal and slots horizontally or vertically	adjacent the
diagonal (i.e., along the subdiagonal and superdiagonal). It	has
the following pattern, where	only the elements marked as "x"	can be
non-zero,

[ x x 0 0 0 ]
[ x x x 0 0 ]
[ 0 x x x 0 ]
[ 0 0 x x x ]
[ 0 0 0 x x ]

is_atridiag()
Returns 1 is	the invocand is	anti-tridiagonal, and 0	otherwise.

\$bool = \$x -> is_tridiag();

A anti-tridiagonal matrix is	a square matrix	with nonzero elements
only	on the anti-diagonal and slots horizontally or vertically
adjacent the	diagonal (i.e.,	along the anti-subdiagonal and anti-
superdiagonal). It has the following	pattern, where only the
elements marked as "x" can be non-zero,

[ 0 0 0 x x ]
[ 0 0 x x x ]
[ 0 x x x 0 ]
[ x x x 0 0 ]
[ x x 0 0 0 ]

Returns 1 is	the invocand is	pentadiagonal, and 0 otherwise.

A pentadiagonal matrix is a square matrix with nonzero elements
only	on the diagonal	and the	two diagonals above and	below the main
diagonal. It	has the	following pattern, where only the elements
marked as "x" can be	non-zero,

[ x x x 0 0 0 ]
[ x x x x 0 0 ]
[ x x x x x 0 ]
[ 0 x x x x x ]
[ 0 0 x x x x ]
[ 0 0 0 x x x ]

Returns 1 is	the invocand is	anti-pentadiagonal, and	0 otherwise.

A anti-pentadiagonal	matrix is a square matrix with nonzero
elements only on the	anti-diagonal and two anti-diagonals above and
below the main anti-diagonal. It has	the following pattern, where
only	the elements marked as "x" can be non-zero,

[ 0 0 0 x x x ]
[ 0 0 x x x x ]
[ 0 x x x x x ]
[ x x x x x 0 ]
[ x x x x 0 0 ]
[ x x x 0 0 0 ]

Returns 1 is	the invocand is	heptadiagonal, and 0 otherwise.

A heptadiagonal matrix is a square matrix with nonzero elements
only	on the diagonal	and the	two diagonals above and	below the main
diagonal. It	has the	following pattern, where only the elements
marked as "x" can be	non-zero,

[ x x x x 0 0 ]
[ x x x x x 0 ]
[ x x x x x x ]
[ x x x x x x ]
[ 0 x x x x x ]
[ 0 0 x x x x ]

Returns 1 is	the invocand is	anti-heptadiagonal, and	0 otherwise.

A anti-heptadiagonal	matrix is a square matrix with nonzero
elements only on the	anti-diagonal and two anti-diagonals above and
below the main anti-diagonal. It has	the following pattern, where
only	the elements marked as "x" can be non-zero,

[ 0 0 x x x x ]
[ 0 x x x x x ]
[ x x x x x x ]
[ x x x x x x ]
[ x x x x x 0 ]
[ x x x x 0 0 ]

is_band()
Returns 1 is	the invocand is	a band matrix with a specified
bandwidth, and 0 otherwise.

\$bool = \$x -> is_band(\$k);

A band matrix is a square matrix with nonzero elements only on the
diagonal and	on the \$k diagonals above and below the	main diagonal.
The bandwidth \$k must be non-negative.

\$bool = \$x -> is_band(0);   # is	\$x diagonal?
\$bool = \$x -> is_band(1);   # is	\$x tridiagonal?
\$bool = \$x -> is_band(2);   # is	\$x pentadiagonal?
\$bool = \$x -> is_band(3);   # is	\$x heptadiagonal?

is_aband()
Returns 1 is	the invocand is	"anti-banded" with a specified
bandwidth, and 0 otherwise.

\$bool = \$x -> is_aband(\$k);

Some	examples

\$bool = \$x -> is_aband(0);  # is	\$x anti-diagonal?
\$bool = \$x -> is_aband(1);  # is	\$x anti-tridiagonal?
\$bool = \$x -> is_aband(2);  # is	\$x anti-pentadiagonal?
\$bool = \$x -> is_aband(3);  # is	\$x anti-heptadiagonal?

A band matrix is a square matrix with nonzero elements only on the
diagonal and	on the \$k diagonals above and below the	main diagonal.
The bandwidth \$k must be non-negative.

A "anti-banded" matrix is a square matrix with nonzero elements
only	on the anti-diagonal and \$k anti-diagonals above and below the
main	anti-diagonal.

is_triu()
Returns 1 is	the invocand is	upper triangular, and 0	otherwise.

\$bool = \$x -> is_triu();

An upper triangular matrix is a square matrix where all non-zero
elements are	on or above the	main diagonal. It has the following
pattern, where only the elements marked as "x" can be non-zero. It
has the following pattern, where only the elements marked as	"x"
can be non-zero,

[ x x x x ]
[ 0 x x x ]
[ 0 0 x x ]
[ 0 0 0 x ]

is_striu()
Returns 1 is	the invocand is	strictly upper triangular, and 0
otherwise.

\$bool = \$x -> is_striu();

A strictly upper triangular matrix is a square matrix where all
non-zero elements are strictly above	the main diagonal. It has the
following pattern, where only the elements marked as	"x" can	be
non-zero,

[ 0 x x x ]
[ 0 0 x x ]
[ 0 0 0 x ]
[ 0 0 0 0 ]

is_tril()
Returns 1 is	the invocand is	lower triangular, and 0	otherwise.

\$bool = \$x -> is_tril();

A lower triangular matrix is	a square matrix	where all non-zero
elements are	on or below the	main diagonal. It has the following
pattern, where only the elements marked as "x" can be non-zero,

[ x 0 0 0 ]
[ x x 0 0 ]
[ x x x 0 ]
[ x x x x ]

is_stril()
Returns 1 is	the invocand is	strictly lower triangular, and 0
otherwise.

\$bool = \$x -> is_stril();

A strictly lower triangular matrix is a square matrix where all
non-zero elements are strictly below	the main diagonal. It has the
following pattern, where only the elements marked as	"x" can	be
non-zero,

[ 0 0 0 0 ]
[ x 0 0 0 ]
[ x x 0 0 ]
[ x x x 0 ]

is_atriu()
Returns 1 is	the invocand is	upper anti-triangular, and 0
otherwise.

\$bool = \$x -> is_atriu();

An upper anti-triangular matrix is a	square matrix where all	non-
zero	elements are on	or above the main anti-diagonal. It has	the
following pattern, where only the elements marked as	"x" can	be
non-zero,

[ x x x x ]
[ x x x 0 ]
[ x x 0 0 ]
[ x 0 0 0 ]

is_satriu()
Returns 1 is	the invocand is	strictly upper anti-triangular,	and 0
otherwise.

\$bool = \$x -> is_satriu();

A strictly anti-triangular matrix is	a square matrix	where all non-
zero	elements are strictly above the	main diagonal. It has the
following pattern, where only the elements marked as	"x" can	be
non-zero,

[ x x x 0 ]
[ x x 0 0 ]
[ x 0 0 0 ]
[ 0 0 0 0 ]

is_atril()
Returns 1 is	the invocand is	lower anti-triangular, and 0
otherwise.

\$bool = \$x -> is_atril();

A lower anti-triangular matrix is a square matrix where all non-
zero	elements are on	or below the main anti-diagonal. It has	the
following pattern, where only the elements marked as	"x" can	be
non-zero,

[ 0 0 0 x ]
[ 0 0 x x ]
[ 0 x x x ]
[ x x x x ]

is_satril()
Returns 1 is	the invocand is	strictly lower anti-triangular,	and 0
otherwise.

\$bool = \$x -> is_satril();

A strictly lower anti-triangular matrix is a	square matrix where
all non-zero	elements are strictly below the	main anti-diagonal. It
has the following pattern, where only the elements marked as	"x"
can be non-zero,

[ 0 0 0 0 ]
[ 0 0 0 x ]
[ 0 0 x x ]
[ 0 x x x ]

Identify elements
This section contains methods for identifying and locating elements

find()
Returns the location	of each	non-zero element.

\$K = \$x -> find();	   # linear index
(\$I, \$J)	= \$x ->	find();	   # subscripts

For example,	to find	the linear index of each element that is
greater than	or equal to 1, use

\$K = \$x -> sge(1) -> find();

is_finite()
Returns a matrix of ones and	zeros. The element is one if the
corresponding element in the	invocand matrix	is finite, and zero
otherwise.

\$y = \$x -> is_finite();

is_inf()
Returns a matrix of ones and	zeros. The element is one if the
corresponding element in the	invocand matrix	is positive or
negative infinity, and zero otherwise.

\$y = \$x -> is_inf();

is_nan()
Returns a matrix of ones and	zeros. The element is one if the
corresponding element in the	invocand matrix	is a NaN (Not-a-
Number), and	zero otherwise.

\$y = \$x -> is_nan();

all()
Tests whether all of	the elements along various dimensions of a
matrix are non-zero.	If the dimension argument is not given,	the
first non-singleton dimension is used.

\$y = \$x -> all(\$dim);
\$y = \$x -> all();

any()
Tests whether any of	the elements along various dimensions of a
matrix are non-zero.	If the dimension argument is not given,	the
first non-singleton dimension is used.

\$y = \$x -> any(\$dim);
\$y = \$x -> any();

cumall()
A cumulative	variant	of "all()". If the dimension argument is not
given, the first non-singleton dimension is used.

\$y = \$x -> cumall(\$dim);
\$y = \$x -> cumall();

cumany()
A cumulative	variant	of "all()". If the dimension argument is not
given, the first non-singleton dimension is used.

\$y = \$x -> cumany(\$dim);
\$y = \$x -> cumany();

Basic properties
size()
You can determine the dimensions of a matrix	by calling:

(\$m, \$n)	= \$a ->	size;

nelm()
Returns the number of elements in the matrix.

\$n = \$x -> nelm();

nrow()
Returns the number of rows.

\$m = \$x -> nrow();

ncol()
Returns the number of columns.

\$n = \$x -> ncol();

npag()
Returns the number of pages.	A non-matrix has one page.

\$n = \$x -> pag();

ndim()
Returns the number of dimensions. This is the number	of dimensions
along which the length is different from one.

\$n = \$x -> ndim();

bandwidth()
Returns the bandwidth of a matrix. In scalar	context, returns the
number of the non-zero diagonal furthest away from the main
diagonal. In	list context, separate values are returned for the
lower and upper bandwidth.

\$n = \$x -> bandwidth();
(\$l, \$u)	= \$x ->	bandwidth();

The bandwidth is a non-negative integer. If the bandwidth is	0, the
matrix is diagonal or zero. If the bandwidth	is 1, the matrix is
tridiagonal.	If the bandwidth is 2, the matrix is pentadiagonal
etc.

A matrix with the following pattern,	where "x" denotes a non-zero
value, would	return 2 in scalar context, and	(1,2) in list context.

[ x x x 0 0 0 ]
[ x x x x 0 0 ]
[ 0 x x x x 0 ]
[ 0 0 x x x x ]
[ 0 0 0 x x x ]
[ 0 0 0 0 x x ]

Manipulate matrices
These methods are for combining matrices, splitting matrices, extracing
parts of	a matrix, inserting new	parts into a matrix, deleting parts of
a matrix	etc.  There are	also methods for shuffling elements around
(relocating elements) inside a matrix.

These methods are not concerned with the	values of the elements.

catrow()
Concatenate rows, i.e., concatenate matrices	vertically. Any	number
of arguments	is allowed. All	non-empty matrices must	have the same
number or columns. The result is a new matrix.

\$x = Math::Matrix -> new([1, 2],	[4, 5]);   # 2-by-2 matrix
\$y = Math::Matrix -> new([3, 6]);	   # 1-by-2 matrix
\$z = \$x -> catrow(\$y);			   # 3-by-2 matrix

catcol()
Concatenate columns,	i.e., matrices horizontally. Any number	of
arguments is	allowed. All non-empty matrices	must have the same
number or rows. The result is a new matrix.

\$x = Math::Matrix -> new([1, 2],	[4, 5]);   # 2-by-2 matrix
\$y = Math::Matrix -> new(, );	   # 2-by-1 matrix
\$z = \$x -> catcol(\$y);			   # 2-by-3 matrix

getrow()
Get the specified row(s). Returns a new matrix with the specified
rows. The number of rows in the output is identical to the number
of elements in the input.

\$y = \$x -> getrow(\$i);		       # get one
\$y = \$x -> getrow([\$i0, \$i1, \$i2]);     # get multiple

getcol()
Get the specified column(s).	Returns	a new matrix with the
specified columns.  The number of columns in	the output is
identical to	the number of elements in the input.

\$y = \$x -> getcol(\$j);		       # get one
\$y = \$x -> getcol([\$j0, \$j1, \$j2]);     # get multiple

delrow()
Delete row(s). Returns a new	matrix identical to the	invocand but
with	the specified row(s) deleted.

\$y = \$x -> delrow(\$i);		       # delete	one
\$y = \$x -> delrow([\$i0, \$i1, \$i2]);     # delete	multiple

delcol()
Delete column(s). Returns a new matrix identical to the invocand
but with the	specified column(s) deleted.

\$y = \$x -> delcol(\$j);		       # delete	one
\$y = \$x -> delcol([\$j0, \$j1, \$j2]);     # delete	multiple

concat()
Concatenate two matrices horizontally. The matrices must have the
same	number of 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(, );	   # 2-by-1 matrix
\$z = \$x -> concat(\$y);			   # 2-by-3 matrix

splicerow()
Row splicing. This is like Perl's built-in splice() function,
except that it works	on the rows of a matrix.

\$y = \$x -> splicerow(\$offset);
\$y = \$x -> splicerow(\$offset, \$length);
\$y = \$x -> splicerow(\$offset, \$length, \$a, \$b, ...);

The built-in	splice() function modifies the first argument and
returns the removed elements, if any. However, since	splicerow()
does	not modify the invocand, it returns the	modified version as
the first output argument and the removed part as a (possibly
empty) second output	argument.

\$x = Math::Matrix -> new([[ 1,  2],
[ 3,  4],
[ 5,  6],
[ 7,  8]]);
\$a = Math::Matrix -> new([[11, 12],
[13, 14]]);
(\$y, \$z)	= \$x ->	splicerow(1, 2,	\$a);

Gives \$y

[  1  2 ]
[ 11 12 ]
[ 13 14 ]
[  7  8 ]

and \$z

[  3  4 ]
[  5  6 ]

splicecol()
Column splicing. This is like Perl's	built-in splice() function,
except that it works	on the columns of a matrix.

\$y = \$x -> splicecol(\$offset);
\$y = \$x -> splicecol(\$offset, \$length);
\$y = \$x -> splicecol(\$offset, \$length, \$a, \$b, ...);

The built-in	splice() function modifies the first argument and
returns the removed elements, if any. However, since	splicecol()
does	not modify the invocand, it returns the	modified version as
the first output argument and the removed part as a (possibly
empty) second output	argument.

\$x = Math::Matrix -> new([[ 1, 3, 5, 7 ],
[ 2, 4, 6, 8 ]]);
\$a = Math::Matrix -> new([[11, 13],
[12, 14]]);
(\$y, \$z)	= \$x ->	splicerow(1, 2,	\$a);

Gives \$y

[ 1  11	13  7 ]
[ 2  12	14  8 ]

and \$z

[ 3  5 ]
[ 4  6 ]

swaprc()
Swap	rows and columns. This method does nothing but shuffle
elements around. For	real numbers, swaprc() is identical to the
transpose() method.

A subclass implementing a matrix of complex numbers should provide
a transpose() method	that also takes	the complex conjugate of each
elements. The swaprc() method, on the other hand, should only
shuffle elements around.

flipud()
Flip	upside-down, i.e., flip	along dimension	1.

\$y = \$x -> flipud();

fliplr()
Flip	left-to-right, i.e., flip along	dimension 2.

\$y = \$x -> fliplr();

flip()
Flip	along various dimensions of a matrix. If the dimension
argument is not given, the first non-singleton dimension is used.

\$y = \$x -> flip(\$dim);
\$y = \$x -> flip();

rot90()
Rotate 90 degrees counterclockwise.

\$y = \$x -> rot90();     # rotate	90 degrees counterclockwise
\$y = \$x -> rot90(\$n);   # rotate	90*\$n degrees counterclockwise

rot180()
Rotate 180 degrees.

\$y = \$x -> rot180();

rot270()
Rotate 270 degrees counterclockwise,	i.e., 90 degrees clockwise.

\$y = \$x -> rot270();

repelm()
Repeat elements.

\$x -> repelm(\$y);

Repeats each	element	in \$x the number of times specified in \$y.

If \$x is the	matrix

[ 4 5 6 ]
[ 7 8 9 ]

and \$y is

[ 3 2 ]

the returned	matrix is

[ 4 4 5 5 6 6 ]
[ 4 4 5 5 6 6 ]
[ 4 4 5 5 6 6 ]
[ 7 7 8 8 9 9 ]
[ 7 7 8 8 9 9 ]
[ 7 7 8 8 9 9 ]

repmat()
Repeat elements.

\$x -> repmat(\$y);

Repeats the matrix \$x the number of times specified in \$y.

If \$x is the	matrix

[ 4 5 6 ]
[ 7 8 9 ]

and \$y is

[ 3 2 ]

the returned	matrix is

[ 4 5 6 4 5 6 ]
[ 7 8 9 7 8 9 ]
[ 4 5 6 4 5 6 ]
[ 7 8 9 7 8 9 ]
[ 4 5 6 4 5 6 ]
[ 7 8 9 7 8 9 ]

reshape()
Returns a reshaped copy of a	matrix.	The reshaping is done by
creating a new matrix and looping over the elements in column major
order. The new matrix must have the same number of elements as the
invocand matrix. The	following returns an \$m-by-\$n matrix,

\$y = \$x -> reshape(\$m, \$n);

The code

\$x = Math::Matrix -> new([[1, 3,	5, 7], [2, 4, 6, 8]]);
\$y = \$x -> reshape(4, 2);

creates the matrix \$x

[ 1  3  5  7 ]
[ 2  4  6  8 ]

and returns a reshaped copy \$y

[ 1  5 ]
[ 2  6 ]
[ 3  7 ]
[ 4  8 ]

to_row()
Reshape to a	row.

\$x -> to_row();

This	method reshapes	the matrix into	a single row. It is
essentially the same	as, but	faster than,

\$x -> reshape(1,	\$x -> nelm());

to_col()
Reshape to a	column.

\$y = \$x -> to_col();

This	method reshapes	the matrix into	a single column. It is
essentially the same	as, but	faster than,

\$x -> reshape(\$x	-> nelm(), 1);

to_permmat()
Permutation vector to permutation matrix. Converts a	vector of
zero-based permutation indices to a permutation matrix.

\$P = \$v -> to_permmat();

For example

\$v = Math::Matrix -> new([[0, 3,	1, 4, 2]]);
\$m = \$v -> to_permmat();

gives the permutation matrix	\$m

[ 1 0 0 0 0 ]
[ 0 0 0 1 0 ]
[ 0 1 0 0 0 ]
[ 0 0 0 0 1 ]
[ 0 0 1 0 0 ]

to_permvec()
Permutation matrix to permutation vector. Converts a	permutation
matrix to a vector of zero-based permutation	indices.

\$v = \$P -> to_permvec();

\$v = Math::Matrix -> new([[0, 3,	1, 4, 2]]);
\$m = \$v -> to_permmat();

Gives the permutation matrix	\$m

[ 1 0 0 0 0 ]
[ 0 0 0 1 0 ]
[ 0 1 0 0 0 ]
[ 0 0 0 0 1 ]
[ 0 0 1 0 0 ]

triu()
Upper triangular part. Extract the upper triangular part of a
matrix and set all other elements to	zero.

\$y = \$x -> triu();
\$y = \$x -> triu(\$n);

The optional	second argument	specifies how many diagonals above or
below the main diagonal should also be set to zero. The default
value of \$n is zero which includes the main diagonal.

tril()
Lower triangular part. Extract the lower triangular part of a
matrix and set all other elements to	zero.

\$y = \$x -> tril();
\$y = \$x -> tril(\$n);

The optional	second argument	specifies how many diagonals above or
below the main diagonal should also be set to zero. The default
value of \$n is zero which includes the main diagonal.

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;

Mathematical	functions

Addition. If	one operands is	a scalar, it is	treated	like a
constant matrix with	the same size as the other operand. Otherwise

is thrown if	the matrices don't have	the same size.

Scalar (element by element) addition	with scalar expansion. This
method places no requirements on the	size of	the input matrices.

Subtraction

sub()
Subtraction.	If one operands	is a scalar, it	is treated as a
constant matrix with	the same size as the other operand. Otherwise,
ordinarly matrix subtraction	is performed.

\$z = \$x -> sub(\$y);

msub()
Matrix subtraction. Subtract	two matrices of	the same size. An
error is thrown if the matrices don't have the same size.

\$z = \$x -> msub(\$y);

ssub()
Scalar (element by element) subtraction with	scalar expansion. This
method places no requirements on the	size of	the input matrices.

\$z = \$x -> ssub(\$y);

subtract()
This	is an alias for	"msub()".

Negation

neg()
Negation. Negate a matrix.

\$y = \$x -> neg();

It is effectively equivalent	to

\$y = \$x -> map(sub { -\$_	});

negative()
This	is an alias for	"neg()".

Multiplication

mul()
Multiplication. If one operands is a	scalar,	it is treated as a
constant matrix with	the same size as the other operand. Otherwise,
ordinary matrix multiplication is performed.

\$z = \$x -> mul(\$y);

mmul()
Matrix multiplication. An error is thrown if	the sizes don't	match;
the number of columns in the	first operand must be equal to the
number of rows in the second	operand.

\$z = \$x -> mmul(\$y);

smul()
Scalar (element by element) multiplication with scalar expansion.
This	method places no requirements on the size of the input
matrices.

\$z = \$x -> smul(\$y);

Matrix fused	multiply and add. If \$x	is a \$p-by-\$q matrix, then \$y
must	be a \$q-by-\$r matrix and \$z must be a \$p-by-\$r matrix. An
error is thrown if the sizes	don't match.

\$w = \$x -> mmuladd(\$y, \$z);

The fused multiply and add is equivalent to,	but computed with
higher accuracy than

\$w = \$x -> mmul(\$y) -> madd(\$z);

This	method can be used to improve the solution of linear systems.

kron()
Kronecker tensor product.

\$A -> kronprod(\$B);

If \$A is an \$m-by-\$n	matrix and \$B is a \$p-by-\$q matrix, then "\$A
-> kron(\$B)"	is an \$m*\$p-by-\$n*\$q matrix formed by taking all
possible products between the elements of \$A	and the	elements of
\$B.

multiply()
This	is an alias for	"mmul()".

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

Powers

pow()
Power function.

This	is an alias for	"mpow()".

mpow()
Matrix power. The second operand must be a non-negative integer.

\$y = \$x -> mpow(\$n);

The following example

\$x = Math::Matrix -> new([[0, -2],[1, 4]]);
\$y = 4;
\$z = \$x -> pow(\$y);

returns the matrix

[ -28  -96 ]
[  48  164 ]

spow()
Scalar (element by element) power function. This method doesn't
require the matrices	to have	the same size.

\$z = \$x -> spow(\$y);

Inversion

inv()
This	is an alias for	"minv()".

invert()
Invert a Matrix using "solve".

minv()
Matrix inverse. Invert a matrix.

\$y = \$x -> inv();

See the section "IMPROVING THE SOLUTION OF LINEAR SYSTEMS" for a
list	of additional parameters that can be used for trying to	obtain
a better solution through iteration.

sinv()
Scalar (element by element) inverse.	Invert each element in a
matrix.

\$y = \$x -> sinv();

mldiv()
Matrix left division. Returns the solution x	of the linear system
of equations	A*x = y, by computing A^(-1)*y.

\$x = \$y -> mldiv(\$A);

This	method also handles overdetermined and underdetermined
systems. There are three cases

o   If A is a square	matrix,	then

x = A\y = inv(A)*y

so that A*x = y to within round-off accuracy.

o   If A is an M-by-N matrix	where M	> N, then A\y is computed as

A\y = (A'*A)\(A'*y) = inv(A'*A)*(A'*y)

where A'	denotes	the transpose of A. The	returned matrix	is the
least squares solution to the linear system of equations	A*x =
y, if it	exists.	The matrix A'*A	must be	non-singular.

o   If A is an where	M < N, then A\y	is computed as

A\y = A'*((A*A')\y)

This solution is	not unique. The	matrix A*A' must be non-
singular.

See the section "IMPROVING THE SOLUTION OF LINEAR SYSTEMS" for a
list	of additional parameters that can be used for trying to	obtain
a better solution through iteration.

sldiv()
Scalar (left) division.

\$x -> sldiv(\$y);

For scalars,	there is no difference between left and	right
division, so	this is	just an	alias for "sdiv()".

mrdiv()
Matrix right	division. Returns the solution x of the	linear system
of equations	x*A = y, by computing x	= y/A =	y*inv(A) = (A'\y')',
where A' and	y' denote the transpose	of A and y, respectively, and
\ is	matrix left division (see "mldiv()").

\$x = \$y -> mrdiv(\$A);

See the section "IMPROVING THE SOLUTION OF LINEAR SYSTEMS" for a
list	of additional parameters that can be used for trying to	obtain
a better solution through iteration.

srdiv()
Scalar (right) division.

\$x -> srdiv(\$y);

For scalars,	there is no difference between left and	right
division, so	this is	just an	alias for "sdiv()".

sdiv()
Scalar division. Performs scalar (element by	element) division.

\$x -> sdiv(\$y);

mpinv()
Matrix pseudo-inverse, "(A'*A)^(-1)*A'", where ""'""	is the
transpose operator.

See the section "IMPROVING THE SOLUTION OF LINEAR SYSTEMS" for a
list	of additional parameters that can be used for trying to	obtain
a better solution through iteration.

pinv()
This	is an alias for	"mpinv()".

pinvert()
This	is an alias for	"mpinv()".

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.

Factorisation

chol()
Cholesky decomposition.

\$L = \$A -> chol();

Every symmetric, positive definite matrix A can be decomposed into
a product of	a unique lower triangular matrix L and its transpose,
so that A = L*L', where L' denotes the transpose of L. L is called
the Cholesky	factor of A.

Miscellaneous matrix functions

transpose()
Returns the transposed matrix. This is the matrix where colums and
rows	of the argument	matrix are swapped.

A subclass implementing matrices of complex numbers should provide
a "transpose()" method that takes the complex conjugate of each
element.

minormatrix()
Minor matrix. The (i,j) minor matrix	of a matrix is identical to
the original	matrix except that row i and column j has been
removed.

\$y = \$x -> minormatrix(\$i, \$j);

minor()
Minor. The (i,j) minor of a matrix is the determinant of the	(i,j)
minor matrix.

\$y = \$x -> minor(\$i, \$j);

cofactormatrix()
Cofactor matrix. Element (i,j) in the cofactor matrix is the	(i,j)
cofactor, which is (-1)^(i+j) multiplied by the determinant of the
(i,j) minor matrix.

\$y = \$x -> cofactormatrix();

cofactor()
Cofactor. The (i,j) cofactor	of a matrix is (-1)**(i+j) times the
(i,j) minor of the matrix.

\$y = \$x -> cofactor(\$i, \$j);

or adjunct, of a square matrix is the transpose of the cofactor
matrix.

det()
Determinant.	Returns	the determinant	of a matrix. The matrix	must
be square.

\$y = \$x -> det();

The matrix is computed by forward elimination, which	might cause
round-off errors. So	for example, the determinant might be a	non-
integer even	for an integer matrix.

determinant()
This	is an alias for	"det()".

detr()
Determinant.	Returns	the determinant	of a matrix. The matrix	must
be square.

\$y = \$x -> determinant();

The determinant is computed by recursion, so	it is generally	much
slower than "det()".

Elementwise mathematical	functions

These method work on each element of a matrix.

int()
Truncate to integer.	Truncates each element to an integer.

\$y = \$x -> int();

This	function is effectivly the same	as

\$y = \$x -> map(sub { int	});

floor()
Round to negative infinity. Rounds each element to negative
infinity.

\$y = \$x -> floor();

ceil()
Round to positive infinity. Rounds each element to positive
infinity.

\$y = \$x -> int();

abs()
Absolute value. The absolute	value of each element.

\$y = \$x -> abs();

This	is effectivly the same as

\$y = \$x -> map(sub { abs	});

sign()
Sign	function. Apply	the sign function to each element.

\$y = \$x -> sign();

This	is effectivly the same as

\$y = \$x -> map(sub { \$_ <=> 0 });

Columnwise or rowwise mathematical functions

These method work along each column or row of a matrix. Some of these
methods return a	matrix with the	same size as the invocand matrix.
Other methods collapse the dimension, so	that, e.g., if the method is
applied to the first dimension a	p-by-q matrix becomes a	1-by-q matrix,
and if applied to the second dimension, it becomes a p-by-1 matrix.
Others, like "diff()", reduces the length along the dimension by	one,
so a p-by-q matrix becomes a (p-1)-by-q or a p-by-(q-1) matrix.

sum()
Sum of elements along various dimensions of a matrix. If the
dimension argument is not given, the	first non-singleton dimension
is used.

\$y = \$x -> sum(\$dim);
\$y = \$x -> sum();

prod()
Product of elements along various dimensions	of a matrix. If	the
dimension argument is not given, the	first non-singleton dimension
is used.

\$y = \$x -> prod(\$dim);
\$y = \$x -> prod();

mean()
Mean	of elements along various dimensions of	a matrix. If the
dimension argument is not given, the	first non-singleton dimension
is used.

\$y = \$x -> mean(\$dim);
\$y = \$x -> mean();

hypot()
Hypotenuse. Computes	the square root	of the sum of the square of
each	element	along various dimensions of a matrix. If the dimension
argument is not given, the first non-singleton dimension is used.

\$y = \$x -> hypot(\$dim);
\$y = \$x -> hypot();

For example,

\$x = Math::Matrix -> new([[3,  4],
[5, 12]]);
\$y = \$x -> hypot(2);

returns the 2-by-1 matrix

[  5 ]
[ 13 ]

min()
Minimum of elements along various dimensions	of a matrix. If	the
dimension argument is not given, the	first non-singleton dimension
is used.

\$y = \$x -> min(\$dim);
\$y = \$x -> min();

max()
Maximum of elements along various dimensions	of a matrix. If	the
dimension argument is not given, the	first non-singleton dimension
is used.

\$y = \$x -> max(\$dim);
\$y = \$x -> max();

median()
Median of elements along various dimensions of a matrix. If the
dimension argument is not given, the	first non-singleton dimension
is used.

\$y = \$x -> median(\$dim);
\$y = \$x -> median();

cumsum()
Returns the cumulative sum along various dimensions of a matrix. If
the dimension argument is not given,	the first non-singleton
dimension is	used.

\$y = \$x -> cumsum(\$dim);
\$y = \$x -> cumsum();

cumprod()
Returns the cumulative product along	various	dimensions of a
matrix. If the dimension argument is	not given, the first non-
singleton dimension is used.

\$y = \$x -> cumprod(\$dim);
\$y = \$x -> cumprod();

cummean()
Returns the cumulative mean along various dimensions	of a matrix.
If the dimension argument is	not given, the first non-singleton
dimension is	used.

\$y = \$x -> cummean(\$dim);
\$y = \$x -> cummean();

diff()
Returns the differences between adjacent elements. If the dimension
argument is not given, the first non-singleton dimension is used.

\$y = \$x -> diff(\$dim);
\$y = \$x -> diff();

vecnorm()
Return the \$p-norm of the elements of \$x. If	the dimension argument
is not given, the first non-singleton dimension is used.

\$y = \$x -> vecnorm(\$p, \$dim);
\$y = \$x -> vecnorm(\$p);
\$y = \$x -> vecnorm();

The \$p-norm of a vector is defined as the \$pth root of the sum of
the absolute	values fo the elements raised to the \$pth power.

apply()
Applies a subroutine	to each	row or column of a matrix. If the
dimension argument is not given, the	first non-singleton dimension
is used.

\$y = \$x -> apply(\$sub, \$dim);
\$y = \$x -> apply(\$sub);

The subroutine is passed a list with	all elements in	a single
column or row.

Comparison
Matrix comparison

Methods matrix comparison. These	methods	return a scalar	value.

meq()
Matrix equal	to. Returns 1 if two matrices are identical and	0
otherwise.

\$bool = \$x -> meq(\$y);

mne()
Matrix not equal to.	Returns	1 if two matrices are different	and 0
otherwise.

\$bool = \$x -> mne(\$y);

equal()
Decide if two matrices are equal. The criterion is, that each pair
of elements differs less than \$Math::Matrix::eps.

\$bool = \$x -> equal(\$y);

Scalar comparison

Each of these methods performs scalar (element by element) comparison
and returns a matrix of ones and	zeros. Scalar expansion	is performed
if necessary.

seq()
Scalar equality. Performs scalar (element by	element) comparison of
two matrices.

\$bool = \$x -> seq(\$y);

sne()
Scalar (element by element) not equal to. Performs scalar (element
by element) comparison of two matrices.

\$bool = \$x -> sne(\$y);

slt()
Scalar (element by element) less than. Performs scalar (element by
element) comparison of two matrices.

\$bool = \$x -> slt(\$y);

sle()
Scalar (element by element) less than or equal to. Performs scalar
(element by element)	comparison of two matrices.

\$bool = \$x -> sle(\$y);

sgt()
Scalar (element by element) greater than. Performs scalar (element
by element) comparison of two matrices.

\$bool = \$x -> sgt(\$y);

sge()
Scalar (element by element) greater than or equal to. Performs
scalar (element by element) comparison of two matrices.

\$bool = \$x -> sge(\$y);

scmp()
Scalar (element by element) comparison. Performs scalar (element by
element) comparison of two matrices.	Each element in	the output
matrix is either -1,	0, or 1	depending on whether the elements are
less	than, equal to,	or greater than	each other.

\$bool = \$x -> scmp(\$y);

Vector functions
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

outer_product()
Compute the outer 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, 7]);
\$p = \$x -> outer_product(\$y);

# Only \$x is an object.
\$p = \$x -> outer_product([4, 5, 6, y]);

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 ]

Conversion
as_string()
Creates a string representation of the matrix and returns it.

\$x = Math::Matrix -> new([1, 2],	[3, 4]);
\$s = \$x -> as_string();

as_array()
Returns the matrix as an unblessed Perl array, i.e.,	and ordinary,
unblessed reference.

\$y = \$x -> as_array();	   # ref(\$y) returns 'ARRAY'

Matrix utilities
Apply a subroutine to each element

map()
Call	a subroutine for every element of a matrix, locally setting \$_
to each element and passing the matrix row and column indices as
input arguments.

# square	each element
\$y = \$x -> map(sub { \$_ ** 2 });

# set strictly lower triangular part to zero
\$y = \$x -> map(sub { \$_ > \$_ ? 0 :	\$_ })'

sapply()
Applies a subroutine	to each	element	of a matrix, or	each set of
corresponding elements if multiple matrices are given, and returns
the result. The first argument is the subroutine to apply. The
following arguments,	if any,	are additional matrices	on which to
apply the subroutine.

\$w = \$x -> sapply(\$sub);		   # single operand
\$w = \$x -> sapply(\$sub, \$y);	   # two operands
\$w = \$x -> sapply(\$sub, \$y, \$z);	   # three operands

Each	matrix element,	or corresponding set of	elements, are passed
to the subroutine as	input arguments.

When	used with a single operand, this method	is similar to the
"map()" method, the syntax is different, since "sapply()" supports
multiple operands.

o   The subroutine is run in	scalar context.

o   No checks are done on the return	value of the subroutine.

o   The number of rows in the output	matrix equals the number of
rows in the operand with	the largest number of rows. Ditto for
columns.	So if \$x is 5-by-2 matrix, and \$y is a 3-by-4 matrix,
the result is a 5-by-4 matrix.

o   For each	operand	that has a number of rows smaller than the
maximum value, the rows are recyled. Ditto for columns.

o   Don't modify the	variables \$_, \$_ etc. inside the
subroutine. Otherwise, there is a risk of modifying the operand
matrices.

o   If the matrix elements are objects that are not cloned when the
"=" (assignment)	operator is used, you might have to explicitly
clone the objects used inside the subroutine. Otherwise,	the
elements	in the output matrix might be references to objects in
the operand matrices, rather than references to new objects.

Some	examples

One operand
With one	operand, i.e., the invocand matrix, the	subroutine is
applied to each element of the invocand matrix. The returned
matrix has the same size	as the invocand. For example,
multiplying the matrix \$x with the scalar \$c

\$sub	= sub {	\$c * \$_ };	   # subroutine	to multiply by \$c
\$z =	\$x -> sapply(\$sub);	   # multiply each element by \$c

Two operands
When two	operands are specfied, the subroutine is applied to
each pair of corresponding elements in the two operands.	For
example,	adding two matrices can	be implemented as

\$sub	= sub {	\$_ *	\$_ };
\$z =	\$x -> sapply(\$sub, \$y);

Note that if the	matrices have different	sizes, the rows	and/or
columns of the smaller are recycled to match the	size of	the
larger. If \$x is	a \$p-by-\$q matrix and \$y is a \$r-by-\$s matrix,
then \$z is a max(\$p,\$r)-by-max(\$q,\$s) matrix, and

\$z -> [\$i][\$j] = \$sub -> (\$x	-> [\$i % \$p][\$j	% \$q],
\$y	-> [\$i % \$r][\$j	% \$s]);

Because of this recycling, multiplying the matrix \$x with the
scalar \$c (see above) can also be implemented as

\$sub	= sub {	\$_ *	\$_ };
\$z =	\$x -> sapply(\$sub, \$c);

Generating a matrix with	all combinations of "\$x**\$y" for \$x
being 4,	5, and 6 and \$y	being 1, 2, 3, and 4 can be done with

\$c =	Math::Matrix ->	new([, , ]);	   # 3-by-1 column
\$r =	Math::Matrix ->	new([[1, 2, 3, 4]]);	   # 1-by-4 row
\$x =	\$c -> sapply(sub { \$_ ** \$_ }, \$r);  # 3-by-4 matrix

Multiple operands
In general, the sapply()	method can have	any number of
arguments. For example, to compute the sum of the four matrices
\$x, \$y, \$z, and \$w,

\$sub	= sub {
\$sum = 0;
for \$val (@_) {
\$sum += \$val;
};
return \$sum;
};
\$sum	= \$x ->	sapply(\$sub, \$y, \$z, \$w);

Forward elimination

These methods take a matrix as input, performs forward elimination, and
returns a matrix	where all elements below the main diagonal are zero.
In list context,	four additional	arguments are returned:	an array with
the row permutations, an	array with the column permutations, an integer
with the	number of row swaps and	an integer with	the number of column
swaps performed during elimination.

The permutation vectors can be converted	to permutation matrices	with
"to_permmat()".

felim_np()
Perform forward elimination with no pivoting.

\$y = \$x -> felim_np();

Forward elimination without pivoting	may fail even when the matrix
is non-singular.

This	method is provided mostly for illustration purposes.

felim_tp()
Perform forward elimination with trivial pivoting, a	variant	of
partial pivoting.

\$y = \$x -> felim_tp();

If A	is a p-by-q matrix, and	the so far remaining unreduced
submatrix starts at element (i,i), the pivot	element	is the first
element in column i that is non-zero.

This	method is provided mostly for illustration purposes.

felim_pp()
Perform forward elimination with (unscaled) partial pivoting.

\$y = \$x -> felim_pp();

If A	is a p-by-q matrix, and	the so far remaining unreduced
submatrix starts at element (i,i), the pivot	element	is the element
in column i that has	the largest absolute value.

This	method is provided mostly for illustration purposes.

felim_sp()
Perform forward elimination with scaled pivoting, a variant of
partial pivoting.

\$y = \$x -> felim_sp();

If A	is a p-by-q matrix, and	the so far remaining unreduced
submatrix starts at element (i,i), the pivot	element	is the element
in column i that has	the largest absolute value relative to the
other elements on the same row.

felim_fp()
Performs forward elimination	with full pivoting.

\$y = \$x -> felim_fp();

The elimination is done with	full pivoting, also called complete
pivoting or total pivoting. If A is a p-by-q	matrix,	and the	so far
remaining unreduced submatrix starts	at element (i,i), the pivot
element is the element in the whole submatrix that has the largest
absolute value. With	full pivoting, both rows and columns might be
swapped.

Back-substitution

bsubs()
Performs back-substitution.

\$y = \$x -> bsubs();

The leftmost	square portion of the matrix must be upper triangular.

Miscellaneous methods
print()
Prints the matrix on	STDOUT.	If the method has additional
parameters, these are printed before	the matrix is printed.

version()
Returns a string contining the package name and version number.

"+" and "+="
Matrix or scalar addition. Unless one or both of the	operands is a
scalar, both	operands must have the same size.

\$C  = \$A	+ \$B;	   # assign \$A + \$B to \$C
\$A += \$B;	   # assign \$A + \$B to \$A

Note	that

"-" and "-="
Matrix or scalar subtraction. Unless	one or both of the operands is
a scalar, both operands must	have the same size.

\$C  = \$A	+ \$B;	   # assign \$A - \$B to \$C
\$A += \$B;	   # assign \$A - \$B to \$A

"*" and "*="
Matrix or scalar multiplication. Unless one or both of the operands
is a	scalar,	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

"**" and	"**="
Matrix power. The second operand must be a scalar.

\$C  = \$A	* \$B;	   # assign \$A ** \$B to	\$C
\$A *= \$B;	   # assign \$A ** \$B to	\$A

"=="
Equal to.

\$A == \$B;	   # is	\$A equal to \$B?

"!="
Not equal to.

\$A != \$B;	   # is	\$A not equal to	\$B?

"neg"
Negation.

\$B = -\$A;	   # \$B	is the negative	of \$A

"~" Transpose.

\$B = ~\$A;	   # \$B	is the transpose of \$A

"abs"
Absolute value.

\$B = abs	\$A;	   # \$B	contains absolute values of \$A

"int"
Truncate to integer.

\$B = int	\$A;	   # \$B	contains only integers

IMPROVING THE SOLUTION OF LINEAR SYSTEMS
The methods that	do an explicit or implicit matrix left division	accept
some additional parameters. If these parameters are specified, the
matrix left division is done repeatedly in an iterative way, which
often gives a better solution.

Background
The linear system of equations

\$A *	\$x = \$y

can be solved for \$x with

\$x =	\$y -> mldiv(\$A);

Ideally "\$A * \$x" should	equal \$y, but due to numerical errors, this is
not always the case. The	following illustrates how to improve the
solution	\$x computed above:

\$r =	\$A -> mmuladd(\$x, -\$y);	   # compute the residual \$A*\$x-\$y
\$d =	\$r -> mldiv(\$A);	   # compute the delta for \$x
\$x -= \$d;			   # improve the solution \$x

This procedure is repeated, and at each step, the absolute error

||\$A*\$x - \$y|| = ||\$r||

and the relative	error

||\$A*\$x - \$y|| / ||\$y|| = ||\$r|| / ||\$y||

are computed and	compared to the	tolerances. Once one of	the stopping
criteria	is satisfied, the algorithm terminates.

Stopping criteria
The algorithm stops when	at least one of	the errors are within the
specified tolerances or the maximum number of iterations	is reached. If
the maximum number of iterations	is reached, but	noen of	the errors are
within the tolerances, a	warning	is displayed and the best solution so
far is returned.

Parameters
MaxIter
The maximum number of iterations to perform.	The value must be a
positive integer. The default is 20.

RelTol
The limit for the relative error. The value must be a non-negative.
The default value is	1e-19 when perl	is compiled with long doubles
or quadruple	precision, and 1e-9 otherwise.

AbsTol
The limit for the absolute error. The value must be a non-negative.
The default value is	0.

Debug
If this parameter does not affect when the algorithm	terminates,
but when set	to non-zero, some information is displayed at each
step.

Example
If

\$A =	[[  8, -8, -5,	6, -1,	3 ],
[ -7, -1,  5, -9,  5,	6 ],
[ -7,	8,  9, -2, -4,	3 ],
[  3, -4,  5,	5,  3,	3 ],
[  9,	8, -3, -4,  1,	6 ],
[ -8,	9, -1,	3,  5,	2 ]];

\$y =	[[  80,	-13 ],
[  -2,	104 ],
[ -57,	-27 ],
[  47,	-28 ],
[   5,	 77 ],
[  91,	133 ]];

the result of "\$x = \$y -> mldiv(\$A);", using double precision
arithmetic, is the approximate solution

\$x =	[[ -2.999999999999998, -5.000000000000000 ],
[ -1.000000000000000,	3.000000000000001 ],
[ -5.999999999999997, -8.999999999999996 ],
[  8.000000000000000, -2.000000000000003 ],
[  6.000000000000003,	9.000000000000002 ],
[  7.999999999999997,	8.999999999999995 ]];

The residual "\$res = \$A -> mmuladd(\$x, -\$y);" is

\$res	= [[  1.24344978758018e-14,  1.77635683940025e-15 ],
[  8.88178419700125e-15, -5.32907051820075e-15 ],
[ -1.24344978758018e-14,  1.77635683940025e-15 ],
[ -7.10542735760100e-15, -4.08562073062058e-14 ],
[ -1.77635683940025e-14, -3.81916720471054e-14 ],
[  1.24344978758018e-14,  8.43769498715119e-15 ]];

and the delta "\$dx = \$res -> mldiv(\$A);"	is

\$dx = [[   -8.592098303124e-16, -2.86724066474914e-15 ],
[ -7.92220125658508e-16, -2.99693950082398e-15 ],
[ -2.22533360993874e-16,  3.03465504177947e-16 ],
[  6.47376093198353e-17, -1.12378127899388e-15 ],
[  6.35204502123966e-16,  2.40938179521241e-15 ],
[  1.55166908001001e-15,  2.08339859425849e-15 ]];

giving the improved, and	in this	case exact, solution "\$x -= \$dx;",

\$x =	[[ -3, -5 ],
[ -1,	3 ],
[ -6, -9 ],
[  8, -2 ],
[  6,	9 ],
[  8,	9 ]];

SUBCLASSING
The methods should work fine with any kind of numerical objects,
provided	that the assignment operator "=" is overloaded,	so that	Perl
knows how to create a copy.

You can check the behaviour of the assignment operator by assigning a
value to	a new variable,	modify the new variable, and check whether
this also modifies the original value. Here is an example:

\$x =	Some::Class -> new(0);		 # create object \$x
\$y =	\$x;				 # create new variable \$y
\$y++;				 # modify \$y
print "it's a clone\n" if \$x	!= \$y;	 # is \$x modified?

The subclass might need to implement some methods of its	own. For
instance, if each element is a complex number, a	transpose() method
needs to	be implemented to take the complex conjugate of	each value. An
as_string() method might	also be	useful for displaying the matrix in a
format more suitable for	the subclass.

Here is an example showing Math::Matrix::Complex, a fully-working
subclass	of Math::Matrix, where each element is a Math::Complex object.

use strict;
use warnings;

package Math::Matrix::Complex;

use Math::Matrix;
use Scalar::Util 'blessed';
use Math::Complex 1.57;     # "=" didn't clone before 1.57

our @ISA = ('Math::Matrix');

# We	need a new() method to make sure every element is an object.

sub new {
my \$self	= shift;
my \$x = \$self ->	SUPER::new(@_);

my \$sub = sub {
defined(blessed(\$_)) && \$_ -> isa('Math::Complex')
? \$_
: Math::Complex ->	new(\$_);
};

return \$x -> sapply(\$sub);
}

# We	need a transpose() method, since the transpose of a matrix
# with complex numbers also takes the conjugate of all elements.

sub transpose {
my \$x = shift;
my \$y = \$x -> SUPER::transpose(@_);

return \$y -> sapply(sub { ~\$_	});
}

# We	need an	as_string() method, since our parent's methods
# doesn't format complex numbers correctly.

sub as_string {
my \$self	= shift;
my \$out = "";
for my \$row (@\$self) {
for my \$elm (@\$row) {
\$out = \$out . sprintf "%10s ", \$elm;
}
\$out	= \$out . sprintf "\n";
}
\$out;
}

1;

BUGS
Please report any bugs or feature requests via
<https://github.com/pjacklam/p5-Math-Matrix/issues>.

Old bug reports and feature requests can	be found at
<http://rt.cpan.org/NoAuth/Bugs.html?Dist=Math-Matrix>.

SUPPORT
You can find documentation for this module with the perldoc command.

perldoc Math::Matrix

You can also look for information at:

o   GitHub Source Repository

<https://github.com/pjacklam/p5-Math-Matrix>

o   MetaCPAN

<https://metacpan.org/release/Math-Matrix>

o   CPAN	Ratings

<http://cpanratings.perl.org/d/Math-Matrix>

o   CPAN	Testers	PASS Matrix

<http://pass.cpantesters.org/distro/A/Math-Matrix.html>

o   CPAN	Testers	Reports

<http://www.cpantesters.org/distro/A/Math-Matrix.html>

o   CPAN	Testers	Matrix

<http://matrix.cpantesters.org/?dist=Math-Matrix>

Copyright (c) 2020-2021,	Peter John Acklam

Copyright (C) 2013, John	M. Gamble <jgamble@ripco.com>, all rights
reserved.

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, UniversitAxt 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> (2020-2021)

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>

perl v5.32.1			  2021-01-13		       Math::Matrix(3)
```

NAME | SYNOPSIS | DESCRIPTION | METHODS | OVERLOADING | IMPROVING THE SOLUTION OF LINEAR SYSTEMS | SUBCLASSING | BUGS | SUPPORT | LICENSE AND COPYRIGHT | AUTHORS

Want to link to this manual page? Use this URL:
<https://www.freebsd.org/cgi/man.cgi?query=Math::Matrix&sektion=3&manpath=FreeBSD+13.0-RELEASE+and+Ports>