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C<< $matrix1->matrixor($matrix2) >>

Finds and returns the matrix whose ($i,$j) element is 
 $matrix1($i,$j) || $matrix2($i,$j).  
The returned matrix is a pattern matrix.


=back

=head2 PROBABILISTIC METHODS

=over 4

=item *

C<< $matrix->normalize()  >>

Returns a matrix $matrix2 scaled so that $matrix2->sum()==1.  

Exits on error if $matrix is not nonnegative.

=item *

C<< $matrix->normalizerows()  >>

As $matrix->normalize(), except that each row is scaled to have a 
sum of 1. 

Exits on error if $matrix is not nonnegative.

=item *

C<< $matrix->normalizecolumns()  >>

As $matrix->normalizerows(), except with columns. Mathematically equivalent to 
$matrix->transpose()->normalizerows()->transpose(). 

Exits on error if $matrix is not nonnegative.

=item *

C<< $matrix->discretepdf() >>

Assuming that $matrix->sum()==1 and $matrix is non-negative, chooses a
random element from $matrix based on the assumption that
the entry at ($i,$j) is the probability of choosing ($i,$j).

=back 

=head2 SOLUTION OF SYSTEMS METHODS

=over 4


=item * 

C<< $matrix->jacobi($constant,$guess, $tol,  $steps) >>

Uses Jacobi iteration to find and return the solution to the
system of equations $matrix * x = $constant, with initial guess
$constant, tolerance $tol,  and maximum iterations $steps.

If $steps is undefined, the default value of 100 is used.

Care should be taken to ensure that $matrix is such that the 
iteration actually converges.

=item * 

C<< $matrix->gaussseidel($constant,$guess,  $tol,  $steps) >>

Uses Gauss-Seidel iteration to find and return the solution to the
system of equations $matrix * x = $constant, with initial guess
$constant, tolerance $tol, and maximum
iterations $steps. This is equivalent to $matrix->SOR with 
relaxation parameter 1.  

Care should be taken to ensure that $matrix is such that the 
iteration actually converges.  

=item * 

C<< $matrix->SOR($constant,$guess, $relax, $tol,  $steps) >>

Uses Successive Over-Relaxation to find and return the solution to the
system of equations $matrix * x = $constant, with initial guess
$constant, relaxation parameter $relax, tolerance $tol, and maximum
iterations $steps.

Care should be taken to ensure that $matrix is such that the 
iteration actually converges.  

=back

=head2 SORTING METHODS

=over 4

=item * 

C<< $matrix->sortbycolumn() >>

C<< $matrix->sortbyrow() >>

Returns an array containing the keys of $matrix, sorted primarily
by either column or row (and secondarily by row or column).  

C<< $matrix->sortbydiagonal() >>

Returns an array containing the keys of $matrix, sorted primarily by
their distance from elements on the main diagonal, lower diagonals
first.  The row of the key is a secondary criterion.

C<< $matrix->sortbyvalue() >>

Returns an array containing the keys of $matrix, sorted primarily by
the value of the element indexed by the key.  Row and column are
secondary and tertiaty criteria. 

These methods are suitable for using inside a loop.  See also 
$matrix->elements() if the order of the elements is not important.

Example:

A 3x3 matrix will be sorted in the following manners:

 sortbycolumn()
 1 4 7 
 2 5 8
 3 6 9

 sortbyrow()
 1 2 3
 4 5 6
 7 8 9

 sortbydiagonal()
 4 7 9
 2 5 8
 1 3 6

=back

=head2 IN-LINE METHODS

The following are as described above, except that they modify their
calling object instead of a copy thereof.