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
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()
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([[1], [2], [3]]); # 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 { $_[0] == $_[1] ? 1 : 0 };
$x = Math::Matrix -> new_from_sub($sub, 4);
Example 2, the code
$x = Math::Matrix -> new_from_sub(sub { 2**$_[1] }, 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);
See also "randi()" and "randn()".
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
See also "rand()" and "randn()".
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);
See also "rand()" and "randi()".
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 ]
is_adiag()
Returns 1 is the invocand is anti-diagonal, and 0 otherwise.
$bool = $x -> is_adiag();
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 ]
is_pentadiag()
Returns 1 is the invocand is pentadiagonal, and 0 otherwise.
$bool = $x -> is_pentadiag();
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 ]
is_apentadiag()
Returns 1 is the invocand is anti-pentadiagonal, and 0 otherwise.
$bool = $x -> is_pentadiag();
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 ]
is_heptadiag()
Returns 1 is the invocand is heptadiagonal, and 0 otherwise.
$bool = $x -> is_heptadiag();
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 ]
is_aheptadiag()
Returns 1 is the invocand is anti-heptadiagonal, and 0 otherwise.
$bool = $x -> is_heptadiag();
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?
See also "is_aband()" and "bandwidth()".
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.
See also "is_band()" and "bandwidth()".
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
within an array. See also "Scalar comparison".
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 ]
See also "is_band()" and "is_aband()".
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([3], [6]); # 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([3], [6]); # 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();
See also "flipud()" and "fliplr()".
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 ]
See also "to_permmat()".
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
add()
Addition. If one operands is a scalar, it is treated like a constant
matrix with the same size as the other operand. Otherwise ordinary
matrix addition is performed.
$z = $x -> add($y);
See also "madd()" and "sadd()".
madd()
Matrix addition. Add two matrices of the same dimensions. An error
is thrown if the matrices don't have the same size.
$z = $x -> madd($y);
See also "add()" and "sadd()".
sadd()
Scalar (element by element) addition with scalar expansion. This
method places no requirements on the size of the input matrices.
$z = $x -> sadd($y);
See also "add()" and "madd()".
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);
See also "msub()" and "ssub()".
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);
See also "sub()" and "ssub()".
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);
See also "sub()" and "msub()".
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);
mmuladd()
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()".
See also "spow()".
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 ]
See also "spow()".
spow()
Scalar (element by element) power function. This method doesn't
require the matrices to have the same size.
$z = $x -> spow($y);
See also "mpow()".
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
* If A is a square matrix, then
x = A\y = inv(A)*y
so that A*x = y to within round-off accuracy.
* 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.
* 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);
See also "minor()".
minor()
Minor. The (i,j) minor of a matrix is the determinant of the (i,j)
minor matrix.
$y = $x -> minor($i, $j);
See also "minormatrix()".
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);
adjugate()
Adjugate of a matrix. The adjugate, also called classical adjoint or
adjunct, of a square matrix is the transpose of the cofactor matrix.
$y = $x -> adjugate();
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] > $_[1] ? 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.
See also "map()".
* The subroutine is run in scalar context.
* No checks are done on the return value of the subroutine.
* 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.
* For each operand that has a number of rows smaller than the
maximum value, the rows are recyled. Ditto for columns.
* Don't modify the variables $_[0], $_[1] etc. inside the
subroutine. Otherwise, there is a risk of modifying the operand
matrices.
* 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 * $_[0] }; # 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 { $_[0] * $_[1] };
$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 { $_[0] * $_[1] };
$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([[4], [5], [6]]); # 3-by-1 column
$r = Math::Matrix -> new([[1, 2, 3, 4]]); # 1-by-4 row
$x = $c -> sapply(sub { $_[0] ** $_[1] }, $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.
OVERLOADING
The following operators are overloaded.
"+" 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 $sub = sub {
defined(blessed($_[0])) && $_[0] -> isa('Math::Complex')
? $_[0]
: Math::Complex -> new($_[0]);
};
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 $y -> sapply(sub { ~$_[0] });
}
# 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 through the web interface at
(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
* RT: CPAN's request tracker
* CPAN Ratings
* MetaCPAN
* CPAN Testers Matrix
LICENSE AND COPYRIGHT
Copyright (c) 2020, Peter John Acklam.
Copyright (C) 2013, John M. Gamble , all rights
reserved.
Copyright (C) 2009, oshalla
https://rt.cpan.org/Public/Bug/Display.html?id=42919
Copyright (C) 2002, Bill Denney , all rights
reserved.
Copyright (C) 2001, Brian J. Watson , all rights
reserved.
Copyright (C) 2001, Ulrich Pfeifer , all rights
reserved. Copyright (C) 1995, Universität Dortmund, all rights reserved.
Copyright (C) 2001, Matthew Brett
This program is free software; you may redistribute it and/or modify it
under the same terms as Perl itself.
AUTHORS
Peter John Acklam (2020)
Ulrich Pfeifer (1995-2013)
Brian J. Watson
Matthew Brett