Java tutorial
/* * This software is a cooperative product of The MathWorks and the National * Institute of Standards and Technology (NIST) which has been released to the * public domain. Neither The MathWorks nor NIST assumes any responsibility * whatsoever for its use by other parties, and makes no guarantees, expressed * or implied, about its quality, reliability, or any other characteristic. */ /* * SingularValueDecomposition.java * Copyright (C) 1999 The Mathworks and NIST * */ package weka.core.matrix; import weka.core.RevisionHandler; import weka.core.RevisionUtils; import java.io.Serializable; /** * Singular Value Decomposition. * <P> * 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'. * <P> * The singular values, sigma[k] = S[k][k], are ordered so that sigma[0] >= * sigma[1] >= ... >= sigma[n-1]. * <P> * The singular value decompostion always exists, so the constructor will never * fail. The matrix condition number and the effective numerical rank can be * computed from this decomposition. * <p/> * Adapted from the <a href="http://math.nist.gov/javanumerics/jama/" target="_blank">JAMA</a> package. * * @author The Mathworks and NIST * @author Fracpete (fracpete at waikato dot ac dot nz) * @author eibe@cs.waikato.ac.nz * @version $Revision$ */ public class SingularValueDecomposition implements Serializable, RevisionHandler { /** for serialization */ private static final long serialVersionUID = -8738089610999867951L; /** * Arrays for internal storage of U and V. * @serial internal storage of U. * @serial internal storage of V. */ private double[][] U, V; /** * Array for internal storage of singular values. * @serial internal storage of singular values. */ private double[] s; /** * Row and column dimensions. * @serial row dimension. * @serial column dimension. */ private int m, n; /** * Construct the singular value decomposition * @param Arg Rectangular matrix */ public SingularValueDecomposition(Matrix Arg) { // Derived from LINPACK code. // Initialize. double[][] A = null; m = Arg.getRowDimension(); n = Arg.getColumnDimension(); /* Apparently the failing cases are only a proper subset of (m<n), so let's not throw error. Correct fix to come later? if (m<n) { throw new IllegalArgumentException("Jama SVD only works for m >= n"); } */ boolean usingTranspose = false; if (m < n) { // Use transpose and convert back at the end // Otherwise m < n case may yield incorrect results (see above comment) A = Arg.transpose().getArrayCopy(); usingTranspose = true; int temp = m; m = n; n = temp; } else { A = Arg.getArrayCopy(); } int nu = Math.min(m, n); s = new double[Math.min(m + 1, n)]; U = new double[m][m]; V = new double[n][n]; double[] e = new double[n]; double[] work = new double[m]; boolean wantu = true; boolean wantv = true; // Reduce A to bidiagonal form, storing the diagonal elements // in s and the super-diagonal elements in e. int nct = Math.min(m - 1, n); int nrt = Math.max(0, Math.min(n - 2, m)); 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. s[k] = 0; for (int i = k; i < m; i++) { s[k] = Maths.hypot(s[k], A[i][k]); } if (s[k] != 0.0) { if (A[k][k] < 0.0) { s[k] = -s[k]; } for (int i = k; i < m; i++) { A[i][k] /= s[k]; } A[k][k] += 1.0; } s[k] = -s[k]; } for (int j = k + 1; j < n; j++) { if ((k < nct) & (s[k] != 0.0)) { // 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 (wantu & (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] = Maths.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 < n; i++) { e[i] /= e[k]; } e[k + 1] += 1.0; } e[k] = -e[k]; if ((k + 1 < m) & (e[k] != 0.0)) { // Apply the transformation. for (int i = k + 1; i < m; i++) { work[i] = 0.0; } 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++) { double t = -e[j] / e[k + 1]; for (int i = k + 1; i < m; i++) { A[i][j] += t * work[i]; } } } if (wantv) { // 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 = Math.min(n, m + 1); if (nct < n) { s[nct] = A[nct][nct]; } if (m < p) { s[p - 1] = 0.0; } if (nrt + 1 < p) { e[nrt] = A[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; i++) { U[i][j] = 0.0; } U[j][j] = 1.0; } for (int k = nct - 1; k >= 0; k--) { if (s[k] != 0.0) { for (int j = k + 1; j < nu; 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] = 1.0 + U[k][k]; for (int i = 0; i < k - 1; i++) { U[i][k] = 0.0; } } else { for (int i = 0; i < m; i++) { U[i][k] = 0.0; } U[k][k] = 1.0; } } } // If required, generate V. if (wantv) { for (int k = n - 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 < 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] = 0.0; } V[k][k] = 1.0; } } // Main iteration loop for the singular values. int pp = p - 1; int iter = 0; double eps = Math.pow(2.0, -52.0); double tiny = Math.pow(2.0, -966.0); while (p > 0) { int k, 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<p // kase = 2 if s(k) is negligible and k<p // kase = 3 if e[k-1] is negligible, k<p, and // s(k), ..., s(p) are not negligible (qr step). // kase = 4 if e(p-1) is negligible (convergence). for (k = p - 2; k >= -1; k--) { if (k == -1) { break; } if (Math.abs(e[k]) <= tiny + eps * (Math.abs(s[k]) + Math.abs(s[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; } double t = (ks != p ? Math.abs(e[ks]) : 0.) + (ks != k + 1 ? Math.abs(e[ks - 1]) : 0.); if (Math.abs(s[ks]) <= tiny + eps * t) { s[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 = Maths.hypot(s[j], f); double cs = s[j] / t; double sn = f / t; s[j] = t; if (j != k) { f = -sn * e[j - 1]; e[j - 1] = cs * e[j - 1]; } if (wantv) { 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] = 0.0; for (int j = k; j < p; j++) { double t = Maths.hypot(s[j], f); double cs = s[j] / t; double sn = f / t; s[j] = t; f = -sn * e[j]; e[j] = cs * e[j]; if (wantu) { 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. double scale = Math.max( Math.max(Math.max(Math.max(Math.abs(s[p - 1]), Math.abs(s[p - 2])), Math.abs(e[p - 2])), Math.abs(s[k])), Math.abs(e[k])); double sp = s[p - 1] / scale; double spm1 = s[p - 2] / scale; double epm1 = e[p - 2] / scale; double sk = s[k] / scale; double ek = e[k] / scale; double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0; 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 = Maths.hypot(f, g); double cs = f / t; double sn = g / t; if (j != k) { e[j - 1] = t; } f = cs * s[j] + sn * e[j]; e[j] = cs * e[j] - sn * s[j]; g = sn * s[j + 1]; s[j + 1] = cs * s[j + 1]; if (wantv) { 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 = Maths.hypot(f, g); cs = f / t; sn = g / t; s[j] = t; f = cs * e[j] + sn * s[j + 1]; s[j + 1] = -sn * e[j] + cs * s[j + 1]; g = sn * e[j + 1]; e[j + 1] = cs * e[j + 1]; if (wantu && (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; iter = iter + 1; } break; // Convergence. case 4: { // Make the singular values positive. if (s[k] <= 0.0) { s[k] = (s[k] < 0.0 ? -s[k] : 0.0); if (wantv) { for (int i = 0; i <= pp; i++) { V[i][k] = -V[i][k]; } } } // Order the singular values. while (k < pp) { if (s[k] >= s[k + 1]) { break; } double t = s[k]; s[k] = s[k + 1]; s[k + 1] = t; if (wantv && (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 (wantu && (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++; } iter = 0; p--; } break; } } if (usingTranspose) { int temp = m; m = n; n = temp; double[][] tempA = U; U = V; V = tempA; } } /** * Return the left singular vectors * @return U */ public Matrix getU() { return new Matrix(U, m, m); } /** * Return the right singular vectors * @return V */ public Matrix getV() { return new Matrix(V, n, n); } /** * Return the one-dimensional array of singular values * @return diagonal of S. */ public double[] getSingularValues() { return s; } /** * Return the diagonal matrix of singular values * @return S */ public Matrix getS() { Matrix X = new Matrix(m, n); double[][] S = X.getArray(); for (int i = 0; i < Math.min(m, n); i++) { S[i][i] = this.s[i]; } return X; } /** * Two norm * @return max(S) */ public double norm2() { return s[0]; } /** * Two norm condition number * @return max(S)/min(S) */ public double cond() { return s[0] / s[Math.min(m, n) - 1]; } /** * Effective numerical matrix rank * @return Number of nonnegligible singular values. */ public int rank() { double eps = Math.pow(2.0, -52.0); double tol = Math.max(m, n) * s[0] * eps; int r = 0; for (int i = 0; i < s.length; i++) { if (s[i] > tol) { r++; } } return r; } /** * Returns the revision string. * * @return the revision */ public String getRevision() { return RevisionUtils.extract("$Revision$"); } }