/*
* Encog(tm) Core v3.4 - Java Version
* http://www.heatonresearch.com/encog/
* https://github.com/encog/encog-java-core
* Copyright 2008-2016 Heaton Research, Inc.
*
* 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.
*
* For more information on Heaton Research copyrights, licenses
* and trademarks visit:
* http://www.heatonresearch.com/copyright
*/
package org.encog.mathutil.matrices.decomposition;
/*
* Encog(tm) Artificial Intelligence Framework v2.3
* Java Version
* http://www.heatonresearch.com/encog/
* http://code.google.com/p/encog-java/
*
* Copyright 2008-2010 by Heaton Research Inc.
*
* Released under the LGPL.
*
* This 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 2.1 of
* the License, or (at your option) any later version.
*
* This software 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 this software; if not, write to the Free
* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA
* 02110-1301 USA, or see the FSF site: http://www.fsf.org.
*
* Encog and Heaton Research are Trademarks of Heaton Research, Inc.
* For information on Heaton Research trademarks, visit:
*
* http://www.heatonresearch.com/copyright.html
*/
import org.encog.mathutil.EncogMath;
import org.encog.mathutil.matrices.Matrix;
/**
* 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.
*
* This file based on a class from the public domain JAMA package.
* http://math.nist.gov/javanumerics/jama/
*/
public class SingularValueDecomposition {
/**
* Arrays for internal storage of U and V.
*/
private double[][] U, V;
/**
* Array for internal storage of singular values.
*/
private double[] s;
/**
* Row and column dimensions.
*/
private int m, n;
/**
* Construct the singular value decomposition
* Structure to access U, S and V.
* @param Arg
* Rectangular matrix
*/
public SingularValueDecomposition(Matrix Arg) {
// Derived from LINPACK code.
// Initialize.
double[][] A = Arg.getArrayCopy();
m = Arg.getRows();
n = Arg.getCols();
/*
* 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"); }
*/
int nu = Math.min(m, n);
s = new double[Math.min(m + 1, n)];
U = new double[m][nu];
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] = EncogMath.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] = EncogMath.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 = EncogMath.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 = EncogMath.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 = EncogMath.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 = EncogMath.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;
}
}
}
/*
* ------------------------ Public Methods ------------------------
*/
/**
* Return the left singular vectors
*
* @return U
*/
public Matrix getU() {
return new Matrix(U);
}
/**
* Return the right singular vectors
*
* @return V
*/
public Matrix getV() {
return new Matrix(V);
}
/**
* 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(n, n);
double[][] S = X.getData();
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
S[i][j] = 0.0;
}
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;
}
}