Author image Lutz Gehlen
and 1 contributors

NAME

Math::Matrix::Banded - non-zero entries confined to a diagonal band

VERSION

Version 0.004

SYNOPSIS

    use Math::Matrix::Banded;

    my $matrix = Math::Matrix::Banded->new(N => 7);
    $matrix->element(0, 0, 5);
    $matrix->element(4, 5, -1);
    # ...

    $matrix->decompose_LU;
    my $x = $matrix->solve_LU([0, 2, 4, 5, 3, 6, -2]);

DESCRIPTION

A banded matrix (or band matrix or band diagonal matrix) is a matrix whose non-zero columns are confined to a diagonal band. Obviously, such a structure allows for more efficient storage and computation than for a general matrix.

CAVEAT: This is a young distribution. It currently focuses on the functionality that I need myself. If you miss certain features, please let me know and I will consider prioritizing them for future releases.

General Remarks

  • Row and column indices start at 0. This might require extra attention since in most mathematics texts, they start at 1.

  • In order to exploit the bandedness of your matrix, you must not set off-band elements even if they are 0. The element method adjusts the band structure of the stored matrix no matter what the value is.

  • Currently, no operators are overloaded. This will probably change in the future in order to at least use * in the natural way.

Constructor Dispatch

This distribution supports both square and non-square matrices. They a stored in different data structures and offer different functionality and are therefore implemented in separate classes. The constructors of Math::Matrix::Banded dispatch to constructors of Math::Matrix::Banded::Square and Math::Matrix::Banded::Rectangular, respectively.

Square Matrices

An NxN matrix A = (a_ij) is a band matrix with lower bandwidth m_below and upper bandwidth m_above if a_ij = 0 for all j < i - m_below and all j > i + m_above. (If you phrase it like this then it might seem more appropriate to speak of left and right bandwidths instead of lower and upper, but the latter are the accepted terms).

According to the above definition, every matrix can be considered as a banded matrix, and this distribution does indeed support arbitrary matrices. However, it only makes sense to treat a matrix as banded if m_below and m_above are much smaller than N.

Symmetric Matrices

Obviously, a symmetric banded matrix can be stored using even less space than a generic banded matrix. Be aware that this potential is not exploited by the current implementation. If you set the symmetric flag (see below) this only leads to that a_ji is set whenever you set a_ij. This might change in a future version.

Non-square matrices

For non-square matrices it is much less well defined in the literature what a banded matrix is. In this distribution it means that each row contains a non-zero block of entries and that start and end of that block are non-decreasing from top to bottom of the matrix. For example, this is a banded rectangular matrix:

    1 2 3 x x x x
    4 5 6 x x x x
    x 7 8 x x x x
    x x 9 1 x x x

Entries are stored as explicit zeroes to enforce this structure:

    1 2 3 x x x x
    4 5 6 x x x x
    x 7 0 x x x x
    x x 9 1 x x x

The 0 at (2, 2) is stored even if it is not set explicitly.

CONSTRUCTORS

new

    $matrix = Math::Matrix::Banded->new(N => 7);
    $matrix = Math::Matrix::Banded->new(
        M => 7,
        N => 11,
    );

The constructor dispatches to the constructor of a square matrix iff N is specified and M is not specified, otherwise it dispatches to the constructor of a rectangular matrix.

Parameters:

N

The number of columns of the matrix. In case of a square matrix, this is also the number of rows. This parameters is mandatory.

M

The number of rows of the matrix. This parameters is mandatory in the case of a non-square matrix.

symmetric

Optional parameter for a square matrix to indicate that the matrix is symmetric. Defaults to 0. If it is set to a true value then a_ji is set whenever you set a_ij.

m_below

Optional parameter to initialize the lower bandwidth of a square matrix. It is adapted as needed when elements are set.

m_above

Optional parameter to initialize the upper bandwidth of a square matrix. It is adapted as needed when elements are set.

ATTRIBUTES

Square Matrices

N

Readonly attribute storing the number of rows and columns of the matrix.

m_below

Readonly attribute storing the lower bandwith of the matrix. It is adapted as needed when elements are set.

m_above

Readonly attribute storing the upper bandwith of the matrix. It is adapted as needed when elements are set.

symmetric

Readonly attribute storing whether the matrix was created as symmetric.

L

Readonly attribute storing the lower diagonal part of the LU decomposition of the matrix.

In the current implementation, the LU decomposition is calculated automatically if this attribute is accessed and is not set. However, this is considered experimental. It is recommended to call decompose_LU explicitly before accessing this attribute.

U

Readonly attribute storing the upper diagonal part of the LU decomposition of the matrix.

In the current implementation, the LU decomposition is calculated automatically if this attribute is accessed and is not set. However, this is considered experimental. It is recommended to call decompose_LU explicitly before accessing this attribute.

Rectangular Matrices

M

Readonly attribute storing the number of rows of the matrix.

N

Readonly attribute storing the number of columns of the matrix.

METHODS

Methods common to square and rectangular Matrices

element

    $matrix->element(0, 1, -5);
    $a = $matrix->element(4, 0);

Sets/gets an element of the matrix. Row and column indices start at 0. The band structure of the matrix is maintained automatically when setting elements.

row

    $row = $matrix->row(0);

Returns the specified row as an array reference. The array is a copy, modifying it will not affect the matrix. This method is mostly meant for debugging purposes because the returned row includes all the vanishing elements that you typically want to skip by using a banded matrix.

column

    $column = $matrix->column(0);

Returns the specified column as an array reference. The array is a copy, modifying it will not affect the matrix. This method is mostly meant for debugging purposes because the returned column includes all the vanishing elements that you typically want to skip by using a banded matrix.

as_string

Returns a string representation of the matrix including all elements. This is for debugging purposes and the exact format might change in the future.

multiply_vector

    $w = $A->multiply_vector([1, 2, 3]);

Expects a vector v as an array reference with N components and returns Av as an array reference.

Square Matrices

multiply_matrix

    $C = $A->multiply_matrix($B);

Expects another banded square matrix B of the same size and returns AB. The input matrices remain unchanged. The lower (upper) bandwidth of the resulting matrix is the sum of the lower (upper) bandwidths of A and B.

CAVEAT: Obviously, it would make mathematical sense to provide a rectangular matrix with N rows and get a rectangular matrix back. However, this is currently not implemented.

decompose_LU

    $success = $A->decompose_LU

Computes the LU decomposition of the matrix without pivoting. Returns 1 on success and undef on failure.

CAVEAT: Be aware that LU decomposition can be numerically unstable or even fail even if the matrix is non-singular. It is only safe to use this method if you know that LU decomposition without pivoting can be used safely on your matrix (e.g. if your matrix is strictly diagonally dominant).

CAVEAT: The LU decomposition is not reset if you change the matrix afterwards by calling the element method. I do not want to call the clearer on every call of element. It is recommended to set all elements before doing anything else. If you cannot ensure that then make sure to call decompose_LU explicitly before using the LU decompositon.

Remark: Straight-forward pivoting destroys the band structure on a matrix, hence it is more complicated to implement than usual. Partial pivoting might be added in a future version of this module.

solve_LU

    my $x = $A->solve_LU($y);

Expects a vector y as an array reference with N components. Uses the LU decomposition of A to determine the solution of the equation system Ax = y. The LU decomposition is computed if that has not happened before. Returns x as array reference or undef if no solution could be computed. See also decompose_LU.

Non-square Matrices

transpose

    $At = $A->transpose

Returns the transposed matrix.

AAt

    $AAt = $A->AAt

Returns the square symmetric NxN matrix that results from multiplying A with its own transpose.

AtA

    $AtA = $A->AtA

Returns the square symmetric MxM matrix that results from multiplying the transpose of A with A. It is identical to

    $AtA = $A->transpose->AAt

AUTHOR

Lutz Gehlen, <perl at lutzgehlen.de>

BUGS

Please report any bugs or feature requests to bug-math-matrix-banded at rt.cpan.org, or through the web interface at https://rt.cpan.org/NoAuth/ReportBug.html?Queue=Math-Matrix-Banded. I will be notified, and then you'll automatically be notified of progress on your bug as I make changes.

LICENSE AND COPYRIGHT

This software is Copyright (c) 2020 by Lutz Gehlen.

This is free software, licensed under:

  The Artistic License 2.0 (GPL Compatible)