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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
+ * 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&times;A = L&times;U. L is lower triangular (with unit diagonal terms), U is upper triangular
+ * and P is a permutation matrix. All matrices are m&times;m.
+ *
+ * <p>As shown by the presence of the P matrix, this decomposition is implemented using partial
+ * pivoting.
+ *
+ * <p>This class is based on the class with similar name from the <a
+ * href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library.
+ *
+ * <ul>
+ * <li>a {@link #getP() getP} method has been added,
+ * <li>the {@code det} method has been renamed as {@link #getDeterminant() getDeterminant},
+ * <li>the {@code getDoublePivot} method has been removed (but the int based {@link #getPivot()
+ * getPivot} method has been kept),
+ * <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the {@code solve} and {@code isNonSingular} methods have been replaced by a {@link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}.
+ * </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
+ *
+ * @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
+ *
+ * @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>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 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 &times; 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));
+ }
+ }
+}
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