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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* 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
* limitations under the License.
*/
package org.apache.commons.math3.linear;
import org.apache.commons.math3.Field;
import org.apache.commons.math3.FieldElement;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.util.MathArrays;
/**
* Calculates the LUP-decomposition of a square matrix.
* The LUP-decomposition of a matrix A consists of three matrices
* L, U and P that satisfy: PA = LU, L is lower triangular, and U is
* upper triangular and P is a permutation matrix. All matrices are
* m×m.
* Since {@link FieldElement field elements} do not provide an ordering
* operator, the permutation matrix is computed here only in order to avoid
* a zero pivot element, no attempt is done to get the largest pivot
* element.
* This class is based on the class with similar name from the
* JAMA library.
*
* - a {@link #getP() getP} method has been added,
* - the {@code det} method has been renamed as {@link #getDeterminant()
* getDeterminant},
* - the {@code getDoublePivot} method has been removed (but the int based
* {@link #getPivot() getPivot} method has been kept),
* - 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}.
*
*
* @param the type of the field elements
* @see MathWorld
* @see Wikipedia
* @since 2.0 (changed to concrete class in 3.0)
*/
public class FieldLUDecomposition> {
/** Field to which the elements belong. */
private final Field field;
/** Entries of LU decomposition. */
private T[][] lu;
/** Pivot permutation associated with LU decomposition. */
private int[] pivot;
/** Parity of the permutation associated with the LU decomposition. */
private boolean even;
/** Singularity indicator. */
private boolean singular;
/** Cached value of L. */
private FieldMatrix cachedL;
/** Cached value of U. */
private FieldMatrix cachedU;
/** Cached value of P. */
private FieldMatrix cachedP;
/**
* Calculates the LU-decomposition of the given matrix.
* @param matrix The matrix to decompose.
* @throws NonSquareMatrixException if matrix is not square
*/
public FieldLUDecomposition(FieldMatrix matrix) {
if (!matrix.isSquare()) {
throw new NonSquareMatrixException(matrix.getRowDimension(),
matrix.getColumnDimension());
}
final int m = matrix.getColumnDimension();
field = matrix.getField();
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++) {
T sum = field.getZero();
// upper
for (int row = 0; row < col; row++) {
final T[] luRow = lu[row];
sum = luRow[col];
for (int i = 0; i < row; i++) {
sum = sum.subtract(luRow[i].multiply(lu[i][col]));
}
luRow[col] = sum;
}
// lower
int nonZero = col; // permutation row
for (int row = col; row < m; row++) {
final T[] luRow = lu[row];
sum = luRow[col];
for (int i = 0; i < col; i++) {
sum = sum.subtract(luRow[i].multiply(lu[i][col]));
}
luRow[col] = sum;
if (lu[nonZero][col].equals(field.getZero())) {
// try to select a better permutation choice
++nonZero;
}
}
// Singularity check
if (nonZero >= m) {
singular = true;
return;
}
// Pivot if necessary
if (nonZero != col) {
T tmp = field.getZero();
for (int i = 0; i < m; i++) {
tmp = lu[nonZero][i];
lu[nonZero][i] = lu[col][i];
lu[col][i] = tmp;
}
int temp = pivot[nonZero];
pivot[nonZero] = pivot[col];
pivot[col] = temp;
even = !even;
}
// Divide the lower elements by the "winning" diagonal elt.
final T luDiag = lu[col][col];
for (int row = col + 1; row < m; row++) {
final T[] luRow = lu[row];
luRow[col] = luRow[col].divide(luDiag);
}
}
}
/**
* Returns the matrix L of the decomposition.
* L is a lower-triangular matrix
* @return the L matrix (or null if decomposed matrix is singular)
*/
public FieldMatrix getL() {
if ((cachedL == null) && !singular) {
final int m = pivot.length;
cachedL = new Array2DRowFieldMatrix(field, m, m);
for (int i = 0; i < m; ++i) {
final T[] luI = lu[i];
for (int j = 0; j < i; ++j) {
cachedL.setEntry(i, j, luI[j]);
}
cachedL.setEntry(i, i, field.getOne());
}
}
return cachedL;
}
/**
* Returns the matrix U of the decomposition.
* U is an upper-triangular matrix
* @return the U matrix (or null if decomposed matrix is singular)
*/
public FieldMatrix getU() {
if ((cachedU == null) && !singular) {
final int m = pivot.length;
cachedU = new Array2DRowFieldMatrix(field, m, m);
for (int i = 0; i < m; ++i) {
final T[] 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 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.
* The positions of the 1 elements are given by the {@link #getPivot()
* pivot permutation vector}.
* @return the P rows permutation matrix (or null if decomposed matrix is singular)
* @see #getPivot()
*/
public FieldMatrix getP() {
if ((cachedP == null) && !singular) {
final int m = pivot.length;
cachedP = new Array2DRowFieldMatrix(field, m, m);
for (int i = 0; i < m; ++i) {
cachedP.setEntry(i, pivot[i], field.getOne());
}
}
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 T getDeterminant() {
if (singular) {
return field.getZero();
} else {
final int m = pivot.length;
T determinant = even ? field.getOne() : field.getZero().subtract(field.getOne());
for (int i = 0; i < m; i++) {
determinant = determinant.multiply(lu[i][i]);
}
return determinant;
}
}
/**
* Get a solver for finding the A × X = B solution in exact linear sense.
* @return a solver
*/
public FieldDecompositionSolver getSolver() {
return new Solver(field, lu, pivot, singular);
}
/** Specialized solver. */
private static class Solver> implements FieldDecompositionSolver {
/** Field to which the elements belong. */
private final Field field;
/** Entries of LU decomposition. */
private final T[][] lu;
/** Pivot permutation associated with LU decomposition. */
private final int[] pivot;
/** Singularity indicator. */
private final boolean singular;
/**
* Build a solver from decomposed matrix.
* @param field field to which the matrix elements belong
* @param lu entries of LU decomposition
* @param pivot pivot permutation associated with LU decomposition
* @param singular singularity indicator
*/
private Solver(final Field field, final T[][] lu,
final int[] pivot, final boolean singular) {
this.field = field;
this.lu = lu;
this.pivot = pivot;
this.singular = singular;
}
/** {@inheritDoc} */
public boolean isNonSingular() {
return !singular;
}
/** {@inheritDoc} */
public FieldVector solve(FieldVector b) {
try {
return solve((ArrayFieldVector) b);
} catch (ClassCastException cce) {
final int m = pivot.length;
if (b.getDimension() != m) {
throw new DimensionMismatchException(b.getDimension(), m);
}
if (singular) {
throw new SingularMatrixException();
}
// Apply permutations to b
final T[] bp = MathArrays.buildArray(field, m);
for (int row = 0; row < m; row++) {
bp[row] = b.getEntry(pivot[row]);
}
// Solve LY = b
for (int col = 0; col < m; col++) {
final T bpCol = bp[col];
for (int i = col + 1; i < m; i++) {
bp[i] = bp[i].subtract(bpCol.multiply(lu[i][col]));
}
}
// Solve UX = Y
for (int col = m - 1; col >= 0; col--) {
bp[col] = bp[col].divide(lu[col][col]);
final T bpCol = bp[col];
for (int i = 0; i < col; i++) {
bp[i] = bp[i].subtract(bpCol.multiply(lu[i][col]));
}
}
return new ArrayFieldVector(field, bp, false);
}
}
/** Solve the linear equation A × X = B.
* The A matrix is implicit here. It is
* @param b right-hand side of the equation A × X = B
* @return a vector X such that A × X = B
* @throws DimensionMismatchException if the matrices dimensions do not match.
* @throws SingularMatrixException if the decomposed matrix is singular.
*/
public ArrayFieldVector solve(ArrayFieldVector b) {
final int m = pivot.length;
final int length = b.getDimension();
if (length != m) {
throw new DimensionMismatchException(length, m);
}
if (singular) {
throw new SingularMatrixException();
}
// Apply permutations to b
final T[] bp = MathArrays.buildArray(field, m);
for (int row = 0; row < m; row++) {
bp[row] = b.getEntry(pivot[row]);
}
// Solve LY = b
for (int col = 0; col < m; col++) {
final T bpCol = bp[col];
for (int i = col + 1; i < m; i++) {
bp[i] = bp[i].subtract(bpCol.multiply(lu[i][col]));
}
}
// Solve UX = Y
for (int col = m - 1; col >= 0; col--) {
bp[col] = bp[col].divide(lu[col][col]);
final T bpCol = bp[col];
for (int i = 0; i < col; i++) {
bp[i] = bp[i].subtract(bpCol.multiply(lu[i][col]));
}
}
return new ArrayFieldVector(bp, false);
}
/** {@inheritDoc} */
public FieldMatrix solve(FieldMatrix 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 T[][] bp = MathArrays.buildArray(field, m, nColB);
for (int row = 0; row < m; row++) {
final T[] 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 T[] bpCol = bp[col];
for (int i = col + 1; i < m; i++) {
final T[] bpI = bp[i];
final T luICol = lu[i][col];
for (int j = 0; j < nColB; j++) {
bpI[j] = bpI[j].subtract(bpCol[j].multiply(luICol));
}
}
}
// Solve UX = Y
for (int col = m - 1; col >= 0; col--) {
final T[] bpCol = bp[col];
final T luDiag = lu[col][col];
for (int j = 0; j < nColB; j++) {
bpCol[j] = bpCol[j].divide(luDiag);
}
for (int i = 0; i < col; i++) {
final T[] bpI = bp[i];
final T luICol = lu[i][col];
for (int j = 0; j < nColB; j++) {
bpI[j] = bpI[j].subtract(bpCol[j].multiply(luICol));
}
}
}
return new Array2DRowFieldMatrix(field, bp, false);
}
/** {@inheritDoc} */
public FieldMatrix getInverse() {
final int m = pivot.length;
final T one = field.getOne();
FieldMatrix identity = new Array2DRowFieldMatrix(field, m, m);
for (int i = 0; i < m; ++i) {
identity.setEntry(i, i, one);
}
return solve(identity);
}
}
}