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* 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
* limitations under the License.
*/
package org.apache.commons.math3.linear;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.util.FastMath;
/**
* Calculates the LUP-decomposition of a square matrix.
* <p>The LUP-decomposition of a matrix A consists of three matrices L, U and
* P that satisfy: P×A = L×U. L is lower triangular (with unit
* diagonal terms), U is upper triangular and P is a permutation matrix. All
* matrices are m×m.</p>
* <p>As shown by the presence of the P matrix, this decomposition is
* implemented using partial pivoting.</p>
* <p>This class is based on the class with similar name from the
* <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library.</p>
* <ul>
* <li>a {@link #getP() getP} method has been added,</li>
* <li>the {@code det} method has been renamed as {@link #getDeterminant()
* getDeterminant},</li>
* <li>the {@code getDoublePivot} method has been removed (but the int based
* {@link #getPivot() getPivot} method has been kept),</li>
* <li>the {@code solve} and {@code isNonSingular} methods have been replaced
* by a {@link #getSolver() getSolver} method and the equivalent methods
* provided by the returned {@link DecompositionSolver}.</li>
* </ul>
*
* @see <a href="http://mathworld.wolfram.com/LUDecomposition.html">MathWorld</a>
* @see <a href="http://en.wikipedia.org/wiki/LU_decomposition">Wikipedia</a>
* @since 2.0 (changed to concrete class in 3.0)
*/
public class LUDecomposition {
/** Default bound to determine effective singularity in LU decomposition. */
private static final double DEFAULT_TOO_SMALL = 1e-11;
/** Entries of LU decomposition. */
private final double[][] lu;
/** Pivot permutation associated with LU decomposition. */
private final int[] pivot;
/** Parity of the permutation associated with the LU decomposition. */
private boolean even;
/** Singularity indicator. */
private boolean singular;
/** Cached value of L. */
private RealMatrix cachedL;
/** Cached value of U. */
private RealMatrix cachedU;
/** Cached value of P. */
private RealMatrix cachedP;
/**
* Calculates the LU-decomposition of the given matrix.
* This constructor uses 1e-11 as default value for the singularity
* threshold.
*
* @param matrix Matrix to decompose.
* @throws NonSquareMatrixException if matrix is not square.
*/
public LUDecomposition(RealMatrix matrix) {
this(matrix, DEFAULT_TOO_SMALL);
}
/**
* Calculates the LU-decomposition of the given matrix.
* @param matrix The matrix to decompose.
* @param singularityThreshold threshold (based on partial row norm)
* under which a matrix is considered singular
* @throws NonSquareMatrixException if matrix is not square
*/
public LUDecomposition(RealMatrix matrix, double singularityThreshold) {
if (!matrix.isSquare()) {
throw new NonSquareMatrixException(matrix.getRowDimension(),
matrix.getColumnDimension());
}
final int m = matrix.getColumnDimension();
lu = matrix.getData();
pivot = new int[m];
cachedL = null;
cachedU = null;
cachedP = null;
// Initialize permutation array and parity
for (int row = 0; row < m; row++) {
pivot[row] = row;
}
even = true;
singular = false;
// Loop over columns
for (int col = 0; col < m; col++) {
// upper
for (int row = 0; row < col; row++) {
final double[] luRow = lu[row];
double sum = luRow[col];
for (int i = 0; i < row; i++) {
sum -= luRow[i] * lu[i][col];
}
luRow[col] = sum;
}
// lower
int max = col; // permutation row
double largest = Double.NEGATIVE_INFINITY;
for (int row = col; row < m; row++) {
final double[] luRow = lu[row];
double sum = luRow[col];
for (int i = 0; i < col; i++) {
sum -= luRow[i] * lu[i][col];
}
luRow[col] = sum;
// maintain best permutation choice
if (FastMath.abs(sum) > largest) {
largest = FastMath.abs(sum);
max = row;
}
}
// Singularity check
if (FastMath.abs(lu[max][col]) < singularityThreshold) {
singular = true;
return;
}
// Pivot if necessary
if (max != col) {
double tmp = 0;
final double[] luMax = lu[max];
final double[] luCol = lu[col];
for (int i = 0; i < m; i++) {
tmp = luMax[i];
luMax[i] = luCol[i];
luCol[i] = tmp;
}
int temp = pivot[max];
pivot[max] = pivot[col];
pivot[col] = temp;
even = !even;
}
// Divide the lower elements by the "winning" diagonal elt.
final double luDiag = lu[col][col];
for (int row = col + 1; row < m; row++) {
lu[row][col] /= luDiag;
}
}
}
/**
* Returns the matrix L of the decomposition.
* <p>L is a lower-triangular matrix</p>
* @return the L matrix (or null if decomposed matrix is singular)
*/
public RealMatrix getL() {
if ((cachedL == null) && !singular) {
final int m = pivot.length;
cachedL = MatrixUtils.createRealMatrix(m, m);
for (int i = 0; i < m; ++i) {
final double[] luI = lu[i];
for (int j = 0; j < i; ++j) {
cachedL.setEntry(i, j, luI[j]);
}
cachedL.setEntry(i, i, 1.0);
}
}
return cachedL;
}
/**
* Returns the matrix U of the decomposition.
* <p>U is an upper-triangular matrix</p>
* @return the U matrix (or null if decomposed matrix is singular)
*/
public RealMatrix getU() {
if ((cachedU == null) && !singular) {
final int m = pivot.length;
cachedU = MatrixUtils.createRealMatrix(m, m);
for (int i = 0; i < m; ++i) {
final double[] luI = lu[i];
for (int j = i; j < m; ++j) {
cachedU.setEntry(i, j, luI[j]);
}
}
}
return cachedU;
}
/**
* Returns the P rows permutation matrix.
* <p>P is a sparse matrix with exactly one element set to 1.0 in
* each row and each column, all other elements being set to 0.0.</p>
* <p>The positions of the 1 elements are given by the {@link #getPivot()
* pivot permutation vector}.</p>
* @return the P rows permutation matrix (or null if decomposed matrix is singular)
* @see #getPivot()
*/
public RealMatrix getP() {
if ((cachedP == null) && !singular) {
final int m = pivot.length;
cachedP = MatrixUtils.createRealMatrix(m, m);
for (int i = 0; i < m; ++i) {
cachedP.setEntry(i, pivot[i], 1.0);
}
}
return cachedP;
}
/**
* Returns the pivot permutation vector.
* @return the pivot permutation vector
* @see #getP()
*/
public int[] getPivot() {
return pivot.clone();
}
/**
* Return the determinant of the matrix
* @return determinant of the matrix
*/
public double getDeterminant() {
if (singular) {
return 0;
} else {
final int m = pivot.length;
double determinant = even ? 1 : -1;
for (int i = 0; i < m; i++) {
determinant *= lu[i][i];
}
return determinant;
}
}
/**
* Get a solver for finding the A × X = B solution in exact linear
* sense.
* @return a solver
*/
public DecompositionSolver getSolver() {
return new Solver(lu, pivot, singular);
}
/** Specialized solver. */
private static class Solver implements DecompositionSolver {
/** Entries of LU decomposition. */
private final double[][] lu;
/** Pivot permutation associated with LU decomposition. */
private final int[] pivot;
/** Singularity indicator. */
private final boolean singular;
/**
* Build a solver from decomposed matrix.
* @param lu entries of LU decomposition
* @param pivot pivot permutation associated with LU decomposition
* @param singular singularity indicator
*/
private Solver(final double[][] lu, final int[] pivot, final boolean singular) {
this.lu = lu;
this.pivot = pivot;
this.singular = singular;
}
/** {@inheritDoc} */
public boolean isNonSingular() {
return !singular;
}
/** {@inheritDoc} */
public RealVector solve(RealVector b) {
final int m = pivot.length;
if (b.getDimension() != m) {
throw new DimensionMismatchException(b.getDimension(), m);
}
if (singular) {
throw new SingularMatrixException();
}
final double[] bp = new double[m];
// Apply permutations to b
for (int row = 0; row < m; row++) {
bp[row] = b.getEntry(pivot[row]);
}
// Solve LY = b
for (int col = 0; col < m; col++) {
final double bpCol = bp[col];
for (int i = col + 1; i < m; i++) {
bp[i] -= bpCol * lu[i][col];
}
}
// Solve UX = Y
for (int col = m - 1; col >= 0; col--) {
bp[col] /= lu[col][col];
final double bpCol = bp[col];
for (int i = 0; i < col; i++) {
bp[i] -= bpCol * lu[i][col];
}
}
return new ArrayRealVector(bp, false);
}
/** {@inheritDoc} */
public RealMatrix solve(RealMatrix b) {
final int m = pivot.length;
if (b.getRowDimension() != m) {
throw new DimensionMismatchException(b.getRowDimension(), m);
}
if (singular) {
throw new SingularMatrixException();
}
final int nColB = b.getColumnDimension();
// Apply permutations to b
final double[][] bp = new double[m][nColB];
for (int row = 0; row < m; row++) {
final double[] bpRow = bp[row];
final int pRow = pivot[row];
for (int col = 0; col < nColB; col++) {
bpRow[col] = b.getEntry(pRow, col);
}
}
// Solve LY = b
for (int col = 0; col < m; col++) {
final double[] bpCol = bp[col];
for (int i = col + 1; i < m; i++) {
final double[] bpI = bp[i];
final double luICol = lu[i][col];
for (int j = 0; j < nColB; j++) {
bpI[j] -= bpCol[j] * luICol;
}
}
}
// Solve UX = Y
for (int col = m - 1; col >= 0; col--) {
final double[] bpCol = bp[col];
final double luDiag = lu[col][col];
for (int j = 0; j < nColB; j++) {
bpCol[j] /= luDiag;
}
for (int i = 0; i < col; i++) {
final double[] bpI = bp[i];
final double luICol = lu[i][col];
for (int j = 0; j < nColB; j++) {
bpI[j] -= bpCol[j] * luICol;
}
}
}
return new Array2DRowRealMatrix(bp, false);
}
/**
* Get the inverse of the decomposed matrix.
*
* @return the inverse matrix.
* @throws SingularMatrixException if the decomposed matrix is singular.
*/
public RealMatrix getInverse() {
return solve(MatrixUtils.createRealIdentityMatrix(pivot.length));
}
}
}