# NAME

Math::GSL::Matrix - Mathematical functions concerning Matrices

# SYNOPSIS

``````    use Math::GSL::Matrix qw/:all/;
my \$matrix1 = Math::GSL::Matrix->new(5,5);  # OO interface
my \$matrix2 = \$matrix1 + 4;                 # You can add or substract values or matrices to OO matrices
my \$matrix3 = \$matrix1 - 4;
my \$matrix4 = \$matrix2 + \$matrix1;
my \$matrix5 = \$matrix2 . \$matrix1;          # This is a scalar product, it simply multiply each element
# with the element of \$matrix1 that have the same position
# See Math::GSL::BLAS if you want scalar product

my \$matrix6 = \$matrix2 . 8;                 # Multiply every elements of \$matrix2 by 8
my \$matrix7 = \$matrix2 * \$matrix1;          # scalar product of two matrices
if(\$matrix1 == \$matrix4) ...
if(\$matrix1 != \$matrix3) ...
my \$matrix8 = gsl_matrix_alloc(5,5);        # standard interface``````

# DESCRIPTION

This module is part of the Math::GSL distribution. It defines a Perl insterface to GNU Scientific Library matrices.

There are two different (but not exclusive) ways to use this module: using the OO API, built manually over the GSL functions, or using directly the functions defined by GSL library.

# OBJECT ORIENTED API

## Constructor

### Math::GSL::Matrix->new()

Creates a new Matrix of the given size.

``    my \$matrix = Math::GSL::Matrix->new(10,10);``

If by any chance you already have a gsl_matrix and want to "objectify" it, you can use the following constructor:

``````    my \$m = gsl_matrix_alloc(10, 20);
# ... something
my \$matrix = Math::GSL::Matrix->new(\$m);``````

Note that `\$m` is NOT copied. The new object will just refer to the old gsl_matrix.

## Getters

### raw()

Get the underlying GSL matrix object created by SWIG, useful for using gsl_matrix_* functions which do not have an OO counterpart.

``````    my \$matrix     = Math::GSL::Matrix->new(3,3);
my \$gsl_matrix = \$matrix->raw;
my \$stuff      = gsl_matrix_get(\$gsl_matrix, 1, 2);``````

### dim()

Returns the number of rows and columns in a matrix as an array.

``   my (\$rows, \$cols) = \$matrix->dim;``

Basically a shortcut to `rows` and `cols` methods.

### rows()

Returns the number of rows in the matrix.

``    my \$rows = \$matrix->rows;``

### cols()

Returns the number of columns in the matrix.

``    my \$cols = \$matrix->cols;``

### as_list()

Get the contents of a Math::GSL::Matrix object as a Perl list.

``````    my \$matrix = Math::GSL::Matrix->new(3,3);
...
my @matrix = \$matrix->as_list;``````

### as_vector()

Returns a 1xN or Nx1 matrix as a Math::GSL::Vector object. Dies if called on a matrix that is not a single row or column. Useful for turning the output of `col()` or `row()` into a vector, like so:

``````    my \$vector1 = \$matrix->col(0)->as_vector;
my \$vector2 = \$matrix->row(1)->as_vector;``````

### get_elem()

Returns an element of a matrix.

``````    my \$matrix = Math::GSL::Matrix->new(3,3);
...
my \$middle = \$matrix->get_elem(1,1);``````

NOTE: just like any other method on this module, rows and arrays start with indice 0.

### row()

Returns a row matrix of the row you enter.

``````    my \$matrix = Math::GSL::Matrix->new(3,3);
...
my \$matrix_row = \$matrix->row(0);``````

### col()

Returns a col matrix of the column you enter.

``````    my \$matrix = Math::GSL::Matrix->new(3,3);
...
my \$matrix_col = \$matrix->col(0);``````

### max()

Computes the maximum value of a matrix. In array context returns the maximum value and the position of that element in the matrix. If the matrix is a vector (being it vertical or horizontal) only one coordinate is returned: the position of the element in the vector.

``````    \$max = \$matrix->max();
(\$max, \$i, \$j) = \$matrix->max();``````

### min()

Computes the minimum value of a matrix. In array context returns the minimum value and the position of that element in the matrix. If the matrix is a vector (being it vertical or horizontal) only one coordinate is returned: the position of the element in the vector.

``````    \$min = \$matrix->min();
(\$min, \$i, \$j) = \$matrix->min();``````

## Setters

### identity()

Set a matrix to the identity matrix, i.e. one on the diagonal and zero elsewhere.

``    my \$I = \$matrix->identity;``

### zero()

Set a matrix to the zero matrix.

``    \$matrix->zero;``

### set_elem()

Sets a specific value in the matrix.

``````    my \$matrix = Math::GSL::Matrix->new(2,2);
\$matrix->set_elem(0, 0, \$value);``````

You can set multiple elements at once with chained calls:

``    \$matrix->set_elem(0,0,1)->set_elem(1,1,1);``

NOTE: just like any other method on this module, rows and arrays start with indice 0.

### set_row()

Sets a the values of a row with the elements of an array.

``````    my \$matrix = Math::GSL::Matrix->new(3,3);
\$matrix->set_row(0, [8, 6, 2]);``````

You can also set multiple rows at once with chained calls:

``````    my \$matrix = Math::GSL::Matrix->new(3,3);
\$matrix->set_row(0, [8, 6, 2])
->set_row(1, [2, 4, 1]);
...``````

### set_col()

Sets a the values of a column with the elements of an array.

``````    my \$matrix = Math::GSL::Matrix->new(3,3);
\$matrix->set_col(0, [8, 6, 2]);``````

You can also set multiple columns at once with chained calls: my \$matrix = Math::GSL::Matrix->new(3,3); \$matrix->set_col(0, [8, 6, 2]) ->set_col(1, [2, 4, 1]); ...

## Utility Functions

### copy()

Returns a copy of the matrix, which has the same size and values but resides at a different location in memory.

``````    my \$matrix = Math::GSL::Matrix->new(5,5);
my \$copy   = \$matrix->copy;``````

### is_square()

Returns true if a matrix is square, i.e. it has the same number of rows as columns, false otherwise.

### det()

Returns the determinant of a matrix (computed by LU decomposition) or dies if called on a non-square matrix.

``    my \$det = \$matrix->det();``

### write

Saves ONE matrix into a file using the GSL binary format.

``     \$matrix->save("matrix.dat");``

Note that this method always overwrite the file if it exists. In case more than one GSL object will be written to the file, please use the `gsl_fopen`, `gsl_matrix_fwrite` and `gsl_fclose` manually. Future versions might include helper for this task.

NOTE: in order to allow the read of one of these files without needing to know in advance the size of the matrix, an horizontal matrix with two elements is saved in the beginning of the file with the number of rows and columns of the saved matrix.

Loads a matrix from a GSL binary file (saved using the `save` method, as it stores some extra information on the file).

``    my \$matrix = Math::GSL::Matrix->read("matrix.dat");``

### lndet()

Returns the natural log of the absolute value of the determinant of a matrix (computed by LU decomposition) or dies if called on a non-square matrix.

``    my \$lndet = \$matrix->lndet();``

### inverse()

Returns the inverse of a matrix or dies when called on a non-square matrix.

``    my \$inverse = \$matrix->inverse;``

### transpose()

Returns the transpose of a matrix or dies when called on a non-square matrix.

``    my \$transposed = \$matrix->transpose;``

### eigenvalues()

Computes the a matrix eigen values.

### ieach()

Applied a function to each element of a matrix.

``````    # compute exp to each element of matrix
\$matrix->ieach( sub { exp(shift) });``````

This method changes the original matrix. See `each` for a non destructive method.

### each()

Applied a function to each element of a matrix.

``````    # compute exp to each element of matrix
\$matrix->each( sub { exp(shift) });``````

This method returns a new matrix.

### vconcat ()

Concatenates vertically a new matrix with the object matrix, where the object matrix is above in the final matrix. Note that both matrices need to have the same number of columns. The method returns a new matrix.

``````    my \$final = \$m1->vconcat(\$m2);
# \$final is \$m1 on top of \$m2``````

### hconcat ()

Concatenates horizontally a new matrix with the object matrix, where the object matrix is at the left in the final matrix. Note that both matrices need to have the same number of rows. The method returns a new matrix.

``````    my \$final = \$m1->hconcat(\$m2);
# \$final is \$m1 at the left of \$m2``````

### random ()

Fills a matrix with random values between 0 and 1

# GSL FUNCTION INTERFACE

Here is a list of all the functions included in this module :

`gsl_matrix_alloc(\$i, \$j)` - Return a gsl_matrix of \$i rows and \$j columns
`gsl_matrix_calloc(\$i, \$j)` - Return a gsl_matrix of \$i rows and \$j columns and initialize all of the elements of the matrix to zero
`gsl_matrix_alloc_from_block` -
`gsl_matrix_free` -
`gsl_matrix_alloc_from_matrix ` -
`gsl_vector_alloc_row_from_matrix` -
`gsl_vector_alloc_col_from_matrix ` -
`gsl_matrix_submatrix(\$m, \$k1, \$k2, \$n1, \$n2)` - Return a matrix view of the matrix \$m. The upper-left element of the submatrix is the element (\$k1,\$k2) of the original matrix. The submatrix has \$n1 rows and \$n2 columns.
`gsl_matrix_row(\$m , \$i)` - Return a vector view of the \$i-th row of the matrix \$m
`gsl_matrix_column(\$m, \$j)` - Return a vector view of the \$j-th column of the matrix \$m
`gsl_matrix_diagonal(\$m)` - Return a vector view of the diagonal of the vector. The matrix doesn't have to be square.
`gsl_matrix_subdiagonal(\$m, \$k)` - Return a vector view of the \$k-th subdiagonal of the matrix \$m. The diagonal of the matrix corresponds to k=0.
`gsl_matrix_superdiagonal(\$m, \$k)` - Return a vector view of the \$k-th superdiagonal of the matrix \$m. The matrix doesn't have to be square.
`gsl_matrix_subrow(\$m, \$i, \$offset, \$n)` - Return a vector view of the \$i-th row of the matrix \$m beginning at offset elements and containing n elements.
`gsl_matrix_subcolumn(\$m, \$j, \$offset, \$n)` - Return a vector view of the \$j-th column of the matrix \$m beginning at offset elements and containing n elements.
`gsl_matrix_view_array(\$base, \$n1, \$n2)` - This function returns a matrix view of the array reference \$base. The matrix has \$n1 rows and \$n2 columns. The physical number of columns in memory is also given by \$n2. Mathematically, the (i,j)-th element of the new matrix is given by, m'(i,j) = \$base->[i*\$n2 + j] where the index i runs from 0 to \$n1-1 and the index j runs from 0 to \$n2-1. The new matrix is only a view of the array reference \$base. When the view goes out of scope the original array reference \$base will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use.
`gsl_matrix_view_array_with_tda(\$base, \$n1, \$n2, \$tda)` - This function returns a matrix view of the array reference \$base with a physical number of columns \$tda which may differ from the corresponding dimension of the matrix. The matrix has \$n1 rows and \$n2 columns, and the physical number of columns in memory is given by \$tda. Mathematically, the (i,j)-th element of the new matrix is given by, m'(i,j) = \$base->[i*\$tda + j] where the index i runs from 0 to \$n1-1 and the index j runs from 0 to \$n2-1. The new matrix is only a view of the array reference \$base. When the view goes out of scope the original array reference \$base will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use.
`gsl_matrix_view_vector` -
`gsl_matrix_view_vector_with_tda` -
`gsl_matrix_const_submatrix` -
`gsl_matrix_get(\$m, \$i, \$j)` - Return the (i,j)-th element of the matrix \$m
`gsl_matrix_set(\$m, \$i, \$j, \$x)` - Set the value of the (i,j)-th element of the matrix \$m to \$x
`gsl_matrix_ptr` -
`gsl_matrix_const_ptr` -
`gsl_matrix_set_zero(\$m)` - Set all the elements of the matrix \$m to zero
`gsl_matrix_set_identity(\$m)` - Set the elements of the matrix \$m to the corresponding elements of the identity matrix
`gsl_matrix_set_all(\$m, \$x)` - Set all the elements of the matrix \$m to the value \$x
`gsl_matrix_fread(\$fh, \$m)` - Read a file which has been written with gsl_matrix_fwrite from the stream \$fh opened with the gsl_fopen function from the Math::GSL module and stores the data inside the matrix \$m
`gsl_matrix_fwrite(\$fh, \$m)` - Write the elements of the matrix \$m in binary format to a stream \$fh opened with the gsl_fopen function from the Math::GSL module
`gsl_matrix_fscanf(\$fh, \$m)` - Read a file which has been written with gsl_matrix_fprintf from the stream \$fh opened with the gsl_fopenfunction from the Math::GSL module and stores the data inside the matrix \$m
`gsl_matrix_fprintf(\$fh, \$m, \$format)` - Write the elements of the matrix \$m in the format \$format (for example "%f" is the format for double) to a stream \$fh opened with the gsl_fopen function from the Math::GSL module
`gsl_matrix_memcpy(\$dest, \$src)` - Copy the elements of the matrix \$src to the matrix \$dest. The two matrices must have the same size.
`gsl_matrix_swap(\$m1, \$m2)` - Exchange the elements of the matrices \$m1 and \$m2 by copying. The two matrices must have the same size.
`gsl_matrix_swap_rows(\$m, \$i, \$j)` - Exchange the \$i-th and \$j-th row of the matrix \$m. The function returns 0 if the operation suceeded, 1 otherwise.
`gsl_matrix_swap_columns(\$m, \$i, \$j)` - Exchange the \$i-th and \$j-th column of the matrix \$m. The function returns 0 if the operation suceeded, 1 otherwise.
`gsl_matrix_swap_rowcol(\$m, \$i, \$j)` - Exchange the \$i-th row and the \$j-th column of the matrix \$m. The matrix must be square. The function returns 0 if the operation suceeded, 1 otherwise.
`gsl_matrix_transpose(\$m)` - This function replaces the matrix m by its transpose by copying the elements of the matrix in-place. The matrix must be square for this operation to be possible.
`gsl_matrix_transpose_memcpy(\$dest, \$src)` - Make the matrix \$dest the transpose of the matrix \$src. This function works for all matrices provided that the dimensions of the matrix dest match the transposed dimensions of the matrix src.
`gsl_matrix_max(\$m)` - Return the maximum value in the matrix \$m
`gsl_matrix_min(\$m)` - Return the minimum value in the matrix \$m
`gsl_matrix_minmax(\$m)` - Return two values, the first is the minimum value of the Matrix \$m and the second is the maximum of the same the same matrix.
`gsl_matrix_max_index(\$m)` - Return two values, the first is the the i indice of the maximum value of the matrix \$m and the second is the j indice of the same value.
`gsl_matrix_min_index(\$m)` - Return two values, the first is the the i indice of the minimum value of the matrix \$m and the second is the j indice of the same value.
`gsl_matrix_minmax_index(\$m)` - Return four values, the first is the i indice of the minimum of the matrix \$m, the second is the j indice of the same value, the third is the i indice of the maximum of the matrix \$m and the fourth is the j indice of the same value
`gsl_matrix_isnull(\$m)` - Return 1 if all the elements of the matrix \$m are zero, 0 otherwise
`gsl_matrix_ispos(\$m)` - Return 1 if all the elements of the matrix \$m are strictly positve, 0 otherwise
`gsl_matrix_isneg(\$m)` - Return 1 if all the elements of the matrix \$m are strictly negative, 0 otherwise
`gsl_matrix_isnonneg(\$m)` - Return 1 if all the elements of the matrix \$m are non-negatuive, 0 otherwise
`gsl_matrix_add(\$a, \$b)` - Add the elements of matrix \$b to the elements of matrix \$a
`gsl_matrix_sub(\$a, \$b)` - Subtract the elements of matrix \$b from the elements of matrix \$a
`gsl_matrix_mul_elements(\$a, \$b)` - Multiplie the elements of matrix \$a by the elements of matrix \$b
`gsl_matrix_div_elements(\$a, \$b)` - Divide the elements of matrix \$a by the elements of matrix \$b
`gsl_matrix_scale(\$a, \$x)` - Multiplie the elements of matrix \$a by the constant factor \$x
`gsl_matrix_add_constant(\$a, \$x)` - Add the constant value \$x to the elements of the matrix \$a
`gsl_matrix_add_diagonal(\$a, \$x)` - Add the constant value \$x to the elements of the diagonal of the matrix \$a
`gsl_matrix_get_row(\$v, \$m, \$i)` - Copy the elements of the \$i-th row of the matrix \$m into the vector \$v. The lenght of the vector must be of the same as the lenght of the row. The function returns 0 if it succeded, 1 otherwise.
`gsl_matrix_get_col(\$v, \$m, \$i)` - Copy the elements of the \$j-th column of the matrix \$m into the vector \$v. The lenght of the vector must be of the same as the lenght of the column. The function returns 0 if it succeded, 1 otherwise.
`gsl_matrix_set_row(\$m, \$i, \$v)` - Copy the elements of vector \$v into the \$i-th row of the matrix \$m The lenght of the vector must be of the same as the lenght of the row. The function returns 0 if it succeded, 1 otherwise.
`gsl_matrix_set_col(\$m, \$j, \$v)` - Copy the elements of vector \$v into the \$j-th row of the matrix \$m The lenght of the vector must be of the same as the lenght of the column. The function returns 0 if it succeded, 1 otherwise.

These are related to constant views of a matrix.

`gsl_matrix_const_row`
`gsl_matrix_const_column`
`gsl_matrix_const_diagonal`
`gsl_matrix_const_subdiagonal`
`gsl_matrix_const_superdiagonal`
`gsl_matrix_const_subrow`
`gsl_matrix_const_subcolumn`
`gsl_matrix_const_view_array`
`gsl_matrix_const_view_array_with_tda`

The following functions are similar to those above but work with `char`'s and `int`'s. We are not quite sure if anyone wants these. Please speak up if you do and/or submit some patches to this documentation, please!

gsl_matrix_const_view_vector
gsl_matrix_const_view_vector_with_tda
gsl_matrix_char_alloc
gsl_matrix_char_calloc
gsl_matrix_char_alloc_from_block
gsl_matrix_char_alloc_from_matrix
gsl_vector_char_alloc_row_from_matrix
gsl_vector_char_alloc_col_from_matrix
gsl_matrix_char_free
gsl_matrix_char_submatrix
gsl_matrix_char_row
gsl_matrix_char_column
gsl_matrix_char_diagonal
gsl_matrix_char_subdiagonal
gsl_matrix_char_superdiagonal
gsl_matrix_char_subrow
gsl_matrix_char_subcolumn
gsl_matrix_char_view_array
gsl_matrix_char_view_array_with_tda
gsl_matrix_char_view_vector
gsl_matrix_char_view_vector_with_tda
gsl_matrix_char_const_submatrix
gsl_matrix_char_const_row
gsl_matrix_char_const_column
gsl_matrix_char_const_diagonal
gsl_matrix_char_const_subdiagonal
gsl_matrix_char_const_superdiagonal
gsl_matrix_char_const_subrow
gsl_matrix_char_const_subcolumn
gsl_matrix_char_const_view_array
gsl_matrix_char_const_view_array_with_tda
gsl_matrix_char_const_view_vector
gsl_matrix_char_const_view_vector_with_tda
gsl_matrix_char_get
gsl_matrix_char_set
gsl_matrix_char_ptr
gsl_matrix_char_const_ptr
gsl_matrix_char_set_zero
gsl_matrix_char_set_identity
gsl_matrix_char_set_all
gsl_matrix_char_fwrite
gsl_matrix_char_fscanf
gsl_matrix_char_fprintf
gsl_matrix_char_memcpy
gsl_matrix_char_swap
gsl_matrix_char_swap_rows
gsl_matrix_char_swap_columns
gsl_matrix_char_swap_rowcol
gsl_matrix_char_transpose
gsl_matrix_char_transpose_memcpy
gsl_matrix_char_max
gsl_matrix_char_min
gsl_matrix_char_minmax
gsl_matrix_char_max_index
gsl_matrix_char_min_index
gsl_matrix_char_minmax_index
gsl_matrix_char_isnull
gsl_matrix_char_ispos
gsl_matrix_char_isneg
gsl_matrix_char_isnonneg
gsl_matrix_char_sub
gsl_matrix_char_mul_elements
gsl_matrix_char_div_elements
gsl_matrix_char_scale
gsl_matrix_int_alloc
gsl_matrix_int_calloc
gsl_matrix_int_alloc_from_block
gsl_matrix_int_alloc_from_matrix
gsl_vector_int_alloc_row_from_matrix
gsl_vector_int_alloc_col_from_matrix
gsl_matrix_int_free
gsl_matrix_int_submatrix
gsl_matrix_int_row
gsl_matrix_int_column
gsl_matrix_int_diagonal
gsl_matrix_int_subdiagonal
gsl_matrix_int_superdiagonal
gsl_matrix_int_subrow
gsl_matrix_int_subcolumn
gsl_matrix_int_view_array
gsl_matrix_int_view_array_with_tda
gsl_matrix_int_view_vector
gsl_matrix_int_view_vector_with_tda
gsl_matrix_int_const_submatrix
gsl_matrix_int_const_row
gsl_matrix_int_const_column
gsl_matrix_int_ptr
gsl_matrix_int_const_ptr
gsl_matrix_int_set_zero
gsl_matrix_int_set_identity
gsl_matrix_int_set_all
gsl_matrix_int_fwrite
gsl_matrix_int_fscanf
gsl_matrix_int_fprintf
gsl_matrix_int_memcpy
gsl_matrix_int_swap
gsl_matrix_int_swap_rows
gsl_matrix_int_swap_columns
gsl_matrix_int_swap_rowcol
gsl_matrix_int_transpose
gsl_matrix_int_transpose_memcpy
gsl_matrix_int_max
gsl_matrix_int_min
gsl_matrix_int_minmax
gsl_matrix_int_max_index
gsl_matrix_int_min_index
gsl_matrix_int_minmax_index
gsl_matrix_int_isnull
gsl_matrix_int_ispos
gsl_matrix_int_isneg
gsl_matrix_int_isnonneg
gsl_matrix_int_sub
gsl_matrix_int_mul_elements
gsl_matrix_int_div_elements
gsl_matrix_int_scale

You have to add the functions you want to use inside the qw /put_funtion_here /. You can also write use Math::GSL::Matrix qw/:all/ to use all avaible functions of the module. Other tags are also avaible, here is a complete list of all tags for this module :

`all`
`int`
`double`
`char`
`complex`

For more informations on the functions, we refer you to the GSL offcial documentation http://www.gnu.org/software/gsl/manual/html_node/

# EXAMPLES

`````` Most of the examples from this section are perl versions of the examples at L<http://www.gnu.org/software/gsl/manual/html_node/Example-programs-for-matrices.html>

The program below shows how to allocate, initialize and read from a matrix using the functions gsl_matrix_alloc, gsl_matrix_set and gsl_matrix_get.

use Math::GSL::Matrix qw/:all/;
my \$m = gsl_matrix_alloc (10,3);
for my \$i (0..9){
for my \$j (0..2){
gsl_matrix_set(\$m, \$i, \$j, 0.23 + 100*\$i + \$j);
}
}

for my \$i (0..99){ # OUT OF RANGE ERROR
for my \$j (0..2){
print "m(\$i, \$j) = " . gsl_matrix_get (\$m, \$i, \$j) . "\n";
}
}
gsl_matrix_free (\$m);

use Math::GSL::Matrix qw/:all/;

my \$m = gsl_matrix_alloc (100, 100);
my \$a = gsl_matrix_alloc (100, 100);

for my \$i (0..99){
for my \$j (0..99){
gsl_matrix_set (\$m, \$i, \$j, 0.23 + \$i + \$j);
}
}

The next program shows how to write a matrix to a file.

my \$out = gsl_fopen("test.dat", "wb");
gsl_matrix_fwrite (\$out, \$m);
gsl_fclose (\$out);

my \$in = gsl_fopen("test.dat", "rb");
gsl_fclose(\$in);

my \$k=0;
for my \$i (0..99){
for my \$j (0..99){
\$mij = gsl_matrix_get (\$m, \$i, \$j);
\$aij = gsl_matrix_get (\$a, \$i, \$j);
\$k++ if (\$mij != \$aij);
}
}

gsl_matrix_free(\$m);
gsl_matrix_free(\$a);

print "differences = \$k (should be zero)\n";``````

# AUTHORS

Jonathan "Duke" Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>