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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
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package org.apache.commons.math3.linear;

import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.Precision;

/**
 * Calculates the compact Singular Value Decomposition of a matrix.
 * 

* The Singular Value Decomposition of matrix A is a set of three matrices: U, * Σ and V such that A = U × Σ × VT. Let A be * a m × n matrix, then U is a m × p orthogonal matrix, Σ is a * p × p diagonal matrix with positive or null elements, V is a p × * n orthogonal matrix (hence VT is also orthogonal) where * p=min(m,n). *

*

This class is similar to the class with similar name from the * JAMA library, with the * following changes:

*
    *
  • the {@code norm2} method which has been renamed as {@link #getNorm() * getNorm},
  • *
  • the {@code cond} method which has been renamed as {@link * #getConditionNumber() getConditionNumber},
  • *
  • the {@code rank} method which has been renamed as {@link #getRank() * getRank},
  • *
  • a {@link #getUT() getUT} method has been added,
  • *
  • a {@link #getVT() getVT} method has been added,
  • *
  • a {@link #getSolver() getSolver} method has been added,
  • *
  • a {@link #getCovariance(double) getCovariance} method has been added.
  • *
* @see MathWorld * @see Wikipedia * @since 2.0 (changed to concrete class in 3.0) */ public class SingularValueDecomposition { /** Relative threshold for small singular values. */ private static final double EPS = 0x1.0p-52; /** Absolute threshold for small singular values. */ private static final double TINY = 0x1.0p-966; /** Computed singular values. */ private final double[] singularValues; /** max(row dimension, column dimension). */ private final int m; /** min(row dimension, column dimension). */ private final int n; /** Indicator for transposed matrix. */ private final boolean transposed; /** Cached value of U matrix. */ private final RealMatrix cachedU; /** Cached value of transposed U matrix. */ private RealMatrix cachedUt; /** Cached value of S (diagonal) matrix. */ private RealMatrix cachedS; /** Cached value of V matrix. */ private final RealMatrix cachedV; /** Cached value of transposed V matrix. */ private RealMatrix cachedVt; /** * Tolerance value for small singular values, calculated once we have * populated "singularValues". **/ private final double tol; /** * Calculates the compact Singular Value Decomposition of the given matrix. * * @param matrix Matrix to decompose. */ public SingularValueDecomposition(final RealMatrix matrix) { final double[][] A; // "m" is always the largest dimension. if (matrix.getRowDimension() < matrix.getColumnDimension()) { transposed = true; A = matrix.transpose().getData(); m = matrix.getColumnDimension(); n = matrix.getRowDimension(); } else { transposed = false; A = matrix.getData(); m = matrix.getRowDimension(); n = matrix.getColumnDimension(); } singularValues = new double[n]; final double[][] U = new double[m][n]; final double[][] V = new double[n][n]; final double[] e = new double[n]; final double[] work = new double[m]; // Reduce A to bidiagonal form, storing the diagonal elements // in s and the super-diagonal elements in e. final int nct = FastMath.min(m - 1, n); final int nrt = FastMath.max(0, n - 2); for (int k = 0; k < FastMath.max(nct, nrt); k++) { if (k < nct) { // Compute the transformation for the k-th column and // place the k-th diagonal in s[k]. // Compute 2-norm of k-th column without under/overflow. singularValues[k] = 0; for (int i = k; i < m; i++) { singularValues[k] = FastMath.hypot(singularValues[k], A[i][k]); } if (singularValues[k] != 0) { if (A[k][k] < 0) { singularValues[k] = -singularValues[k]; } for (int i = k; i < m; i++) { A[i][k] /= singularValues[k]; } A[k][k] += 1; } singularValues[k] = -singularValues[k]; } for (int j = k + 1; j < n; j++) { if (k < nct && singularValues[k] != 0) { // Apply the transformation. double t = 0; for (int i = k; i < m; i++) { t += A[i][k] * A[i][j]; } t = -t / A[k][k]; for (int i = k; i < m; i++) { A[i][j] += t * A[i][k]; } } // Place the k-th row of A into e for the // subsequent calculation of the row transformation. e[j] = A[k][j]; } if (k < nct) { // Place the transformation in U for subsequent back // multiplication. for (int i = k; i < m; i++) { U[i][k] = A[i][k]; } } if (k < nrt) { // Compute the k-th row transformation and place the // k-th super-diagonal in e[k]. // Compute 2-norm without under/overflow. e[k] = 0; for (int i = k + 1; i < n; i++) { e[k] = FastMath.hypot(e[k], e[i]); } if (e[k] != 0) { if (e[k + 1] < 0) { e[k] = -e[k]; } for (int i = k + 1; i < n; i++) { e[i] /= e[k]; } e[k + 1] += 1; } e[k] = -e[k]; if (k + 1 < m && e[k] != 0) { // Apply the transformation. for (int i = k + 1; i < m; i++) { work[i] = 0; } for (int j = k + 1; j < n; j++) { for (int i = k + 1; i < m; i++) { work[i] += e[j] * A[i][j]; } } for (int j = k + 1; j < n; j++) { final double t = -e[j] / e[k + 1]; for (int i = k + 1; i < m; i++) { A[i][j] += t * work[i]; } } } // Place the transformation in V for subsequent // back multiplication. for (int i = k + 1; i < n; i++) { V[i][k] = e[i]; } } } // Set up the final bidiagonal matrix or order p. int p = n; if (nct < n) { singularValues[nct] = A[nct][nct]; } if (m < p) { singularValues[p - 1] = 0; } if (nrt + 1 < p) { e[nrt] = A[nrt][p - 1]; } e[p - 1] = 0; // Generate U. for (int j = nct; j < n; j++) { for (int i = 0; i < m; i++) { U[i][j] = 0; } U[j][j] = 1; } for (int k = nct - 1; k >= 0; k--) { if (singularValues[k] != 0) { for (int j = k + 1; j < n; j++) { double t = 0; for (int i = k; i < m; i++) { t += U[i][k] * U[i][j]; } t = -t / U[k][k]; for (int i = k; i < m; i++) { U[i][j] += t * U[i][k]; } } for (int i = k; i < m; i++) { U[i][k] = -U[i][k]; } U[k][k] = 1 + U[k][k]; for (int i = 0; i < k - 1; i++) { U[i][k] = 0; } } else { for (int i = 0; i < m; i++) { U[i][k] = 0; } U[k][k] = 1; } } // Generate V. for (int k = n - 1; k >= 0; k--) { if (k < nrt && e[k] != 0) { for (int j = k + 1; j < n; j++) { double t = 0; for (int i = k + 1; i < n; i++) { t += V[i][k] * V[i][j]; } t = -t / V[k + 1][k]; for (int i = k + 1; i < n; i++) { V[i][j] += t * V[i][k]; } } } for (int i = 0; i < n; i++) { V[i][k] = 0; } V[k][k] = 1; } // Main iteration loop for the singular values. final int pp = p - 1; while (p > 0) { int k; int kase; // Here is where a test for too many iterations would go. // This section of the program inspects for // negligible elements in the s and e arrays. On // completion the variables kase and k are set as follows. // kase = 1 if s(p) and e[k-1] are negligible and k

= 0; k--) { final double threshold = TINY + EPS * (FastMath.abs(singularValues[k]) + FastMath.abs(singularValues[k + 1])); // the following condition is written this way in order // to break out of the loop when NaN occurs, writing it // as "if (FastMath.abs(e[k]) <= threshold)" would loop // indefinitely in case of NaNs because comparison on NaNs // always return false, regardless of what is checked // see issue MATH-947 if (!(FastMath.abs(e[k]) > threshold)) { e[k] = 0; break; } } if (k == p - 2) { kase = 4; } else { int ks; for (ks = p - 1; ks >= k; ks--) { if (ks == k) { break; } final double t = (ks != p ? FastMath.abs(e[ks]) : 0) + (ks != k + 1 ? FastMath.abs(e[ks - 1]) : 0); if (FastMath.abs(singularValues[ks]) <= TINY + EPS * t) { singularValues[ks] = 0; break; } } if (ks == k) { kase = 3; } else if (ks == p - 1) { kase = 1; } else { kase = 2; k = ks; } } k++; // Perform the task indicated by kase. switch (kase) { // Deflate negligible s(p). case 1: { double f = e[p - 2]; e[p - 2] = 0; for (int j = p - 2; j >= k; j--) { double t = FastMath.hypot(singularValues[j], f); final double cs = singularValues[j] / t; final double sn = f / t; singularValues[j] = t; if (j != k) { f = -sn * e[j - 1]; e[j - 1] = cs * e[j - 1]; } for (int i = 0; i < n; i++) { t = cs * V[i][j] + sn * V[i][p - 1]; V[i][p - 1] = -sn * V[i][j] + cs * V[i][p - 1]; V[i][j] = t; } } } break; // Split at negligible s(k). case 2: { double f = e[k - 1]; e[k - 1] = 0; for (int j = k; j < p; j++) { double t = FastMath.hypot(singularValues[j], f); final double cs = singularValues[j] / t; final double sn = f / t; singularValues[j] = t; f = -sn * e[j]; e[j] = cs * e[j]; for (int i = 0; i < m; i++) { t = cs * U[i][j] + sn * U[i][k - 1]; U[i][k - 1] = -sn * U[i][j] + cs * U[i][k - 1]; U[i][j] = t; } } } break; // Perform one qr step. case 3: { // Calculate the shift. final double maxPm1Pm2 = FastMath.max(FastMath.abs(singularValues[p - 1]), FastMath.abs(singularValues[p - 2])); final double scale = FastMath.max(FastMath.max(FastMath.max(maxPm1Pm2, FastMath.abs(e[p - 2])), FastMath.abs(singularValues[k])), FastMath.abs(e[k])); final double sp = singularValues[p - 1] / scale; final double spm1 = singularValues[p - 2] / scale; final double epm1 = e[p - 2] / scale; final double sk = singularValues[k] / scale; final double ek = e[k] / scale; final double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0; final double c = (sp * epm1) * (sp * epm1); double shift = 0; if (b != 0 || c != 0) { shift = FastMath.sqrt(b * b + c); if (b < 0) { shift = -shift; } shift = c / (b + shift); } double f = (sk + sp) * (sk - sp) + shift; double g = sk * ek; // Chase zeros. for (int j = k; j < p - 1; j++) { double t = FastMath.hypot(f, g); double cs = f / t; double sn = g / t; if (j != k) { e[j - 1] = t; } f = cs * singularValues[j] + sn * e[j]; e[j] = cs * e[j] - sn * singularValues[j]; g = sn * singularValues[j + 1]; singularValues[j + 1] = cs * singularValues[j + 1]; for (int i = 0; i < n; i++) { t = cs * V[i][j] + sn * V[i][j + 1]; V[i][j + 1] = -sn * V[i][j] + cs * V[i][j + 1]; V[i][j] = t; } t = FastMath.hypot(f, g); cs = f / t; sn = g / t; singularValues[j] = t; f = cs * e[j] + sn * singularValues[j + 1]; singularValues[j + 1] = -sn * e[j] + cs * singularValues[j + 1]; g = sn * e[j + 1]; e[j + 1] = cs * e[j + 1]; if (j < m - 1) { for (int i = 0; i < m; i++) { t = cs * U[i][j] + sn * U[i][j + 1]; U[i][j + 1] = -sn * U[i][j] + cs * U[i][j + 1]; U[i][j] = t; } } } e[p - 2] = f; } break; // Convergence. default: { // Make the singular values positive. if (singularValues[k] <= 0) { singularValues[k] = singularValues[k] < 0 ? -singularValues[k] : 0; for (int i = 0; i <= pp; i++) { V[i][k] = -V[i][k]; } } // Order the singular values. while (k < pp) { if (singularValues[k] >= singularValues[k + 1]) { break; } double t = singularValues[k]; singularValues[k] = singularValues[k + 1]; singularValues[k + 1] = t; if (k < n - 1) { for (int i = 0; i < n; i++) { t = V[i][k + 1]; V[i][k + 1] = V[i][k]; V[i][k] = t; } } if (k < m - 1) { for (int i = 0; i < m; i++) { t = U[i][k + 1]; U[i][k + 1] = U[i][k]; U[i][k] = t; } } k++; } p--; } break; } } // Set the small value tolerance used to calculate rank and pseudo-inverse tol = FastMath.max(m * singularValues[0] * EPS, FastMath.sqrt(Precision.SAFE_MIN)); if (!transposed) { cachedU = MatrixUtils.createRealMatrix(U); cachedV = MatrixUtils.createRealMatrix(V); } else { cachedU = MatrixUtils.createRealMatrix(V); cachedV = MatrixUtils.createRealMatrix(U); } } /** * Returns the matrix U of the decomposition. *

U is an orthogonal matrix, i.e. its transpose is also its inverse.

* @return the U matrix * @see #getUT() */ public RealMatrix getU() { // return the cached matrix return cachedU; } /** * Returns the transpose of the matrix U of the decomposition. *

U is an orthogonal matrix, i.e. its transpose is also its inverse.

* @return the U matrix (or null if decomposed matrix is singular) * @see #getU() */ public RealMatrix getUT() { if (cachedUt == null) { cachedUt = getU().transpose(); } // return the cached matrix return cachedUt; } /** * Returns the diagonal matrix Σ of the decomposition. *

Σ is a diagonal matrix. The singular values are provided in * non-increasing order, for compatibility with Jama.

* @return the Σ matrix */ public RealMatrix getS() { if (cachedS == null) { // cache the matrix for subsequent calls cachedS = MatrixUtils.createRealDiagonalMatrix(singularValues); } return cachedS; } /** * Returns the diagonal elements of the matrix Σ of the decomposition. *

The singular values are provided in non-increasing order, for * compatibility with Jama.

* @return the diagonal elements of the Σ matrix */ public double[] getSingularValues() { return singularValues.clone(); } /** * Returns the matrix V of the decomposition. *

V is an orthogonal matrix, i.e. its transpose is also its inverse.

* @return the V matrix (or null if decomposed matrix is singular) * @see #getVT() */ public RealMatrix getV() { // return the cached matrix return cachedV; } /** * Returns the transpose of the matrix V of the decomposition. *

V is an orthogonal matrix, i.e. its transpose is also its inverse.

* @return the V matrix (or null if decomposed matrix is singular) * @see #getV() */ public RealMatrix getVT() { if (cachedVt == null) { cachedVt = getV().transpose(); } // return the cached matrix return cachedVt; } /** * Returns the n × n covariance matrix. *

The covariance matrix is V × J × VT * where J is the diagonal matrix of the inverse of the squares of * the singular values.

* @param minSingularValue value below which singular values are ignored * (a 0 or negative value implies all singular value will be used) * @return covariance matrix * @exception IllegalArgumentException if minSingularValue is larger than * the largest singular value, meaning all singular values are ignored */ public RealMatrix getCovariance(final double minSingularValue) { // get the number of singular values to consider final int p = singularValues.length; int dimension = 0; while (dimension < p && singularValues[dimension] >= minSingularValue) { ++dimension; } if (dimension == 0) { throw new NumberIsTooLargeException(LocalizedFormats.TOO_LARGE_CUTOFF_SINGULAR_VALUE, minSingularValue, singularValues[0], true); } final double[][] data = new double[dimension][p]; getVT().walkInOptimizedOrder(new DefaultRealMatrixPreservingVisitor() { /** {@inheritDoc} */ @Override public void visit(final int row, final int column, final double value) { data[row][column] = value / singularValues[row]; } }, 0, dimension - 1, 0, p - 1); RealMatrix jv = new Array2DRowRealMatrix(data, false); return jv.transpose().multiply(jv); } /** * Returns the L2 norm of the matrix. *

The L2 norm is max(|A × u|2 / * |u|2), where |.|2 denotes the vectorial 2-norm * (i.e. the traditional euclidian norm).

* @return norm */ public double getNorm() { return singularValues[0]; } /** * Return the condition number of the matrix. * @return condition number of the matrix */ public double getConditionNumber() { return singularValues[0] / singularValues[n - 1]; } /** * Computes the inverse of the condition number. * In cases of rank deficiency, the {@link #getConditionNumber() condition * number} will become undefined. * * @return the inverse of the condition number. */ public double getInverseConditionNumber() { return singularValues[n - 1] / singularValues[0]; } /** * Return the effective numerical matrix rank. *

The effective numerical rank is the number of non-negligible * singular values. The threshold used to identify non-negligible * terms is max(m,n) × ulp(s1) where ulp(s1) * is the least significant bit of the largest singular value.

* @return effective numerical matrix rank */ public int getRank() { int r = 0; for (int i = 0; i < singularValues.length; i++) { if (singularValues[i] > tol) { r++; } } return r; } /** * Get a solver for finding the A × X = B solution in least square sense. * @return a solver */ public DecompositionSolver getSolver() { return new Solver(singularValues, getUT(), getV(), getRank() == m, tol); } /** Specialized solver. */ private static class Solver implements DecompositionSolver { /** Pseudo-inverse of the initial matrix. */ private final RealMatrix pseudoInverse; /** Singularity indicator. */ private boolean nonSingular; /** * Build a solver from decomposed matrix. * * @param singularValues Singular values. * @param uT UT matrix of the decomposition. * @param v V matrix of the decomposition. * @param nonSingular Singularity indicator. * @param tol tolerance for singular values */ private Solver(final double[] singularValues, final RealMatrix uT, final RealMatrix v, final boolean nonSingular, final double tol) { final double[][] suT = uT.getData(); for (int i = 0; i < singularValues.length; ++i) { final double a; if (singularValues[i] > tol) { a = 1 / singularValues[i]; } else { a = 0; } final double[] suTi = suT[i]; for (int j = 0; j < suTi.length; ++j) { suTi[j] *= a; } } pseudoInverse = v.multiply(new Array2DRowRealMatrix(suT, false)); this.nonSingular = nonSingular; } /** * Solve the linear equation A × X = B in least square sense. *

* The m×n matrix A may not be square, the solution X is such that * ||A × X - B|| is minimal. *

* @param b Right-hand side of the equation A × X = B * @return a vector X that minimizes the two norm of A × X - B * @throws org.apache.commons.math3.exception.DimensionMismatchException * if the matrices dimensions do not match. */ public RealVector solve(final RealVector b) { return pseudoInverse.operate(b); } /** * Solve the linear equation A × X = B in least square sense. *

* The m×n matrix A may not be square, the solution X is such that * ||A × X - B|| is minimal. *

* * @param b Right-hand side of the equation A × X = B * @return a matrix X that minimizes the two norm of A × X - B * @throws org.apache.commons.math3.exception.DimensionMismatchException * if the matrices dimensions do not match. */ public RealMatrix solve(final RealMatrix b) { return pseudoInverse.multiply(b); } /** * Check if the decomposed matrix is non-singular. * * @return {@code true} if the decomposed matrix is non-singular. */ public boolean isNonSingular() { return nonSingular; } /** * Get the pseudo-inverse of the decomposed matrix. * * @return the inverse matrix. */ public RealMatrix getInverse() { return pseudoInverse; } } }




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