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* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
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package org.apache.commons.math3.linear;
/**
* Calculates the rectangular Cholesky decomposition of a matrix.
* <p>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<sup>T</sup>, depending
* on a user-defined tolerance. In a sense, this is the square root of A.</p>
* <p>The difference with respect to the regular {@link 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 {@link
* org.apache.commons.math3.random.CorrelatedRandomVectorGenerator 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.</p>
* <p>Rectangular Cholesky decomposition is <em>not</em> suited for solving
* linear systems, so it does not provide any {@link DecompositionSolver
* decomposition solver}.</p>
*
* @see <a href="http://mathworld.wolfram.com/CholeskyDecomposition.html">MathWorld</a>
* @see <a href="http://en.wikipedia.org/wiki/Cholesky_decomposition">Wikipedia</a>
* @since 2.0 (changed to concrete class in 3.0)
*/
public class RectangularCholeskyDecomposition {
/** Permutated Cholesky root of the symmetric positive semidefinite matrix. */
private final RealMatrix root;
/** Rank of the symmetric positive semidefinite matrix. */
private int rank;
/**
* Decompose a symmetric positive semidefinite matrix.
* <p>
* <b>Note:</b> this constructor follows the linpack method to detect dependent
* columns by proceeding with the Cholesky algorithm until a nonpositive diagonal
* element is encountered.
*
* @see <a href="http://eprints.ma.man.ac.uk/1193/01/covered/MIMS_ep2008_56.pdf">
* Analysis of the Cholesky Decomposition of a Semi-definite Matrix</a>
*
* @param matrix Symmetric positive semidefinite matrix.
* @exception NonPositiveDefiniteMatrixException if the matrix is not
* positive semidefinite.
* @since 3.1
*/
public RectangularCholeskyDecomposition(RealMatrix matrix)
throws NonPositiveDefiniteMatrixException {
this(matrix, 0);
}
/**
* Decompose a symmetric positive semidefinite matrix.
*
* @param matrix Symmetric positive semidefinite matrix.
* @param small Diagonal elements threshold under which columns are
* considered to be dependent on previous ones and are discarded.
* @exception NonPositiveDefiniteMatrixException if the matrix is not
* positive semidefinite.
*/
public RectangularCholeskyDecomposition(RealMatrix matrix, double small)
throws NonPositiveDefiniteMatrixException {
final int order = matrix.getRowDimension();
final double[][] c = matrix.getData();
final double[][] b = new double[order][order];
int[] index = new int[order];
for (int i = 0; i < order; ++i) {
index[i] = i;
}
int r = 0;
for (boolean loop = true; loop;) {
// find maximal diagonal element
int swapR = r;
for (int i = r + 1; i < order; ++i) {
int ii = index[i];
int isr = index[swapR];
if (c[ii][ii] > c[isr][isr]) {
swapR = i;
}
}
// swap elements
if (swapR != r) {
final int tmpIndex = index[r];
index[r] = index[swapR];
index[swapR] = tmpIndex;
final double[] tmpRow = b[r];
b[r] = b[swapR];
b[swapR] = tmpRow;
}
// check diagonal element
int ir = index[r];
if (c[ir][ir] <= small) {
if (r == 0) {
throw new NonPositiveDefiniteMatrixException(c[ir][ir], ir, small);
}
// check remaining diagonal elements
for (int i = r; i < order; ++i) {
if (c[index[i]][index[i]] < -small) {
// there is at least one sufficiently negative diagonal element,
// the symmetric positive semidefinite matrix is wrong
throw new NonPositiveDefiniteMatrixException(c[index[i]][index[i]], i, small);
}
}
// all remaining diagonal elements are close to zero, we consider we have
// found the rank of the symmetric positive semidefinite matrix
loop = false;
} else {
// transform the matrix
final double sqrt = Math.sqrt(c[ir][ir]);
b[r][r] = sqrt;
final double inverse = 1 / sqrt;
final double inverse2 = 1 / c[ir][ir];
for (int i = r + 1; i < order; ++i) {
final int ii = index[i];
final double e = inverse * c[ii][ir];
b[i][r] = e;
c[ii][ii] -= c[ii][ir] * c[ii][ir] * inverse2;
for (int j = r + 1; j < i; ++j) {
final int ij = index[j];
final double f = c[ii][ij] - e * b[j][r];
c[ii][ij] = f;
c[ij][ii] = f;
}
}
// prepare next iteration
loop = ++r < order;
}
}
// build the root matrix
rank = r;
root = MatrixUtils.createRealMatrix(order, r);
for (int i = 0; i < order; ++i) {
for (int j = 0; j < r; ++j) {
root.setEntry(index[i], j, b[i][j]);
}
}
}
/** Get the root of the covariance matrix.
* The root is the rectangular matrix <code>B</code> such that
* the covariance matrix is equal to <code>B.B<sup>T</sup></code>
* @return root of the square matrix
* @see #getRank()
*/
public RealMatrix getRootMatrix() {
return root;
}
/** 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.
* @return r of the square matrix.
* @see #getRootMatrix()
*/
public int getRank() {
return rank;
}
}