/*
* Copyright (C) 2008-2015 by Holger Arndt
*
* This file is part of the Universal Java Matrix Package (UJMP).
* See the NOTICE file distributed with this work for additional
* information regarding copyright ownership and licensing.
*
* UJMP is free software; you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as
* published by the Free Software Foundation; either version 2
* of the License, or (at your option) any later version.
*
* UJMP is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with UJMP; if not, write to the
* Free Software Foundation, Inc., 51 Franklin St, Fifth Floor,
* Boston, MA 02110-1301 USA
*/
package org.ujmp.core.doublematrix.calculation.general.decomposition;
import org.ujmp.core.Matrix;
import org.ujmp.core.doublematrix.DenseDoubleMatrix2D;
import org.ujmp.core.util.DecompositionOps;
import org.ujmp.core.util.MathUtil;
import org.ujmp.core.util.UJMPSettings;
/**
* Singular Value Decomposition.
*
* For an m-by-n matrix A, the singular value decomposition is an m-by-(m or n)
* orthogonal matrix U, a (m or 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 decompostion always exists, so the constructor will never
* fail. The matrix condition number and the effective numerical rank can be
* computed from this decomposition.
*/
public interface SVD {
public T[] calc(T source);
public static int THRESHOLD = 100;
public static final SVD MATRIX = new SVD() {
public final Matrix[] calc(Matrix source) {
if (UJMPSettings.getInstance().getNumberOfThreads() == 1) {
if (source.getRowCount() >= THRESHOLD && source.getColumnCount() >= THRESHOLD) {
return MATRIXLARGESINGLETHREADED.calc(source);
} else {
return MATRIXSMALLSINGLETHREADED.calc(source);
}
} else {
if (source.getRowCount() >= THRESHOLD && source.getColumnCount() >= THRESHOLD) {
return MATRIXLARGEMULTITHREADED.calc(source);
} else {
return MATRIXSMALLMULTITHREADED.calc(source);
}
}
}
};
public static final SVD INSTANCE = MATRIX;
public static final SVD UJMP = new SVD() {
public final Matrix[] calc(Matrix source) {
SVDMatrix svd = new SVDMatrix(source);
return new Matrix[] { svd.getU(), svd.getS(), svd.getV() };
}
};
public static final SVD MATRIXSMALLSINGLETHREADED = UJMP;
public static final SVD MATRIXLARGESINGLETHREADED = new SVD() {
public final Matrix[] calc(Matrix source) {
SVD svd = null;
if (UJMPSettings.getInstance().isUseMTJ()) {
svd = DecompositionOps.SVD_MTJ;
}
if (svd == null && UJMPSettings.getInstance().isUseOjalgo()) {
svd = DecompositionOps.SVD_OJALGO;
}
if (svd == null && UJMPSettings.getInstance().isUseEJML()) {
svd = DecompositionOps.SVD_EJML;
}
if (svd == null) {
svd = UJMP;
}
return svd.calc(source);
}
};
public static final SVD MATRIXSMALLMULTITHREADED = UJMP;
public static final SVD MATRIXLARGEMULTITHREADED = new SVD() {
public final Matrix[] calc(Matrix source) {
SVD svd = null;
if (UJMPSettings.getInstance().isUseOjalgo()) {
svd = DecompositionOps.SVD_OJALGO;
}
if (svd == null && UJMPSettings.getInstance().isUseMTJ()) {
svd = DecompositionOps.SVD_MTJ;
}
if (svd == null && UJMPSettings.getInstance().isUseEJML()) {
svd = DecompositionOps.SVD_EJML;
}
if (svd == null) {
svd = UJMP;
}
return svd.calc(source);
}
};
final class SVDMatrix {
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[][] U, V;
/**
* Array for internal storage of singular values.
*
* @serial internal storage of singular values.
*/
private final double[] s;
/**
* Row and column dimensions.
*
* @serial row dimension.
* @serial column dimension.
* @serial U column dimension.
*/
private final int m, n, ncu;
/**
* Column specification of matrix U
*
* @serial U column dimension toggle
*/
private final boolean thin;
/*
* ------------------------ Old Constructor ------------------------
*/
/**
* Construct the singular value decomposition
*
* @param Arg
* Rectangular matrix
*/
public SVDMatrix(Matrix Arg) {
this(Arg, true, true, true);
}
/*
* ------------------------ Constructor ------------------------
*/
/**
* Construct the singular value decomposition
*
* @param Arg
* Rectangular matrix
* @param thin
* If true U is economy sized
* @param wantu
* If true generate the U matrix
* @param wantv
* If true generate the V matrix
*/
public SVDMatrix(Matrix Arg, boolean thin, boolean wantu, boolean wantv) {
if (Arg.containsMissingValues()) {
throw new RuntimeException("matrix contains missing values");
}
// Derived from LINPACK code.
// Initialize.
final double[][] A = Arg.toDoubleArray();
m = (int) Arg.getRowCount();
n = (int) Arg.getColumnCount();
this.thin = thin;
ncu = thin ? Math.min(m, n) : m;
s = new double[Math.min(m + 1, n)];
U = new double[m][ncu];
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 = Math.min(m - 1, n);
final int nrt = Math.max(0, Math.min(n - 2, m));
final int lu = Math.max(nct, nrt);
for (int k = 0; k < lu; 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] = MathUtil.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] = MathUtil.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 < ncu; 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 < ncu; 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;
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 < 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] = 0.0;
}
V[k][k] = 1.0;
}
}
// Main iteration loop for the singular values.
final int pp = p - 1;
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= -1; k--) {
if (k == -1) {
break;
}
if (Math.abs(e[k]) <= TINY + EPSILON * (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 + EPSILON * 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 = MathUtil.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 = MathUtil.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.
final 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]));
final double sp = s[p - 1] / scale;
final double spm1 = s[p - 2] / scale;
final double epm1 = e[p - 2] / scale;
final double sk = s[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 = MathUtil.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 = MathUtil.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;
}
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 < n; 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++;
}
p--;
}
break;
}
}
}
/*
* ------------------------ Public Methods ------------------------
*/
/**
* Return the left singular vectors
*
* @return U
*/
public final DenseDoubleMatrix2D getU() {
final double[][] x = new double[m][m >= n ? (thin ? Math.min(m + 1, n) : ncu) : ncu];
for (int r = 0; r < m; r++) {
for (int c = x[0].length; --c >= 0;) {
x[r][c] = U[r][c];
}
}
return Matrix.Factory.linkToArray(x);
}
/**
* Return the right singular vectors
*
* @return V
*/
public final DenseDoubleMatrix2D getV() {
return V == null ? null : Matrix.Factory.linkToArray(V);
}
/**
* Return the one-dimensional array of singular values
*
* @return diagonal of S.
*/
public final double[] getSingularValues() {
return s;
}
/**
* Return the diagonal matrix of singular values
*
* @return S
*/
public final DenseDoubleMatrix2D getS() {
final double[][] X = new double[m >= n ? (thin ? n : ncu) : ncu][n];
for (int i = Math.min(m, n); --i >= 0;)
X[i][i] = s[i];
return Matrix.Factory.linkToArray(X);
}
/**
* Return the diagonal matrix of the reciprocals of the singular values
*
* @return S+
*/
public final DenseDoubleMatrix2D getreciprocalS() {
final double[][] X = new double[n][m >= n ? (thin ? n : ncu) : ncu];
for (int i = Math.min(m, n) - 1; i >= 0; i--)
X[i][i] = s[i] == 0.0 ? 0.0 : 1.0 / s[i];
return Matrix.Factory.linkToArray(X);
}
/**
* Return the Moore-Penrose (generalized) inverse Slightly modified
* version of Kim van der Linde's code
*
* @param omit
* if true tolerance based omitting of negligible singular
* values
* @return A+
*/
public final DenseDoubleMatrix2D inverse(boolean omit) {
final double[][] inverse = new double[n][m];
if (rank() > 0) {
final double[] reciprocalS = new double[s.length];
if (omit) {
double tol = Math.max(m, n) * s[0] * EPSILON;
for (int i = s.length - 1; i >= 0; i--)
reciprocalS[i] = Math.abs(s[i]) < tol ? 0.0 : 1.0 / s[i];
} else
for (int i = s.length - 1; i >= 0; i--)
reciprocalS[i] = s[i] == 0.0 ? 0.0 : 1.0 / s[i];
int min = Math.min(n, ncu);
for (int i = n - 1; i >= 0; i--)
for (int j = m - 1; j >= 0; j--)
for (int k = min - 1; k >= 0; k--)
inverse[i][j] += V[i][k] * reciprocalS[k] * U[j][k];
}
return Matrix.Factory.linkToArray(inverse);
}
/**
* Two norm
*
* @return max(S)
*/
public final double norm2() {
return s[0];
}
/**
* Two norm condition number
*
* @return max(S)/min(S)
*/
public final double cond() {
return s[0] / s[Math.min(m, n) - 1];
}
/**
* Effective numerical matrix rank
*
* @return Number of nonnegligible singular values.
*/
public final int rank() {
final double tol = Math.max(m, n) * s[0] * EPSILON;
int r = 0;
for (int i = 0; i < s.length; i++) {
if (s[i] > tol) {
r++;
}
}
return r;
}
}
}
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