org.apache.commons.math3.linear
Class RectangularCholeskyDecomposition

java.lang.Object
  org.apache.commons.math3.linear.RectangularCholeskyDecomposition

public class RectangularCholeskyDecomposition
extends Object

Calculates the rectangular Cholesky decomposition of a matrix.

The rectangular Cholesky decomposition of a real symmetric positive semidefinite matrix A consists of a rectangular matrix B with the same number of rows such that: A is almost equal to BB^T, depending on a user-defined tolerance. In a sense, this is the square root of A.

The difference with respect to the regular CholeskyDecomposition is that rows/columns may be permuted (hence the rectangular shape instead of the traditional triangular shape) and there is a threshold to ignore small diagonal elements. This is used for example to generate correlated random n-dimensions vectors in a p-dimension subspace (p < n). In other words, it allows generating random vectors from a covariance matrix that is only positive semidefinite, and not positive definite.

Rectangular Cholesky decomposition is not suited for solving linear systems, so it does not provide any decomposition solver.

Since:

2.0 (changed to concrete class in 3.0)

Version:

$Id: RectangularCholeskyDecomposition.java 7721 2013-02-14 14:07:13Z CardosoP $

See Also:

MathWorld, Wikipedia

Constructor Summary

RectangularCholeskyDecomposition(RealMatrix matrix)
Decompose a symmetric positive semidefinite matrix.

RectangularCholeskyDecomposition(RealMatrix matrix, double small)
Decompose a symmetric positive semidefinite matrix.

Method Summary

int	getRank() Get the rank of the symmetric positive semidefinite matrix.
RealMatrix	getRootMatrix() Get the root of the covariance matrix.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

RectangularCholeskyDecomposition

public RectangularCholeskyDecomposition(RealMatrix matrix)
                                 throws NonPositiveDefiniteMatrixException

Decompose a symmetric positive semidefinite matrix.

Note: this constructor follows the linpack method to detect dependent columns by proceeding with the Cholesky algorithm until a nonpositive diagonal element is encountered.

Parameters:

matrix - Symmetric positive semidefinite matrix.

Throws:

NonPositiveDefiniteMatrixException - if the matrix is not positive semidefinite.

Since:

3.1

See Also:

Analysis of the Cholesky Decomposition of a Semi-definite Matrix

RectangularCholeskyDecomposition

public RectangularCholeskyDecomposition(RealMatrix matrix,
                                        double small)
                                 throws NonPositiveDefiniteMatrixException

Decompose a symmetric positive semidefinite matrix.

Parameters:

matrix - Symmetric positive semidefinite matrix.

small - Diagonal elements threshold under which columns are considered to be dependent on previous ones and are discarded.

Throws:

NonPositiveDefiniteMatrixException - if the matrix is not positive semidefinite.

Method Detail

getRootMatrix

public RealMatrix getRootMatrix()

Get the root of the covariance matrix. The root is the rectangular matrix B such that the covariance matrix is equal to B.BT

Returns:

root of the square matrix

See Also:

getRank()

getRank

public int getRank()

Get the rank of the symmetric positive semidefinite matrix. The r is the number of independent rows in the symmetric positive semidefinite matrix, it is also the number of columns of the rectangular matrix of the decomposition.

Returns:

r of the square matrix.

See Also:

getRootMatrix()