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/**
 * Copyright (C) 2014-2020 Philip Helger (www.helger.com)
 * philip[at]helger[dot]com
 *
 * Licensed 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 com.helger.matrix;

import java.io.Serializable;
import java.util.Arrays;

import javax.annotation.Nonnull;

import com.helger.commons.annotation.ReturnsMutableCopy;
import com.helger.commons.math.MathHelper;

import edu.umd.cs.findbugs.annotations.SuppressFBWarnings;

/**
 * Singular Value Decomposition.
 * 

* For an m-by-n matrix A with m >= n, the singular value decomposition is an * m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and an n-by-n * orthogonal matrix V so that A = U*S*V'. *

*

* The singular values, sigma[k] = S[k][k], are ordered so that sigma[0] >= * sigma[1] >= ... >= sigma[n-1]. *

*

* The singular value decomposition always exists, so the constructor will never * fail. The matrix condition number and the effective numerical rank can be * computed from this decomposition. *

*/ public class SingularValueDecomposition implements Serializable { private static final double EPSILON = Math.pow (2.0, -52.0); private static final double TINY = Math.pow (2.0, -966.0); /** * Arrays for internal storage of U and V. * * @serial internal storage of U. * @serial internal storage of V. */ private final double [] [] m_aU; private final double [] [] m_aV; /** * Array for internal storage of singular values. * * @serial internal storage of singular values. */ private final double [] m_aData; /** * Row dimensions. * * @serial row dimension. */ private final int m_nRows; /** * Column dimension. * * @serial column dimension. */ private final int m_nCols; /** * Construct the singular value decomposition Structure to access U, S and V. * * @param aMatrix * Rectangular matrix */ public SingularValueDecomposition (@Nonnull final Matrix aMatrix) { // Derived from LINPACK code. // Initialize. final double [] [] aArray = aMatrix.getArrayCopy (); m_nRows = aMatrix.getRowDimension (); m_nCols = aMatrix.getColumnDimension (); /* * Apparently the failing cases are only a proper subset of (m= n"); } */ final int nu = Math.min (m_nRows, m_nCols); m_aData = new double [Math.min (m_nRows + 1, m_nCols)]; m_aU = new double [m_nRows] [nu]; m_aV = new double [m_nCols] [m_nCols]; final double [] e = new double [m_nCols]; final double [] work = new double [m_nRows]; final boolean wantu = true; final boolean wantv = true; // Reduce A to bidiagonal form, storing the diagonal elements // in s and the super-diagonal elements in e. final int nct = Math.min (m_nRows - 1, m_nCols); final int nrt = Math.max (0, Math.min (m_nCols - 2, m_nRows)); for (int k = 0; k < Math.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. m_aData[k] = 0; for (int i = k; i < m_nRows; i++) m_aData[k] = MathHelper.hypot (m_aData[k], aArray[i][k]); if (m_aData[k] != 0.0) { if (aArray[k][k] < 0.0) m_aData[k] = -m_aData[k]; for (int i = k; i < m_nRows; i++) aArray[i][k] /= m_aData[k]; aArray[k][k] += 1.0; } m_aData[k] = -m_aData[k]; } for (int j = k + 1; j < m_nCols; j++) { if (k < nct && m_aData[k] != 0.0) { // Apply the transformation. double t = 0; for (int i = k; i < m_nRows; i++) t += aArray[i][k] * aArray[i][j]; t = -t / aArray[k][k]; for (int i = k; i < m_nRows; i++) aArray[i][j] += t * aArray[i][k]; } // Place the k-th row of A into e for the // subsequent calculation of the row transformation. e[j] = aArray[k][j]; } if (wantu && k < nct) { // Place the transformation in U for subsequent back // multiplication. for (int i = k; i < m_nRows; i++) m_aU[i][k] = aArray[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 < m_nCols; i++) e[k] = MathHelper.hypot (e[k], e[i]); if (e[k] != 0.0) { if (e[k + 1] < 0.0) e[k] = -e[k]; for (int i = k + 1; i < m_nCols; i++) e[i] /= e[k]; e[k + 1] += 1.0; } e[k] = -e[k]; if ((k + 1 < m_nRows) && (e[k] != 0.0)) { // Apply the transformation. for (int i = k + 1; i < m_nRows; i++) work[i] = 0.0; for (int j = k + 1; j < m_nCols; j++) for (int i = k + 1; i < m_nRows; i++) work[i] += e[j] * aArray[i][j]; for (int j = k + 1; j < m_nCols; j++) { final double t = -e[j] / e[k + 1]; for (int i = k + 1; i < m_nRows; i++) aArray[i][j] += t * work[i]; } } if (wantv) { // Place the transformation in V for subsequent // back multiplication. for (int i = k + 1; i < m_nCols; i++) m_aV[i][k] = e[i]; } } } // Set up the final bidiagonal matrix or order p. int p = Math.min (m_nCols, m_nRows + 1); if (nct < m_nCols) m_aData[nct] = aArray[nct][nct]; if (m_nRows < p) m_aData[p - 1] = 0.0; if (nrt + 1 < p) e[nrt] = aArray[nrt][p - 1]; e[p - 1] = 0.0; // If required, generate U. if (wantu) { for (int j = nct; j < nu; j++) { for (int i = 0; i < m_nRows; i++) m_aU[i][j] = 0.0; m_aU[j][j] = 1.0; } for (int k = nct - 1; k >= 0; k--) { if (m_aData[k] != 0.0) { for (int j = k + 1; j < nu; j++) { double t = 0; for (int i = k; i < m_nRows; i++) t += m_aU[i][k] * m_aU[i][j]; t = -t / m_aU[k][k]; for (int i = k; i < m_nRows; i++) m_aU[i][j] += t * m_aU[i][k]; } for (int i = k; i < m_nRows; i++) m_aU[i][k] = -m_aU[i][k]; m_aU[k][k] = 1.0 + m_aU[k][k]; for (int i = 0; i < k - 1; i++) m_aU[i][k] = 0.0; } else { for (int i = 0; i < m_nRows; i++) m_aU[i][k] = 0.0; m_aU[k][k] = 1.0; } } } // If required, generate V. if (wantv) { for (int k = m_nCols - 1; k >= 0; k--) { if ((k < nrt) && (e[k] != 0.0)) { for (int j = k + 1; j < nu; j++) { double t = 0; for (int i = k + 1; i < m_nCols; i++) t += m_aV[i][k] * m_aV[i][j]; t = -t / m_aV[k + 1][k]; for (int i = k + 1; i < m_nCols; i++) m_aV[i][j] += t * m_aV[i][k]; } } for (int i = 0; i < m_nCols; i++) m_aV[i][k] = 0.0; m_aV[k][k] = 1.0; } } // 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

= -1; k--) { if (k == -1) break; if (MathHelper.abs (e[k]) <= TINY + EPSILON * (MathHelper.abs (m_aData[k]) + MathHelper.abs (m_aData[k + 1]))) { e[k] = 0.0; 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 ? MathHelper.abs (e[ks]) : 0.) + (ks != k + 1 ? MathHelper.abs (e[ks - 1]) : 0.); if (MathHelper.abs (m_aData[ks]) <= TINY + EPSILON * t) { m_aData[ks] = 0.0; 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] = 0.0; for (int j = p - 2; j >= k; j--) { double t = MathHelper.hypot (m_aData[j], f); final double cs = m_aData[j] / t; final double sn = f / t; m_aData[j] = t; if (j != k) { f = -sn * e[j - 1]; e[j - 1] *= cs; } if (wantv) { for (int i = 0; i < m_nCols; i++) { final double q = m_aV[i][p - 1]; t = cs * m_aV[i][j] + sn * q; m_aV[i][p - 1] = -sn * m_aV[i][j] + cs * q; m_aV[i][j] = t; } } } break; } // Split at negligible s(k). case 2: { double f = e[k - 1]; e[k - 1] = 0.0; for (int j = k; j < p; j++) { double t = MathHelper.hypot (m_aData[j], f); final double cs = m_aData[j] / t; final double sn = f / t; m_aData[j] = t; f = -sn * e[j]; e[j] = cs * e[j]; if (wantu) { for (int i = 0; i < m_nRows; i++) { final double q = m_aU[i][k - 1]; t = cs * m_aU[i][j] + sn * q; m_aU[i][k - 1] = -sn * m_aU[i][j] + cs * q; m_aU[i][j] = t; } } } break; } // Perform one qr step. case 3: { // Calculate the shift. final double scale = MathHelper.getMaxDouble (MathHelper.abs (m_aData[p - 1]), MathHelper.abs (m_aData[p - 2]), MathHelper.abs (e[p - 2]), MathHelper.abs (m_aData[k]), MathHelper.abs (e[k])); final double sp = m_aData[p - 1] / scale; final double spm1 = m_aData[p - 2] / scale; final double epm1 = e[p - 2] / scale; final double sk = m_aData[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 = 0.0; if ((b != 0.0) || (c != 0.0)) { shift = Math.sqrt (b * b + c); if (b < 0.0) 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 = MathHelper.hypot (f, g); double cs = f / t; double sn = g / t; if (j != k) e[j - 1] = t; f = cs * m_aData[j] + sn * e[j]; e[j] = cs * e[j] - sn * m_aData[j]; g = sn * m_aData[j + 1]; m_aData[j + 1] = cs * m_aData[j + 1]; if (wantv) { for (int i = 0; i < m_nCols; i++) { t = cs * m_aV[i][j] + sn * m_aV[i][j + 1]; m_aV[i][j + 1] = -sn * m_aV[i][j] + cs * m_aV[i][j + 1]; m_aV[i][j] = t; } } t = MathHelper.hypot (f, g); cs = f / t; sn = g / t; m_aData[j] = t; f = cs * e[j] + sn * m_aData[j + 1]; m_aData[j + 1] = -sn * e[j] + cs * m_aData[j + 1]; g = sn * e[j + 1]; e[j + 1] = cs * e[j + 1]; if (wantu && (j < m_nRows - 1)) { for (int i = 0; i < m_nRows; i++) { final double q = m_aU[i][j + 1]; t = cs * m_aU[i][j] + sn * q; m_aU[i][j + 1] = -sn * m_aU[i][j] + cs * q; m_aU[i][j] = t; } } } e[p - 2] = f; break; } // Convergence. case 4: { // Make the singular values positive. if (m_aData[k] <= 0.0) { m_aData[k] = m_aData[k] < 0.0 ? -m_aData[k] : 0.0; if (wantv) { for (int i = 0; i <= pp; i++) m_aV[i][k] = -m_aV[i][k]; } } // Order the singular values. while (k < pp) { if (m_aData[k] >= m_aData[k + 1]) break; double t = m_aData[k]; m_aData[k] = m_aData[k + 1]; m_aData[k + 1] = t; if (wantv && (k < m_nCols - 1)) { for (int i = 0; i < m_nCols; i++) { t = m_aV[i][k + 1]; m_aV[i][k + 1] = m_aV[i][k]; m_aV[i][k] = t; } } if (wantu && (k < m_nRows - 1)) { for (int i = 0; i < m_nRows; i++) { t = m_aU[i][k + 1]; m_aU[i][k + 1] = m_aU[i][k]; m_aU[i][k] = t; } } k++; } p--; break; } default: throw new IllegalStateException (); } } } /* * ------------------------ Public Methods ------------------------ */ /** * Return the left singular vectors * * @return U */ @Nonnull @ReturnsMutableCopy public Matrix getU () { return new Matrix (m_aU, m_nRows, Math.min (m_nRows + 1, m_nCols)); } /** * Return the right singular vectors * * @return V */ @Nonnull @ReturnsMutableCopy public Matrix getV () { return new Matrix (m_aV, m_nCols, m_nCols); } /** * Return the one-dimensional array of singular values * * @return diagonal of S. */ @Nonnull @SuppressFBWarnings ("EI_EXPOSE_REP") public double [] getSingularValues () { return m_aData; } /** * Return the diagonal matrix of singular values * * @return S */ @Nonnull @ReturnsMutableCopy public Matrix getS () { final Matrix aNewMatrix = new Matrix (m_nCols, m_nCols); final double [] [] aNewArray = aNewMatrix.internalGetArray (); for (int i = 0; i < m_nCols; i++) { Arrays.fill (aNewArray[i], 0.0); aNewArray[i][i] = m_aData[i]; } return aNewMatrix; } /** * Two norm * * @return max(S) */ public double norm2 () { return m_aData[0]; } /** * Two norm condition number * * @return max(S)/min(S) */ public double cond () { return m_aData[0] / m_aData[Math.min (m_nRows, m_nCols) - 1]; } /** * Effective numerical matrix rank * * @return Number of nonnegligible singular values. */ public int rank () { final double tol = Math.max (m_nRows, m_nCols) * m_aData[0] * EPSILON; int r = 0; for (final double element : m_aData) if (element > tol) r++; return r; } }





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