/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* 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
* 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.
*
* Copyright 1999 CERN - European Organization for Nuclear Research.
* Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose
* is hereby granted without fee, provided that the above copyright notice appear in all copies and
* that both that copyright notice and this permission notice appear in supporting documentation.
* CERN makes no representations about the suitability of this software for any purpose.
* It is provided "as is" without expressed or implied warranty.
*/
package org.apache.mahout.math;
public class SingularValueDecomposition implements java.io.Serializable {
/** Arrays for internal storage of U and V. */
private final double[][] u;
private final double[][] v;
/** Array for internal storage of singular values. */
private final double[] s;
/** Row and column dimensions. */
private final int m;
private final int n;
/**To handle the case where numRows() < numCols() and to use the fact that SVD(A')=VSU'=> SVD(A')'=SVD(A)**/
private boolean transpositionNeeded = false;
/**
* Constructs and returns a new singular value decomposition object; The
* decomposed matrices can be retrieved via instance methods of the returned
* decomposition object.
*
* @param arg
* A rectangular matrix.
*/
public SingularValueDecomposition(Matrix arg) {
if (arg.numRows() < arg.numCols()) {
transpositionNeeded = true;
}
// Derived from LINPACK code.
// Initialize.
double[][] a;
if (transpositionNeeded) {
//use the transpose Matrix
m = arg.numCols();
n = arg.numRows();
a = new double[m][n];
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
a[i][j] = arg.get(j, i);
}
}
} else {
m = arg.numRows();
n = arg.numCols();
a = new double[m][n];
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
a[i][j] = arg.get(i, j);
}
}
}
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] = Algebra.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] = Algebra.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);
while (p > 0) {
int k;
// 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]) <= eps * (Math.abs(s[k]) + Math.abs(s[k + 1]))) {
e[k] = 0.0;
break;
}
}
int kase;
if (k == p - 2) {
kase = 4;
} else {
int ks;
for (ks = p - 1; ks >= k; ks--) {
if (ks == k) {
break;
}
double t = (ks == p ? 0.0 : Math.abs(e[ks])) + (ks == k + 1 ? 0.0 : Math.abs(e[ks - 1]));
if (Math.abs(s[ks]) <= 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 = Algebra.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 = Algebra.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 = Algebra.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 = Algebra.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 += 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;
default:
throw new IllegalStateException();
}
}
}
/**
* Returns the two norm condition number, which is <tt>max(S) / min(S)</tt>.
*/
public double cond() {
return s[0] / s[Math.min(m, n) - 1];
}
/**
* @return the diagonal matrix of singular values.
*/
public Matrix getS() {
double[][] s = new double[n][n];
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 new DenseMatrix(s);
}
/**
* Returns the diagonal of <tt>S</tt>, which is a one-dimensional array of
* singular values
*
* @return diagonal of <tt>S</tt>.
*/
public double[] getSingularValues() {
return s;
}
/**
* Returns the left singular vectors <tt>U</tt>.
*
* @return <tt>U</tt>
*/
public Matrix getU() {
if (transpositionNeeded) { //case numRows() < numCols()
return new DenseMatrix(v);
} else {
int numCols = Math.min(m + 1, n);
Matrix r = new DenseMatrix(m, numCols);
for (int i = 0; i < m; i++) {
for (int j = 0; j < numCols; j++) {
r.set(i, j, u[i][j]);
}
}
return r;
}
}
/**
* Returns the right singular vectors <tt>V</tt>.
*
* @return <tt>V</tt>
*/
public Matrix getV() {
if (transpositionNeeded) { //case numRows() < numCols()
int numCols = Math.min(m + 1, n);
Matrix r = new DenseMatrix(m, numCols);
for (int i = 0; i < m; i++) {
for (int j = 0; j < numCols; j++) {
r.set(i, j, u[i][j]);
}
}
return r;
} else {
return new DenseMatrix(v);
}
}
/** Returns the two norm, which is <tt>max(S)</tt>. */
public double norm2() {
return s[0];
}
/**
* Returns the effective numerical matrix rank, which is the 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 (double value : s) {
if (value > tol) {
r++;
}
}
return r;
}
/**
* @parameter minSingularValue
* minSingularValue - value below which singular values are ignored (a 0 or negative
* value implies all singular value will be used)
* @return 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.
*/
Matrix getCovariance(double minSingularValue) {
Matrix j = new DenseMatrix(s.length,s.length);
Matrix vMat = new DenseMatrix(this.v);
for (int i = 0; i < s.length; i++) {
j.set(i, i, s[i] >= minSingularValue ? 1 / (s[i] * s[i]) : 0.0);
}
return vMat.times(j).times(vMat.transpose());
}
/**
* Returns a String with (propertyName, propertyValue) pairs. Useful for
* debugging or to quickly get the rough picture. For example,
*
* <pre>
* rank : 3
* trace : 0
* </pre>
*/
@Override
public String toString() {
StringBuilder buf = new StringBuilder();
buf.append("---------------------------------------------------------------------\n");
buf.append("SingularValueDecomposition(A) --> cond(A), rank(A), norm2(A), U, S, V\n");
buf.append("---------------------------------------------------------------------\n");
buf.append("cond = ");
String unknown = "Illegal operation or error: ";
try {
buf.append(String.valueOf(this.cond()));
} catch (IllegalArgumentException exc) {
buf.append(unknown).append(exc.getMessage());
}
buf.append("\nrank = ");
try {
buf.append(String.valueOf(this.rank()));
} catch (IllegalArgumentException exc) {
buf.append(unknown).append(exc.getMessage());
}
buf.append("\nnorm2 = ");
try {
buf.append(String.valueOf(this.norm2()));
} catch (IllegalArgumentException exc) {
buf.append(unknown).append(exc.getMessage());
}
buf.append("\n\nU = ");
try {
buf.append(String.valueOf(this.getU()));
} catch (IllegalArgumentException exc) {
buf.append(unknown).append(exc.getMessage());
}
buf.append("\n\nS = ");
try {
buf.append(String.valueOf(this.getS()));
} catch (IllegalArgumentException exc) {
buf.append(unknown).append(exc.getMessage());
}
buf.append("\n\nV = ");
try {
buf.append(String.valueOf(this.getV()));
} catch (IllegalArgumentException exc) {
buf.append(unknown).append(exc.getMessage());
}
return buf.toString();
}
}