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Jonathan Leto >
Math-MatrixReal-1.9
> Math::MatrixReal
Module Version: 1.9   Source

Download: Math-MatrixReal-1.9.tar.gz

NAME

DESCRIPTION

SYNOPSIS

Constructor
Methods And Such

Matrix
Row, Column and Element operations

Matrix
Operations

Arithmetic
Operations

Boolean
Matrix Operations

Eigensystems

Miscellaneous

OVERLOADED
OPERATORS

SYNOPSIS

DESCRIPTION

SEE
ALSO

VERSION

AUTHORS

CREDITS

COPYRIGHT

LICENSE
AGREEMENT

for(my $i = 0;$i < $rows; $i++ ){
$big = abs($matrix->[0][$i][0]);
for(my $j=0;$j < $cols; $j++ ){
$big = $big < abs($matrix->[0][$i][$j])
? abs($matrix->[0][$i][$j]) : $big;
}
next unless $big;
# now $big is biggest element in row
for(my $j = 0;$j < $cols; $j++ ){
$matrix->[0][$i][$j] /= $big;
}
# now all elements are between [-1,1] and
# dependence can be easily checked
}

for(my $i = 0;$i < $rows; $i++ ){
for(my $k = 0; $k < $rows; $k++ ){
next if ($i == $k);
print "i,k = $i,$k\n";
$tmprow = $matrix->row($i+1) + $matrix->row($k+1);
print "tmprow= $tmprow\n";
print "tmprow has " . $tmprow->_count_zeroes_row . " zeroes.\n";
print "i+1 has " . $matrix->row($i+1)->_count_zeroes_row . " zeroes\n";
print "k+1 has " . $matrix->row($k+1)->_count_zeroes_row . " zeroes\n";

$tmprow = $tmprow->_normalize_row;
if( $tmprow->norm_sum == 0 ){
for(my $j=0;$j<$cols;$j++){
$matrix->[0][$k][$j] = 0;
}
} elsif ( $tmprow->_count_zeroes_row > $matrix->row($i+1)->_count_zeroes_row ){
print "tmprow has more zeroes, replacing row " . ($i+1) . "\n";
## assign row
$matrix->assign_row($i+1,$tmprow);
} elsif ( $tmprow->_count_zeroes_row > $matrix->row($k+1)->_count_zeroes_row ){
print "tmprow has more zeroes, replacing row " . ($k+1) . "\n";
$matrix->assign_row($k+1,$tmprow);
}

}
}
return $matrix;
}

NAME

Math::MatrixReal - Matrix of Reals

Implements the data type "matrix of reals" (and consequently also "vector of
reals").

DESCRIPTION

Implements the data type "matrix of reals", which can be used almost like any
other basic Perl type thanks to OPERATOR OVERLOADING, i.e.,

$product = $matrix1 * $matrix2;

does what you would like it to do (a matrix multiplication).

Also features many important operations and methods: matrix norm, matrix
transposition, matrix inverse, determinant of a matrix, order and numerical
condition of a matrix, scalar product of vectors, vector product of vectors,
vector length, projection of row and column vectors, a comfortable way for
reading in a matrix from a file, the keyboard or your code, and many more.

Allows to solve linear equation systems using an efficient algorithm known as
"L-R-decomposition" and several approximative (iterative) methods.

Features an implementation of Kleene's algorithm to compute the minimal costs
for all paths in a graph with weighted edges (the "weights" being the costs
associated with each edge).

SYNOPSIS

Constructor Methods And Such

use Math::MatrixReal;

Makes the methods and overloaded operators of this module available to your
program.

$new_matrix = new Math::MatrixReal($rows,$columns);

The matrix object constructor method. A new matrix of size $rows by
$columns will be created, with the value 0.0 for all
elements.

Note that this method is implicitly called by many of the other methods in
this module.

$new_matrix =
$some_matrix->new($rows,$columns);

Another way of calling the matrix object constructor method.

Matrix "$some_matrix" is not changed by this in any way.

$new_matrix = $matrix->new_from_cols( [
$column_vector|$array_ref|$string, ... ] )

Creates a new matrix given a reference to an array of any of the
following:

column vectors ( n by 1 Math::MatrixReal matrices )

references to arrays

strings properly formatted to create a column with Math::MatrixReal's
new_from_string command

You may mix and match these as you wish. However, all must be of the same
dimension--no padding happens automatically. Example:

my $matrix = Math::MatrixReal->new_from_cols( [ [1,2], [3,4] ] );
print $matrix;

will print

[ 1.000000000000E+00 3.000000000000E+00 ]
[ 2.000000000000E+00 4.000000000000E+00 ]

new_from_rows( [ $row_vector|$array_ref|$string, ... ] )

Creates a new matrix given a reference to an array of any of the
following:

row vectors ( 1 by n Math::MatrixReal matrices )

references to arrays

strings properly formatted to create a row with Math::MatrixReal's
new_from_string command

You may mix and match these as you wish. However, all must be of the same
dimension--no padding happens automatically. Example:

my $matrix = Math::MatrixReal->new_from_rows( [ [1,2], [3,4] ] );
print $matrix;

will print

[ 1.000000000000E+00 2.000000000000E+00 ]
[ 3.000000000000E+00 4.000000000000E+00 ]

$new_matrix = Math::MatrixReal->new_diag( $array_ref );

This method allows you to create a diagonal matrix by only specifying the
diagonal elements. Example:

$matrix = Math::MatrixReal->new_diag( [ 1,2,3,4 ] );
print $matrix;

will print

[ 1.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 ]
[ 0.000000000000E+00 2.000000000000E+00 0.000000000000E+00 0.000000000000E+00 ]
[ 0.000000000000E+00 0.000000000000E+00 3.000000000000E+00 0.000000000000E+00 ]
[ 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 4.000000000000E+00 ]

$new_matrix =
Math::MatrixReal->new_from_string($string);

This method allows you to read in a matrix from a string (for instance,
from the keyboard, from a file or from your code).

The syntax is simple: each row must start with "[ " and end
with " ]\n" ("\n" being the newline character and
" " a space or tab) and contain one or more numbers, all
separated from each other by spaces or tabs.

Additional spaces or tabs can be added at will, but no comments.

Examples:

$string = "[ 1 2 3 ]\n[ 2 2 -1 ]\n[ 1 1 1 ]\n";
$matrix = Math::MatrixReal->new_from_string($string);
print "$matrix";

By the way, this prints

[ 1.000000000000E+00 2.000000000000E+00 3.000000000000E+00 ]
[ 2.000000000000E+00 2.000000000000E+00 -1.000000000000E+00 ]
[ 1.000000000000E+00 1.000000000000E+00 1.000000000000E+00 ]

But you can also do this in a much more comfortable way using the
shell-like "here-document" syntax:

$matrix = Math::MatrixReal->new_from_string(<<'MATRIX');
[ 1 0 0 0 0 0 1 ]
[ 0 1 0 0 0 0 0 ]
[ 0 0 1 0 0 0 0 ]
[ 0 0 0 1 0 0 0 ]
[ 0 0 0 0 1 0 0 ]
[ 0 0 0 0 0 1 0 ]
[ 1 0 0 0 0 0 -1 ]
MATRIX

You can even use variables in the matrix:

$c1 = 2 / 3;
$c2 = -2 / 5;
$c3 = 26 / 9;

$matrix = Math::MatrixReal->new_from_string(<<"MATRIX");

[ 3 2 0 ]
[ 0 3 2 ]
[ $c1 $c2 $c3 ]

MATRIX

(Remember that you may use spaces and tabs to format the matrix to your
taste)

Note that this method uses exactly the same representation for a matrix as
the "stringify" operator "": this means that you can convert any matrix into a
string with $string = "$matrix"; and read it back in later (for
instance from a file!).

Note however that you may suffer a precision loss in this process because
only 13 digits are supported in the mantissa when printed!!

If the string you supply (or someone else supplies) does not obey the
syntax mentioned above, an exception is raised, which can be caught by "eval"
as follows:

print "Please enter your matrix (in one line): ";
$string = <STDIN>;
$string =~ s/\\n/\n/g;
eval { $matrix = Math::MatrixReal->new_from_string($string); };
if ($@)
{
print "$@";
# ...
# (error handling)
}
else
{
# continue...
}

or as follows:

eval { $matrix = Math::MatrixReal->new_from_string(<<"MATRIX"); };
[ 3 2 0 ]
[ 0 3 2 ]
[ $c1 $c2 $c3 ]
MATRIX
if ($@)
# ...

Actually, the method shown above for reading a matrix from the keyboard is
a little awkward, since you have to enter a lot of "\n"'s for the
newlines.

A better way is shown in this piece of code:

while (1)
{
print "\nPlease enter your matrix ";
print "(multiple lines, <ctrl-D> = done):\n";
eval { $new_matrix =
Math::MatrixReal->new_from_string(join('',<STDIN>)); };
if ($@)
{
$@ =~ s/\s+at\b.*?$//;
print "${@}Please try again.\n";
}
else { last; }
}

Possible error messages of the "new_from_string()" method are:

Math::MatrixReal::new_from_string(): syntax error in input string
Math::MatrixReal::new_from_string(): empty input string

If the input string has rows with varying numbers of columns, the following
warning will be printed to STDERR:

Math::MatrixReal::new_from_string(): missing elements will be set to zero!

If everything is okay, the method returns an object reference to the (newly
allocated) matrix containing the elements you specified.

$new_matrix = $some_matrix->shadow();

Returns an object reference to a NEW but EMPTY matrix (filled
with zero's) of the SAME SIZE as matrix
"$some_matrix".

Matrix "$some_matrix" is not changed by this in any way.

$matrix1->copy($matrix2);

Copies the contents of matrix "$matrix2" to an ALREADY
EXISTING matrix "$matrix1" (which must have the same size as
matrix "$matrix2"!).

Matrix "$matrix2" is not changed by this in any way.

$twin_matrix = $some_matrix->clone();

Returns an object reference to a NEW matrix of the SAME SIZE
as matrix "$some_matrix". The contents of matrix
"$some_matrix" have ALREADY BEEN COPIED to the new matrix
"$twin_matrix". This is the method that the operator "=" is
overloaded to when you type $a = $b, when $a and
$b are matrices.

Matrix "$some_matrix" is not changed by this in any
way.

Matrix Row, Column and
Element operations

$row_vector = $matrix->row($row);

This is a projection method which returns an object reference to a
NEW matrix (which in fact is a (row) vector since it has only one row)
to which row number "$row" of matrix "$matrix" has
already been copied.

Matrix "$matrix" is not changed by this in any way.

$column_vector = $matrix->column($column);

This is a projection method which returns an object reference to a
NEW matrix (which in fact is a (column) vector since it has only one
column) to which column number "$column" of matrix
"$matrix" has already been copied.

Matrix "$matrix" is not changed by this in any way.

$matrix->assign($row,$column,$value);

Explicitly assigns a value "$value" to a single element of the
matrix "$matrix", located in row "$row" and column
"$column", thereby replacing the value previously stored
there.

$value = $matrix->element($row,$column);

Returns the value of a specific element of the matrix
"$matrix", located in row "$row" and column
"$column".

$new_matrix = $matrix->each( \&function );

Creates a new matrix by evaluating a code reference on each element of the
given matrix. The function is passed the element, the row index and the column
index, in that order. The value the function returns ( or the value of the
last executed statement ) is the value given to the corresponding element in
$new_matrix.

Example:

# add 1 to every element in the matrix
$matrix = $matrix->each ( sub { (shift) + 1 } );

Example:

my $cofactor = $matrix->each( sub { my(undef,$i,$j) = @_;
($i+$j) % 2 == 0 ? $matrix->minor($i,$j)->det()
: -1*$matrix->minor($i,$j)->det();
} );

This code needs some explanation. For each element of $matrix, it throws
away the actual value and stores the row and column indexes in $i and $j. Then
it sets element [$i,$j] in $cofactor to the determinant of
$matrix->minor($i,$j) if it is an "even" element, or
-1*$matrix->minor($i,$j) if it is an "odd" element.

$new_matrix = $matrix->each_diag( \&function );

Creates a new matrix by evaluating a code reference on each diagonal
element of the given matrix. The function is passed the element, the row index
and the column index, in that order. The value the function returns ( or the
value of the last executed statement ) is the value given to the corresponding
element in $new_matrix.

$matrix->swap_col( $col1, $col2 );

This method takes two one-based column numbers and swaps the values of each
element in each column. $matrix->swap_col(2,3) would replace
column 2 in $matrix with column 3, and replace column 3 with column 2.

$matrix->swap_row( $row1, $row2 );

This method takes two one-based row numbers and swaps the values of each
element in each row. $matrix->swap_row(2,3) would replace row
2 in $matrix with row 3, and replace row 3 with row 2.

Matrix Operations

$det = $matrix->det();

Returns the determinant of the matrix, without going through the rigamarole
of computing a LR decomposition. This method should be much faster than LR
decomposition if the matrix is diagonal or triangular. Otherwise, it is just a
wrapper for $matrix->decompose_LR->det_LR. If the
determinant is zero, there is no inverse and vice-versa. Only quadratic
matrices have determinants.

$inverse = $matrix->inverse();

Returns the inverse of a matrix, without going through the rigamarole of
computing a LR decomposition. If no inverse exists, undef is returned and an
error is printed via carp(). This is nothing but a wrapper for
$matrix->decompose_LR->invert_LR.

($rows,$columns) = $matrix->dim();

Returns a list of two items, representing the number of rows and columns
the given matrix "$matrix" contains.

$norm_one = $matrix->norm_one();

Returns the "one"-norm of the given matrix "$matrix".

The "one"-norm is defined as follows:

For each column, the sum of the absolute values of the elements in the
different rows of that column is calculated. Finally, the maximum of these
sums is returned.

Note that the "one"-norm and the "maximum"-norm are mathematically
equivalent, although for the same matrix they usually yield a different
value.

Therefore, you should only compare values that have been calculated using
the same norm!

Throughout this package, the "one"-norm is (arbitrarily) used for all
comparisons, for the sake of uniformity and comparability, except for the
iterative methods "solve_GSM()", "solve_SSM()" and "solve_RM()" which use
either norm depending on the matrix itself.

$norm_max = $matrix->norm_max();

Returns the "maximum"-norm of the given matrix $matrix.

The "maximum"-norm is defined as follows:

For each row, the sum of the absolute values of the elements in the
different columns of that row is calculated. Finally, the maximum of these
sums is returned.

Note that the "maximum"-norm and the "one"-norm are mathematically
equivalent, although for the same matrix they usually yield a different
value.

Therefore, you should only compare values that have been calculated using
the same norm!

Throughout this package, the "one"-norm is (arbitrarily) used for all
comparisons, for the sake of uniformity and comparability, except for the
iterative methods "solve_GSM()", "solve_SSM()" and "solve_RM()" which use
either norm depending on the matrix itself.

$norm_sum = $matrix->norm_sum();

This is a very simple norm which is defined as the sum of the absolute
values of every element.

$p_norm = $matrix->norm_p($n);>

This function returns the "p-norm" of a vector. The argument $n must be a
number greater than or equal to 1 or the string "Inf". The p-norm is defined
as (sum(x_i^p))^(1/p). In words, it raised each element to the p-th power,
adds them up, and then takes the p-th root of that number. If the string "Inf"
is passed, the "infinity-norm" is computed, which is really the limit of the
p-norm as p goes to infinity. It is defined as the maximum element of the
vector. Also, note that the familiar Euclidean distance between two vectors is
just a special case of a p-norm, when p is equal to 2.

Example: $a = Math::MatrixReal->new_from_cols([[1,2,3]]); $p1 =
$a->norm_p(1); $p2 = $a->norm_p(2); $p3 = $a->norm_p(3); $pinf =
$a->norm_p("Inf");

print "(1,2,3,Inf) norm:\n$p1\n$p2\n$p3\n$pinf\n";

$i1 = $a->new_from_rows([[1,0]]);
$i2 = $a->new_from_rows([[0,1]]);

# this should be sqrt(2) since it is the same as the
# hypotenuse of a 1 by 1 right triangle

$dist = ($i1-$i2)->norm_p(2);
print "Distance is $dist, which should be " . sqrt(2) . "\n";

Output:

(1,2,3,Inf) norm:
6
3.74165738677394139
3.30192724889462668
3

Distance is 1.41421356237309505, which should be 1.41421356237309505

$frob_norm = $matrix->norm_frobenius();

This norm is similar to that of a p-norm where p is 2, except it acts on a
matrix, not a vector. Each element of the matrix is squared, this is
added up, and then a square root is taken.

$matrix->spectral_radius();

Returns the maximum value of the absolute value of all eigenvalues.
Currently this computes all eigenvalues, then sifts through them to
find the largest in absolute value. Needless to say, this is very inefficient,
and in the future an algorithm that computes only the largest eigenvalue may
be implemented.

$matrix1->transpose($matrix2);

Calculates the transposed matrix of matrix $matrix2 and stores the result
in matrix "$matrix1" (which must already exist and have the same
size as matrix "$matrix2"!).

This operation can also be carried out "in-place", i.e., input and output
matrix may be identical.

Transposition is a symmetry operation: imagine you rotate the matrix along
the axis of its main diagonal (going through elements (1,1), (2,2), (3,3) and
so on) by 180 degrees.

Another way of looking at it is to say that rows and columns are swapped.
In fact the contents of element (i,j) are swapped with those of
element (j,i).

Note that (especially for vectors) it makes a big difference if you have a
row vector, like this:

[ -1 0 1 ]

or a column vector, like this:

[ -1 ]
[ 0 ]
[ 1 ]

the one vector being the transposed of the other!

This is especially true for the matrix product of two vectors:

[ -1 ]
[ -1 0 1 ] * [ 0 ] = [ 2 ] , whereas
[ 1 ]

* [ -1 0 1 ]
[ -1 ] [ 1 0 -1 ]
[ 0 ] * [ -1 0 1 ] = [ -1 ] [ 1 0 -1 ] = [ 0 0 0 ]
[ 1 ] [ 0 ] [ 0 0 0 ] [ -1 0 1 ]
[ 1 ] [ -1 0 1 ]

So be careful about what you really mean!

Hint: throughout this module, whenever a vector is explicitly required for
input, a COLUMN vector is expected!

$trace = $matrix->trace();

This returns the trace of the matrix, which is defined as the sum of the
diagonal elements. The matrix must be quadratic.

$minor = $matrix->minor($row,$col);

Returns the minor matrix corresponding to $row and $col. $matrix must be
quadratic. If $matrix is n rows by n cols, the minor of $row and $col will be
an (n-1) by (n-1) matrix. The minor is defined as crossing out the row and the
col specified and returning the remaining rows and columns as a matrix. This
method is used by cofactor().

$cofactor = $matrix->cofactor();

The cofactor matrix is constructed as follows:

For each element, cross out the row and column that it sits in. Now, take
the determinant of the matrix that is left in the other rows and columns.
Multiply the determinant by (-1)^(i+j), where i is the row index, and j is the
column index. Replace the given element with this value.

The cofactor matrix can be used to find the inverse of the matrix. One
formula for the inverse of a matrix is the cofactor matrix transposed divided
by the original determinant of the matrix.

The following two inverses should be exactly the same:

my $inverse1 = $matrix->inverse;
my $inverse2 = ~($matrix->cofactor)->each( sub { (shift)/$matrix->det() } );

Caveat: Although the cofactor matrix is simple algorithm to compute the
inverse of a matrix, and can be used with pencil and paper for small matrices,
it is comically slower than the native inverse() function. Here
is a small benchmark:

# $matrix1 is 15x15
$det = $matrix1->det;
timethese( 10,
{'inverse' => sub { $matrix1->inverse(); },
'cofactor' => sub { (~$matrix1->cofactor)->each ( sub { (shift)/$det; } ) }
} );

Benchmark: timing 10 iterations of LR, cofactor, inverse...
inverse: 1 wallclock secs ( 0.56 usr + 0.00 sys = 0.56 CPU) @ 17.86/s (n=10)
cofactor: 36 wallclock secs (36.62 usr + 0.01 sys = 36.63 CPU) @ 0.27/s (n=10)

$adjoint = $matrix->adjoint();

The adjoint is just the transpose of the cofactor matrix. This method is
just an alias for ~($matrix->cofactor).

Arithmetic Operations

$matrix1->add($matrix2,$matrix3);

Calculates the sum of matrix "$matrix2" and matrix
"$matrix3" and stores the result in matrix
"$matrix1" (which must already exist and have the same size as
matrix "$matrix2" and matrix "$matrix3"!).

This operation can also be carried out "in-place", i.e., the output and one
(or both) of the input matrices may be identical.

$matrix1->subtract($matrix2,$matrix3);

Calculates the difference of matrix "$matrix2" minus matrix
"$matrix3" and stores the result in matrix
"$matrix1" (which must already exist and have the same size as
matrix "$matrix2" and matrix "$matrix3"!).

This operation can also be carried out "in-place", i.e., the output and one
(or both) of the input matrices may be identical.

Note that this operation is the same as
$matrix1->add($matrix2,-$matrix3);, although the latter is a
little less efficient.

$matrix1->multiply_scalar($matrix2,$scalar);

Calculates the product of matrix "$matrix2" and the number
"$scalar" (i.e., multiplies each element of matrix
"$matrix2" with the factor "$scalar") and stores the
result in matrix "$matrix1" (which must already exist and have
the same size as matrix "$matrix2"!).

This operation can also be carried out "in-place", i.e., input and output
matrix may be identical.

$product_matrix = $matrix1->multiply($matrix2);

Calculates the product of matrix "$matrix1" and matrix
"$matrix2" and returns an object reference to a new matrix
"$product_matrix" in which the result of this operation has been
stored.

Note that the dimensions of the two matrices "$matrix1" and
"$matrix2" (i.e., their numbers of rows and columns) must
harmonize in the following way (example):

[ 2 2 ]
[ 2 2 ]
[ 2 2 ]

[ 1 1 1 ] [ * * ]
[ 1 1 1 ] [ * * ]
[ 1 1 1 ] [ * * ]
[ 1 1 1 ] [ * * ]

I.e., the number of columns of matrix "$matrix1" has to be the
same as the number of rows of matrix "$matrix2".

The number of rows and columns of the resulting matrix
"$product_matrix" is determined by the number of rows of matrix
"$matrix1" and the number of columns of matrix
"$matrix2", respectively.

$matrix1->negate($matrix2);

Calculates the negative of matrix "$matrix2" (i.e., multiplies
all elements with "-1") and stores the result in matrix
"$matrix1" (which must already exist and have the same size as
matrix "$matrix2"!).

This operation can also be carried out "in-place", i.e., input and output
matrix may be identical.

$matrix_to_power = $matrix1->exponent($integer);

Raises the matrix to the $integer power. Obviously,
$integer must be an integer. If it is zero, the identity matrix
is returned. If a negative integer is given, the inverse will be computed (if
it exists) and then raised the the absolute value of $integer.
The matrix must be quadratic.

Boolean Matrix Operations

$matrix->is_quadratic();

Returns a boolean value indicating if the given matrix is quadratic (also
know as "square" or "n by n"). A matrix is quadratic if it has the same number
of rows as it does columns.

$matrix->is_square();

This is an alias for is_quadratic().

$matrix->is_symmetric();

Returns a boolean value indicating if the given matrix is symmetric. By
definition, a matrix is symmetric if and only if
(M[i,j]=M[j,i]). This is equivalent
to ($matrix == ~$matrix) but without memory allocation. Only
quadratic matrices can be symmetric.

Notes: A symmetric matrix always has real eigenvalues/eigenvectors. A
matrix plus its transpose is always symmetric.

$matrix->is_skew_symmetric();

Returns a boolean value indicating if the given matrix is skew symmetric.
By definition, a matrix is symmetric if and only if
(M[i,j]=-M[j,i]). This is equivalent
to ($matrix == -(~$matrix)) but without memory allocation. Only
quadratic matrices can be skew symmetric.

$matrix->is_diagonal();

Returns a boolean value indicating if the given matrix is diagonal, i.e.
all of the nonzero elements are on the main diagonal. Only quadratic matrices
can be diagonal.

$matrix->is_tridiagonal();

Returns a boolean value indicating if the given matrix is tridiagonal, i.e.
all of the nonzero elements are on the main diagonal or the diagonals above
and below the main diagonal. Only quadratic matrices can be tridiagonal.

$matrix->is_upper_triangular();

Returns a boolean value indicating if the given matrix is upper triangular,
i.e. all of the nonzero elements not on the main diagonal are above it. Only
quadratic matrices can be upper triangular. Note: diagonal matrices are both
upper and lower triangular.

$matrix->is_lower_triangular();

Returns a boolean value indicating if the given matrix is lower triangular,
i.e. all of the nonzero elements not on the main diagonal are below it. Only
quadratic matrices can be lower triangular. Note: diagonal matrices are both
upper and lower triangular.

$matrix->is_orthogonal();

Returns a boolean value indicating if the given matrix is orthogonal. An
orthogonal matrix is has the property that the transpose equals the inverse of
the matrix. Instead of computing each and comparing them, this method
multiplies the matrix by it's transpose, and returns true if this turns out to
be the identity matrix, false otherwise. Only quadratic matrices can
orthogonal.

$matrix->is_binary();

Returns a boolean value indicating if the given matrix is binary. A matrix
is binary if it contains only zeroes or ones.

$matrix->is_gramian();

Returns a boolean value indicating if the give matrix is Gramian. A matrix
$A is Gramian if and only if there exists a square matrix
$B such that $A = ~$B*$B. This is equivalent to
checking if $A is symmetric and has all nonnegative eigenvalues,
which is what Math::MatrixReal uses to check for this property.

$matrix->is_LR();

Returns a boolean value indicating if the matrix is an LR decomposition
matrix.

$matrix->is_positive();

Returns a boolean value indicating if the matrix contains only positive
entries. Note that a zero entry is not positive and will cause
is_positive() to return false.

$matrix->is_negative();

Returns a boolean value indicating if the matrix contains only negative
entries. Note that a zero entry is not negative and will cause
is_negative() to return false.

$matrix->is_periodic($k);

Returns a boolean value indicating if the matrix is periodic with period
$k. This is true if $matrix ** ($k+1) == $matrix. When $k
== 1, this reduces down to the is_idempotent()
function.

$matrix->is_idempotent();

Returns a boolean value indicating if the matrix is idempotent, which is
defined as the square of the matrix being equal to the original matrix, i.e
$matrix ** 2 == $matrix.

$matrix->is_row_vector();

Returns a boolean value indicating if the matrix is a row vector. A row
vector is a matrix which is 1xn. Note that the 1x1 matrix is both a row and
column vector.

$matrix->is_col_vector();

Returns a boolean value indicating if the matrix is a col vector. A col
vector is a matrix which is nx1. Note that the 1x1 matrix is both a row and
column vector.

Eigensystems

($l, $V) = $matrix->sym_diagonalize();

This method performs the diagonalization of the quadratic symmetric
matrix M stored in $matrix. On output, l is a column vector
containing all the eigenvalues of M and V is an orthogonal
matrix which columns are the corresponding normalized eigenvectors. The
primary property of an eigenvalue l and an eigenvector x is of
course that: M * x = l * x.

The method uses a Householder reduction to tridiagonal form followed by a
QL algoritm with implicit shifts on this tridiagonal. (The tridiagonal matrix
is kept internally in a compact form in this routine to save memory.) In fact,
this routine wraps the householder() and tri_diagonalize() methods described
below when their intermediate results are not desired. The overall algorithmic
complexity of this technique is O(N^3). According to several books, the
coefficient hidden by the 'O' is one of the best possible for general
(symmetric) matrixes.

($T, $Q) = $matrix->householder();

This method performs the Householder algorithm which reduces the n
by n real symmetric matrix M contained in $matrix to
tridiagonal form. On output, T is a symmetric tridiagonal matrix (only
diagonal and off-diagonal elements are non-zero) and Q is an
orthogonal matrix performing the tranformation between M and
T ($M == $Q * $T * ~$Q).

($l, $V) = $T->tri_diagonalize([$Q]);

This method diagonalizes the symmetric tridiagonal matrix T. On
output, $l and $V are similar to the output values described for
sym_diagonalize().

The optional argument $Q corresponds to an orthogonal transformation matrix
Q that should be used additionally during V (eigenvectors)
computation. It should be supplied if the desired eigenvectors correspond to a
more general symmetric matrix M previously reduced by the householder()
method, not a mere tridiagonal. If T is really a tridiagonal matrix,
Q can be omitted (it will be internally created in fact as an identity
matrix). The method uses a QL algorithm (with implicit shifts).

$l = $matrix->sym_eigenvalues();

This method computes the eigenvalues of the quadratic symmetric
matrix M stored in $matrix. On output, l is a column vector
containing all the eigenvalues of M. Eigenvectors are not computed (on
the contrary of sym_diagonalize()) and this method is more
efficient (even though it uses a similar algorithm with two phases). However,
understand that the algorithmic complexity of this technique is still also
O(N^3). But the coefficient hidden by the 'O' is better by a factor of...,
well, see your benchmark, it's wiser.

This routine wraps the householder_tridiagonal() and tri_eigenvalues()
methods described below when the intermediate tridiagonal matrix is not
needed.

$T = $matrix->householder_tridiagonal();

This method performs the Householder algorithm which reduces the n
by n real symmetric matrix M contained in $matrix to
tridiagonal form. On output, T is the obtained symmetric tridiagonal
matrix (only diagonal and off-diagonal elements are non-zero). The operation
is similar to the householder() method, but potentially a little more
efficient as the transformation matrix is not computed.

$l = $T->tri_eigenvalues();

This method computesthe eigenvalues of the symmetric tridiagonal matrix
T. On output, $l is a vector containing the eigenvalues (similar to
sym_eigenvalues()). This method is much more efficient than
tri_diagonalize() when eigenvectors are not needed.

Miscellaneous

$matrix->zero();

Assigns a zero to every element of the matrix "$matrix", i.e.,
erases all values previously stored there, thereby effectively transforming
the matrix into a "zero"-matrix or "null"-matrix, the neutral element of the
addition operation in a Ring.

(For instance the (quadratic) matrices with "n" rows and columns and matrix
addition and multiplication form a Ring. Most prominent characteristic of a
Ring is that multiplication is not commutative, i.e., in general,
"matrix1 * matrix2" is not the same as "matrix2 *
matrix1"!)

$matrix->one();

Assigns one's to the elements on the main diagonal (elements (1,1), (2,2),
(3,3) and so on) of matrix "$matrix" and zero's to all others,
thereby erasing all values previously stored there and transforming the matrix
into a "one"-matrix, the neutral element of the multiplication operation in a
Ring.

(If the matrix is quadratic (which this method doesn't require, though),
then multiplying this matrix with itself yields this same matrix again, and
multiplying it with some other matrix leaves that other matrix unchanged!)

$latex_string = $matrix->as_latex( align=> "c", format =>
"%s", name => "" );

This function returns the matrix as a LaTeX string. It takes a hash as an
argument which is used to control the style of the output. The hash element
align may be "c","l" or "r", corresponding to center, left and
right, respectively. The format element is a format string that
is given to sprintf to control the style of number format, such a
floating point or scientific notation. The name element can be
used so that a LaTeX string of "$name = " is prepended to the string.

Example:

my $a = Math::MatrixReal->new_from_cols([[ 1.234, 5.678, 9.1011],[1,2,3]] );
print $a->as_latex( ( format => "%.2f", align => "l",name => "A" ) );

Output: $A = $ $ \left( \begin{array}{ll} 1.23&1.00 \\ 5.68&2.00 \\
9.10&3.00 \end{array} \right) $

$yacas_string = $matrix->as_yacas( format => "%s", name =>
"", semi => 0 );

This function returns the matrix as a string that can be read by Yacas. It
takes a hash as an an argument which controls the style of the output. The
format element is a format string that is given to
sprintf to control the style of number format, such a floating
point or scientific notation. The name element can be used so
that "$name = " is prepended to the string. The <semi> element can be
set to 1 to that a semicolon is appended (so Matlab does not print out the
matrix.)

Example:

$a = Math::MatrixReal->new_from_cols([[ 1.234, 5.678, 9.1011],[1,2,3]] );
print $a->as_yacas( ( format => "%.2f", align => "l",name => "A" ) );

Output:

A := {{1.23,1.00},{5.68,2.00},{9.10,3.00}}

$matlab_string = $matrix->as_matlab( format => "%s", name
=> "", semi => 0 );

This function returns the matrix as a string that can be read by Matlab. It
takes a hash as an an argument which controls the style of the output. The
format element is a format string that is given to
sprintf to control the style of number format, such a floating
point or scientific notation. The name element can be used so
that "$name = " is prepended to the string. The <semi> element can be
set to 1 to that a semicolon is appended (so Matlab does not print out the
matrix.)

Example:

my $a = Math::MatrixReal->new_from_rows([[ 1.234, 5.678, 9.1011],[1,2,3]] );
print $a->as_matlab( ( format => "%.3f", name => "A",semi => 1 ) );

Output: A = [ 1.234 5.678 9.101; 1.000 2.000 3.000];

$scilab_string = $matrix->as_scilab( format => "%s", name
=> "", semi => 0 );

This function is just an alias for as_matlab(), since both
Scilab and Matlab have the same matrix format.

$minimum = Math::MatrixReal::min($number1,$number2);

Returns the minimum of the two numbers "number1" and
"number2".

$maximum = Math::MatrixReal::max($number1,$number2);

Returns the maximum of the two numbers "number1" and
"number2".

$minimal_cost_matrix = $cost_matrix->kleene();

Copies the matrix "$cost_matrix" (which has to be quadratic!)
to a new matrix of the same size (i.e., "clones" the input matrix) and applies
Kleene's algorithm to it.

See Math::Kleene(3) for
more details about this algorithm!

The method returns an object reference to the new matrix.

Matrix "$cost_matrix" is not changed by this method in any
way.

($norm_matrix,$norm_vector) = $matrix->normalize($vector);

This method is used to improve the numerical stability when solving linear
equation systems.

Suppose you have a matrix "A" and a vector "b" and you want to find out a
vector "x" so that A * x = b, i.e., the vector "x" which solves
the equation system represented by the matrix "A" and the vector "b".

Applying this method to the pair (A,b) yields a pair (A',b') where each row
has been divided by (the absolute value of) the greatest coefficient appearing
in that row. So this coefficient becomes equal to "1" (or "-1") in the new
pair (A',b') (all others become smaller than one and greater than minus
one).

Note that this operation does not change the equation system itself because
the same division is carried out on either side of the equation sign!

The method requires a quadratic (!) matrix "$matrix" and a
vector "$vector" for input (the vector must be a column vector
with the same number of rows as the input matrix) and returns a list of two
items which are object references to a new matrix and a new vector, in this
order.

The output matrix and vector are clones of the input matrix and vector to
which the operation explained above has been applied.

The input matrix and vector are not changed by this in any way.

Example of how this method can affect the result of the methods to solve
equation systems (explained immediately below following this method):

Consider the following little program:

#!perl -w

use Math::MatrixReal qw(new_from_string);

$A = Math::MatrixReal->new_from_string(<<"MATRIX");
[ 1 2 3 ]
[ 5 7 11 ]
[ 23 19 13 ]
MATRIX

$b = Math::MatrixReal->new_from_string(<<"MATRIX");
[ 0 ]
[ 1 ]
[ 29 ]
MATRIX

$LR = $A->decompose_LR();
if (($dim,$x,$B) = $LR->solve_LR($b))
{
$test = $A * $x;
print "x = \n$x";
print "A * x = \n$test";
}

($A_,$b_) = $A->normalize($b);

$LR = $A_->decompose_LR();
if (($dim,$x,$B) = $LR->solve_LR($b_))
{
$test = $A * $x;
print "x = \n$x";
print "A * x = \n$test";
}

This will print:

x =
[ 1.000000000000E+00 ]
[ 1.000000000000E+00 ]
[ -1.000000000000E+00 ]
A * x =
[ 4.440892098501E-16 ]
[ 1.000000000000E+00 ]
[ 2.900000000000E+01 ]
x =
[ 1.000000000000E+00 ]
[ 1.000000000000E+00 ]
[ -1.000000000000E+00 ]
A * x =
[ 0.000000000000E+00 ]
[ 1.000000000000E+00 ]
[ 2.900000000000E+01 ]

You can see that in the second example (where "normalize()" has been used),
the result is "better", i.e., more accurate!

$LR_matrix = $matrix->decompose_LR();

This method is needed to solve linear equation systems.

Suppose you have a matrix "A" and a vector "b" and you want to find out a
vector "x" so that A * x = b, i.e., the vector "x" which solves
the equation system represented by the matrix "A" and the vector "b".

You might also have a matrix "A" and a whole bunch of different vectors
"b1".."bk" for which you need to find vectors "x1".."xk" so that A * xi
= bi, for i=1..k.

Using Gaussian transformations (multiplying a row or column with a factor,
swapping two rows or two columns and adding a multiple of one row or column to
another), it is possible to decompose any matrix "A" into two triangular
matrices, called "L" and "R" (for "Left" and "Right").

"L" has one's on the main diagonal (the elements (1,1), (2,2), (3,3) and so
so), non-zero values to the left and below of the main diagonal and all zero's
in the upper right half of the matrix.

"R" has non-zero values on the main diagonal as well as to the right and
above of the main diagonal and all zero's in the lower left half of the
matrix, as follows:

[ 1 0 0 0 0 ] [ x x x x x ]
[ x 1 0 0 0 ] [ 0 x x x x ]
L = [ x x 1 0 0 ] R = [ 0 0 x x x ]
[ x x x 1 0 ] [ 0 0 0 x x ]
[ x x x x 1 ] [ 0 0 0 0 x ]

Note that "L * R" is equivalent to matrix "A" in the sense
that L * R * x = b <==> A * x = b for all vectors "x",
leaving out of account permutations of the rows and columns (these are taken
care of "magically" by this module!) and numerical errors.

Trick:

Because we know that "L" has one's on its main diagonal, we can store both
matrices together in the same array without information loss! I.e.,

[ R R R R R ]
[ L R R R R ]
LR = [ L L R R R ]
[ L L L R R ]
[ L L L L R ]

Beware, though, that "LR" and "L * R" are not the same!!!

Note also that for the same reason, you cannot apply the method
"normalize()" to an "LR" decomposition matrix. Trying to do so will yield
meaningless rubbish!

(You need to apply "normalize()" to each pair (Ai,bi) BEFORE
decomposing the matrix "Ai'"!)

Now what does all this help us in solving linear equation systems?

It helps us because a triangular matrix is the next best thing that can
happen to us besides a diagonal matrix (a matrix that has non-zero values only
on its main diagonal - in which case the solution is trivial, simply divide
"b[i]" by "A[i,i]" to get "x[i]"!).

To find the solution to our problem "A * x = b", we divide
this problem in parts: instead of solving A * x = b directly, we
first decompose "A" into "L" and "R" and then solve "L * y = b"
and finally "R * x = y" (motto: divide and rule!).

From the illustration above it is clear that solving "L * y =
b" and "R * x = y" is straightforward: we immediately know
that y[1] = b[1]. We then deduce swiftly that

y[2] = b[2] - L[2,1] * y[1]

(and we know "y[1]" by now!), that

y[3] = b[3] - L[3,1] * y[1] - L[3,2] * y[2]

and so on.

Having effortlessly calculated the vector "y", we now proceed to calculate
the vector "x" in a similar fashion: we see immediately that x[n] = y[n]
/ R[n,n]. It follows that

x[n-1] = ( y[n-1] - R[n-1,n] * x[n] ) / R[n-1,n-1]

and

x[n-2] = ( y[n-2] - R[n-2,n-1] * x[n-1] - R[n-2,n] * x[n] )
/ R[n-2,n-2]

and so on.

You can see that - especially when you have many vectors "b1".."bk" for
which you are searching solutions to A * xi = bi - this scheme is
much more efficient than a straightforward, "brute force" approach.

This method requires a quadratic matrix as its input matrix.

If you don't have that many equations, fill up with zero's (i.e., do
nothing to fill the superfluous rows if it's a "fresh" matrix, i.e., a matrix
that has been created with "new()" or "shadow()").

The method returns an object reference to a new matrix containing the
matrices "L" and "R".

The input matrix is not changed by this method in any way.

Note that you can "copy()" or "clone()" the result of this method without
losing its "magical" properties (for instance concerning the hidden
permutations of its rows and columns).

However, as soon as you are applying any method that alters the contents of
the matrix, its "magical" properties are stripped off, and the matrix
immediately reverts to an "ordinary" matrix (with the values it just happens
to contain at that moment, be they meaningful as an ordinary matrix or
not!).

($dimension,$x_vector,$base_matrix) =
$LR_matrix->solve_LR($b_vector);

Use this method to actually solve an equation system.

Matrix "$LR_matrix" must be a (quadratic) matrix returned by
the method "decompose_LR()", the LR decomposition matrix of the matrix "A" of
your equation system A * x = b.

The input vector "$b_vector" is the vector "b" in your
equation system A * x = b, which must be a column vector and have
the same number of rows as the input matrix "$LR_matrix".

The method returns a list of three items if a solution exists or an empty
list otherwise (!).

Therefore, you should always use this method like this:

if ( ($dim,$x_vec,$base) = $LR->solve_LR($b_vec) )
{
# do something with the solution...
}
else
{
# do something with the fact that there is no solution...
}

The three items returned are: the dimension "$dimension" of
the solution space (which is zero if only one solution exists, one if the
solution is a straight line, two if the solution is a plane, and so on), the
solution vector "$x_vector" (which is the vector "x" of your
equation system A * x = b) and a matrix
"$base_matrix" representing a base of the solution space (a set
of vectors which put up the solution space like the spokes of an
umbrella).

Only the first "$dimension" columns of this base matrix
actually contain entries, the remaining columns are all zero.

Now what is all this stuff with that "base" good for?

The output vector "x" is ALWAYS a solution of your equation system
A * x = b.

But also any vector "$vector"

$vector = $x_vector->clone();

$machine_infinity = 1E+99; # or something like that

for ( $i = 1; $i <= $dimension; $i++ )
{
$vector += rand($machine_infinity) * $base_matrix->column($i);
}

is a solution to your problem A * x = b, i.e., if
"$A_matrix" contains your matrix "A", then

print abs( $A_matrix * $vector - $b_vector ), "\n";

should print a number around 1E-16 or so!

By the way, note that you can actually calculate those vectors
"$vector" a little more efficient as follows:

$rand_vector = $x_vector->shadow();

$machine_infinity = 1E+99; # or something like that

for ( $i = 1; $i <= $dimension; $i++ )
{
$rand_vector->assign($i,1, rand($machine_infinity) );
}

$vector = $x_vector + ( $base_matrix * $rand_vector );

Note that the input matrix and vector are not changed by this method in any
way.

$inverse_matrix = $LR_matrix->invert_LR();

Use this method to calculate the inverse of a given matrix
"$LR_matrix", which must be a (quadratic) matrix returned by the
method "decompose_LR()".

The method returns an object reference to a new matrix of the same size as
the input matrix containing the inverse of the matrix that you initially fed
into "decompose_LR()" IF THE INVERSE EXISTS, or an empty list
otherwise.

Therefore, you should always use this method in the following way:

if ( $inverse_matrix = $LR->invert_LR() )
{
# do something with the inverse matrix...
}
else
{
# do something with the fact that there is no inverse matrix...
}

Note that by definition (disregarding numerical errors), the product of the
initial matrix and its inverse (or vice-versa) is always a matrix containing
one's on the main diagonal (elements (1,1), (2,2), (3,3) and so on) and zero's
elsewhere.

The input matrix is not changed by this method in any way.

$condition = $matrix->condition($inverse_matrix);

In fact this method is just a shortcut for

abs($matrix) * abs($inverse_matrix)

Both input matrices must be quadratic and have the same size, and the
result is meaningful only if one of them is the inverse of the other (for
instance, as returned by the method "invert_LR()").

The number returned is a measure of the "condition" of the given matrix
"$matrix", i.e., a measure of the numerical stability of the
matrix.

This number is always positive, and the smaller its value, the better the
condition of the matrix (the better the stability of all subsequent
computations carried out using this matrix).

Numerical stability means for example that if

abs( $vec_correct - $vec_with_error ) < $epsilon

holds, there must be a "$delta" which doesn't depend on the
vector "$vec_correct" (nor "$vec_with_error", by the
way) so that

abs( $matrix * $vec_correct - $matrix * $vec_with_error ) < $delta

also holds.

$determinant = $LR_matrix->det_LR();

Calculates the determinant of a matrix, whose LR decomposition matrix
"$LR_matrix" must be given (which must be a (quadratic) matrix
returned by the method "decompose_LR()").

In fact the determinant is a by-product of the LR decomposition: It is (in
principle, that is, except for the sign) simply the product of the elements on
the main diagonal (elements (1,1), (2,2), (3,3) and so on) of the LR
decomposition matrix.

(The sign is taken care of "magically" by this module)

$order = $LR_matrix->order_LR();

Calculates the order (called "Rang" in German) of a matrix, whose LR
decomposition matrix "$LR_matrix" must be given (which must be a
(quadratic) matrix returned by the method "decompose_LR()").

This number is a measure of the number of linear independent row and column
vectors (= number of linear independent equations in the case of a matrix
representing an equation system) of the matrix that was initially fed into
"decompose_LR()".

If "n" is the number of rows and columns of the (quadratic!) matrix, then
"n - order" is the dimension of the solution space of the associated equation
system.

$rank = $LR_matrix->rank_LR();

This is an alias for the order_LR() function. The "order" is
usually called the "rank" in the United States.

$scalar_product = $vector1->scalar_product($vector2);

Returns the scalar product of vector "$vector1" and vector
"$vector2".

Both vectors must be column vectors (i.e., a matrix having several rows but
only one column).

This is a (more efficient!) shortcut for

$temp = ~$vector1 * $vector2;
$scalar_product = $temp->element(1,1);

or the sum i=1..n of the products vector1[i] *
vector2[i].

Provided none of the two input vectors is the null vector, then the two
vectors are orthogonal, i.e., have an angle of 90 degrees between them,
exactly when their scalar product is zero, and vice-versa.

$vector_product = $vector1->vector_product($vector2);

Returns the vector product of vector "$vector1" and vector
"$vector2".

Both vectors must be column vectors (i.e., a matrix having several rows but
only one column).

Currently, the vector product is only defined for 3 dimensions (i.e.,
vectors with 3 rows); all other vectors trigger an error message.

In 3 dimensions, the vector product of two vectors "x" and "y" is defined
as

| x[1] y[1] e[1] |
determinant | x[2] y[2] e[2] |
| x[3] y[3] e[3] |

where the "x[i]" and "y[i]" are the components of
the two vectors "x" and "y", respectively, and the "e[i]" are
unity vectors (i.e., vectors with a length equal to one) with a one in row "i"
and zero's elsewhere (this means that you have numbers and vectors as elements
in this matrix!).

This determinant evaluates to the rather simple formula

z[1] = x[2] * y[3] - x[3] * y[2]
z[2] = x[3] * y[1] - x[1] * y[3]
z[3] = x[1] * y[2] - x[2] * y[1]

A characteristic property of the vector product is that the resulting
vector is orthogonal to both of the input vectors (if neither of both is the
null vector, otherwise this is trivial), i.e., the scalar product of each of
the input vectors with the resulting vector is always zero.

$length = $vector->length();

This is actually a shortcut for

$length = sqrt( $vector->scalar_product($vector) );

and returns the length of a given (column!) vector
"$vector".

Note that the "length" calculated by this method is in fact the "two"-norm
of a vector "$vector"!

The general definition for norms of vectors is the following:

sub vector_norm
{
croak "Usage: \$norm = \$vector->vector_norm(\$n);"
if (@_ != 2);

my($vector,$n) = @_;
my($rows,$cols) = ($vector->[1],$vector->[2]);
my($k,$comp,$sum);

croak "Math::MatrixReal::vector_norm(): vector is not a column vector"
unless ($cols == 1);

croak "Math::MatrixReal::vector_norm(): norm index must be > 0"
unless ($n > 0);

croak "Math::MatrixReal::vector_norm(): norm index must be integer"
unless ($n == int($n));

$sum = 0;
for ( $k = 0; $k < $rows; $k++ )
{
$comp = abs( $vector->[0][$k][0] );
$sum += $comp ** $n;
}
return( $sum ** (1 / $n) );
}

Note that the case "n = 1" is the "one"-norm for matrices applied to a
vector, the case "n = 2" is the euclidian norm or length of a vector, and if
"n" goes to infinity, you have the "infinity"- or "maximum"-norm for matrices
applied to a vector!

$xn_vector =
$matrix->solve_GSM($x0_vector,$b_vector,$epsilon);

$xn_vector =
$matrix->solve_SSM($x0_vector,$b_vector,$epsilon);

$xn_vector =
$matrix->solve_RM($x0_vector,$b_vector,$weight,$epsilon);

In some cases it might not be practical or desirable to solve an equation
system "A * x = b" using an analytical algorithm like the
"decompose_LR()" and "solve_LR()" method pair.

In fact in some cases, due to the numerical properties (the "condition") of
the matrix "A", the numerical error of the obtained result can be greater than
by using an approximative (iterative) algorithm like one of the three
implemented here.

All three methods, GSM ("Global Step Method" or "Gesamtschrittverfahren"),
SSM ("Single Step Method" or "Einzelschrittverfahren") and RM ("Relaxation
Method" or "Relaxationsverfahren"), are fix-point iterations, that is, can be
described by an iteration function "x(t+1) = Phi( x(t) )" which
has the property:

Phi(x) = x <==> A * x = b

We can define "Phi(x)" as follows:

Phi(x) := ( En - A ) * x + b

where "En" is a matrix of the same size as "A" ("n" rows and columns) with
one's on its main diagonal and zero's elsewhere.

This function has the required property.

Proof:

A * x = b

<==> -( A * x ) = -b

<==> -( A * x ) + x = -b + x

<==> -( A * x ) + x + b = x

<==> x - ( A * x ) + b = x

<==> ( En - A ) * x + b = x

This last step is true because

x[i] - ( a[i,1] x[1] + ... + a[i,i] x[i] + ... + a[i,n] x[n] ) + b[i]

is the same as

( -a[i,1] x[1] + ... + (1 - a[i,i]) x[i] + ... + -a[i,n] x[n] ) + b[i]

qed

Note that actually solving the equation system "A * x = b"
means to calculate

a[i,1] x[1] + ... + a[i,i] x[i] + ... + a[i,n] x[n] = b[i]

<==> a[i,i] x[i] =
b[i]
- ( a[i,1] x[1] + ... + a[i,i] x[i] + ... + a[i,n] x[n] )
+ a[i,i] x[i]

<==> x[i] =
( b[i]
- ( a[i,1] x[1] + ... + a[i,i] x[i] + ... + a[i,n] x[n] )
+ a[i,i] x[i]
) / a[i,i]

<==> x[i] =
( b[i] -
( a[i,1] x[1] + ... + a[i,i-1] x[i-1] +
a[i,i+1] x[i+1] + ... + a[i,n] x[n] )
) / a[i,i]

There is one major restriction, though: a fix-point iteration is guaranteed
to converge only if the first derivative of the iteration function has an
absolute value less than one in an area around the point "x(*)"
for which "Phi( x(*) ) = x(*)" is to be true, and if the start
vector "x(0)" lies within that area!

This is best verified grafically, which unfortunately is impossible to do
in this textual documentation!

See literature on Numerical Analysis for details!

In our case, this restriction translates to the following three
conditions:

There must exist a norm so that the norm of the matrix of the iteration
function, ( En - A ), has a value less than one, the matrix "A"
may not have any zero value on its main diagonal and the initial vector
"x(0)" must be "good enough", i.e., "close enough" to the
solution "x(*)".

(Remember school math: the first derivative of a straight line given by
"y = a * x + b" is "a"!)

The three methods expect a (quadratic!) matrix "$matrix" as
their first argument, a start vector "$x0_vector", a vector
"$b_vector" (which is the vector "b" in your equation system
"A * x = b"), in the case of the "Relaxation Method" ("RM"), a
real number "$weight" best between zero and two, and finally an
error limit (real number) "$epsilon".

(Note that the weight "$weight" used by the "Relaxation
Method" ("RM") is NOT checked to lie within any reasonable range!)

The three methods first test the first two conditions of the three
conditions listed above and return an empty list if these conditions are not
fulfilled.

Therefore, you should always test their return value using some code
like:

if ( $xn_vector = $A_matrix->solve_GSM($x0_vector,$b_vector,1E-12) )
{
# do something with the solution...
}
else
{
# do something with the fact that there is no solution...
}

Otherwise, they iterate until abs( Phi(x) - x ) <
epsilon.

(Beware that theoretically, infinite loops might result if the starting
vector is too far "off" the solution! In practice, this shouldn't be a
problem. Anyway, you can always press <ctrl-C> if you think that the
iteration takes too long!)

The difference between the three methods is the following:

In the "Global Step Method" ("GSM"), the new vector "x(t+1)"
(called "y" here) is calculated from the vector "x(t)" (called
"x" here) according to the formula:

y[i] =
( b[i]
- ( a[i,1] x[1] + ... + a[i,i-1] x[i-1] +
a[i,i+1] x[i+1] + ... + a[i,n] x[n] )
) / a[i,i]

In the "Single Step Method" ("SSM"), the components of the vector
"x(t+1)" which have already been calculated are used to calculate
the remaining components, i.e.

y[i] =
( b[i]
- ( a[i,1] y[1] + ... + a[i,i-1] y[i-1] + # note the "y[]"!
a[i,i+1] x[i+1] + ... + a[i,n] x[n] ) # note the "x[]"!
) / a[i,i]

In the "Relaxation method" ("RM"), the components of the vector
"x(t+1)" are calculated by "mixing" old and new value (like cold
and hot water), and the weight "$weight" determines the
"aperture" of both the "hot water tap" as well as of the "cold water tap",
according to the formula:

y[i] =
( b[i]
- ( a[i,1] y[1] + ... + a[i,i-1] y[i-1] + # note the "y[]"!
a[i,i+1] x[i+1] + ... + a[i,n] x[n] ) # note the "x[]"!
) / a[i,i]
y[i] = weight * y[i] + (1 - weight) * x[i]

Note that the weight "$weight" should be greater than zero and
less than two (!).

The three methods are supposed to be of different efficiency.
Experiment!

Remember that in most cases, it is probably advantageous to first
"normalize()" your equation system prior to solving it!

OVERLOADED OPERATORS

SYNOPSIS

Unary operators:

"-", "~", "abs", test,
"!", '""'

Binary (arithmetic) operators:

"+", "-", "*", "**",
"+=", "-=", "*=",
"**="

Binary (relational) operators:

"==", "!=", "<",
"<=", ">", ">="

"eq", "ne", "lt", "le",
"gt", "ge"

Note that the latter ("eq", "ne", ... ) are just
synonyms of the former ("==", "!=", ... ), defined
for convenience only.

DESCRIPTION

'-'

Unary minus

Returns the negative of the given matrix, i.e., the matrix with all
elements multiplied with the factor "-1".

Example:

$matrix = -$matrix;

'~'

Transposition

Returns the transposed of the given matrix.

Examples:

$temp = ~$vector * $vector;
$length = sqrt( $temp->element(1,1) );

if (~$matrix == $matrix) { # matrix is symmetric ... }

abs

Norm

Returns the "one"-Norm of the given matrix.

Example:

$error = abs( $A * $x - $b );

test

Boolean test

Tests wether there is at least one non-zero element in the matrix.

Example:

if ($xn_vector) { # result of iteration is not zero ... }

'!'

Negated boolean test

Tests wether the matrix contains only zero's.

Examples:

if (! $b_vector) { # heterogenous equation system ... }
else { # homogenous equation system ... }

unless ($x_vector) { # not the null-vector! }

'""""'

"Stringify" operator

Converts the given matrix into a string.

Uses scientific representation to keep precision loss to a minimum in case
you want to read this string back in again later with "new_from_string()".

Uses a 13-digit mantissa and a 20-character field for each element so that
lines will wrap nicely on an 80-column screen.

Examples:

$matrix = Math::MatrixReal->new_from_string(<<"MATRIX");
[ 1 0 ]
[ 0 -1 ]
MATRIX
print "$matrix";

[ 1.000000000000E+00 0.000000000000E+00 ]
[ 0.000000000000E+00 -1.000000000000E+00 ]

$string = "$matrix";
$test = Math::MatrixReal->new_from_string($string);
if ($test == $matrix) { print ":-)\n"; } else { print ":-(\n"; }

'+'

Addition

Returns the sum of the two given matrices.

Examples:

$matrix_S = $matrix_A + $matrix_B;

$matrix_A += $matrix_B;

'-'

Subtraction

Returns the difference of the two given matrices.

Examples:

$matrix_D = $matrix_A - $matrix_B;

$matrix_A -= $matrix_B;

Note that this is the same as:

$matrix_S = $matrix_A + -$matrix_B;

$matrix_A += -$matrix_B;

(The latter are less efficient, though)

'*'

Multiplication

Returns the matrix product of the two given matrices or the product of the
given matrix and scalar factor.

Examples:

$matrix_P = $matrix_A * $matrix_B;

$matrix_A *= $matrix_B;

$vector_b = $matrix_A * $vector_x;

$matrix_B = -1 * $matrix_A;

$matrix_B = $matrix_A * -1;

$matrix_A *= -1;

'**'

Exponentiation

Returns the matrix raised to an integer power. If 0 is passed, the identity
matrix is returned. If a negative integer is passed, it computes the inverse
(if it exists) and then raised the inverse to the absolute value of the
integer. The matrix must be quadratic.

Examples:

$matrix2 = $matrix ** 2;

$matrix **= 2;

$inv2 = $matrix ** -2;

$ident = $matrix ** 0;

'=='

Equality

Tests two matrices for equality.

Example:

if ( $A * $x == $b ) { print "EUREKA!\n"; }

Note that in most cases, due to numerical errors (due to the finite
precision of computer arithmetics), it is a bad idea to compare two matrices
or vectors this way.

Better use the norm of the difference of the two matrices you want to
compare and compare that norm with a small number, like this:

if ( abs( $A * $x - $b ) < 1E-12 ) { print "BINGO!\n"; }

'!='

Inequality

Tests two matrices for inequality.

Example:

while ($x0_vector != $xn_vector) { # proceed with iteration ... }

(Stops when the iteration becomes stationary)

Note that (just like with the '==' operator), it is usually a bad idea to
compare matrices or vectors this way. Compare the norm of the difference of
the two matrices with a small number instead.

'<'

Less than

Examples:

if ( $matrix1 < $matrix2 ) { # ... }

if ( $vector < $epsilon ) { # ... }

if ( 1E-12 < $vector ) { # ... }

if ( $A * $x - $b < 1E-12 ) { # ... }

These are just shortcuts for saying:

if ( abs($matrix1) < abs($matrix2) ) { # ... }

if ( abs($vector) < abs($epsilon) ) { # ... }

if ( abs(1E-12) < abs($vector) ) { # ... }

if ( abs( $A * $x - $b ) < abs(1E-12) ) { # ... }

Uses the "one"-norm for matrices and Perl's built-in "abs()" for
scalars.

'<='

Less than or equal

As with the '<' operator, this is just a shortcut for the same
expression with "abs()" around all arguments.

Example:

if ( $A * $x - $b <= 1E-12 ) { # ... }

which in fact is the same as:

if ( abs( $A * $x - $b ) <= abs(1E-12) ) { # ... }

Uses the "one"-norm for matrices and Perl's built-in "abs()" for
scalars.

'>'

Greater than

As with the '<' and '<=' operator, this

if ( $xn - $x0 > 1E-12 ) { # ... }

is just a shortcut for:

if ( abs( $xn - $x0 ) > abs(1E-12) ) { # ... }

Uses the "one"-norm for matrices and Perl's built-in "abs()" for
scalars.

'>='

Greater than or equal

As with the '<', '<=' and '>' operator, the following

if ( $LR >= $A ) { # ... }

is simply a shortcut for:

if ( abs($LR) >= abs($A) ) { # ... }

Uses the "one"-norm for matrices and Perl's built-in "abs()" for
scalars.

SEE ALSO

Math::VectorReal,
Math::PARI, Math::MatrixBool,
DFA::Kleene, Math::Kleene, Set::IntegerRange,
Set::IntegerFast
.

VERSION

This man page documents Math::MatrixReal version 1.9.

The latest version can always be found at http://leto.net/code/Math-MatrixReal/

AUTHORS

Steffen Beyer <[email-masked]>, Rodolphe Ortalo
<[email-masked]>, Jonathan Leto <[email-masked]>.

Currently maintained by Jonathan Leto, send all bugs/patches to me.

CREDITS

Many thanks to Prof. Pahlings for stoking the fire of my enthusiasm for
Algebra and Linear Algebra at the university (RWTH Aachen, Germany), and to
Prof. Esser and his assistant, Mr. Jarausch, for their fascinating lectures in
Numerical Analysis!

COPYRIGHT

Copyright (c) 1996-2002 by Steffen Beyer, Rodolphe Ortalo, Jonathan Leto. All
rights reserved.

LICENSE AGREEMENT

This package is free software; you can redistribute it and/or modify it under
the same terms as Perl itself.

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