/*
Copyright (C) 2008 Richard Gomes
This source code is release under the BSD License.
This file is part of JQuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://jquantlib.org/
JQuantLib is free software: you can redistribute it and/or modify it
under the terms of the JQuantLib license. You should have received a
copy of the license along with this program; if not, please email
. The license is also available online at
.
This program 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 license for more details.
JQuantLib is based on QuantLib. http://quantlib.org/
When applicable, the original copyright notice follows this notice.
*/
package org.jquantlib.math.matrixutilities;
import org.jquantlib.lang.annotation.QualityAssurance;
import org.jquantlib.lang.annotation.QualityAssurance.Quality;
import org.jquantlib.lang.annotation.QualityAssurance.Version;
/**
* 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.data[S.addr.op(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.
*
* @note This class was adapted from JAMA
* @see JAMA
*
* @author Richard Gomes
*/
@QualityAssurance(quality = Quality.Q1_TRANSLATION, version = Version.OTHER, reviewers = { "Richard Gomes" })
public class SVD {
private final Matrix U;
private final Matrix V;
private final Matrix S;
private final Array s;
private final int m;
private final int n;
//
// public constructors
//
/**
* Construct the singular value decomposition
*
* @param A is a rectangular matrix
* @return Structure to access U, S and V.
*/
public SVD(final Matrix A) {
this.m = A.rows();
this.n = A.cols();
final int nu = Math.min(m, n);
this.U = new Matrix(m, nu);
this.V = new Matrix(n, n);
this.S = new Matrix(n, n);
this.s = new Array(Math.min(m + 1, n));
final double[] e = new double[n];
final double[] work = new double[m];
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 - 1, n);
final 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.data[s.addr.op(k)].
// Compute 2-norm of k-th column without under/overflow.
s.$[s.addr.op(k)] = 0;
for (int i = k; i < m; i++) {
s.$[s.addr.op(k)] = Math.hypot(s.$[s.addr.op(k)], A.$[A.addr.op(i, k)]);
}
if (s.$[s.addr.op(k)] != 0.0) {
if (A.$[A.addr.op(k, k)] < 0.0) {
s.$[s.addr.op(k)] = -s.$[s.addr.op(k)];
}
for (int i = k; i < m; i++) {
A.$[A.addr.op(i, k)] /= s.$[s.addr.op(k)];
}
A.$[A.addr.op(k, k)] += 1.0;
}
s.$[s.addr.op(k)] = -s.$[s.addr.op(k)];
}
for (int j = k + 1; j < n; j++) {
if ((k < nct) & (s.$[s.addr.op(k)] != 0.0)) {
// Apply the transformation.
double t = 0;
for (int i = k; i < m; i++) {
t += A.$[A.addr.op(i, k)] * A.$[A.addr.op(i, j)];
}
t = -t / A.$[A.addr.op(k, k)];
for (int i = k; i < m; i++) {
A.$[A.addr.op(i, j)] += t * A.$[A.addr.op(i, k)];
}
}
// Place the k-th row of A into e for the
// subsequent calculation of the row transformation.
e[j] = A.$[A.addr.op(k, j)];
}
if (wantu & (k < nct)) {
// Place the transformation in U for subsequent back
// multiplication.
for (int i = k; i < m; i++) {
U.$[U.addr.op(i, k)] = A.$[A.addr.op(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] = Math.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.$[A.addr.op(i, j)];
}
}
for (int j = k + 1; j < n; j++) {
final double t = -e[j] / e[k + 1];
for (int i = k + 1; i < m; i++) {
A.$[A.addr.op(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.$[V.addr.op(i, k)] = e[i];
}
}
}
}
// Set up the final bidiagonal matrix or order p.
int p = Math.min(n, m + 1);
if (nct < n) {
s.$[nct] = A.$[A.addr.op(nct, nct)];
}
if (m < p) {
s.$[p - 1] = 0.0;
}
if (nrt + 1 < p) {
e[nrt] = A.$[A.addr.op(nrt, p - 1)];
}
e[p - 1] = 0.0;
// If required, generate U.
if (wantu) {
for (int j = nct; j < nu; j++) {
for (int i = 0; i < m; i++) {
U.$[U.addr.op(i, j)] = 0.0;
}
U.$[U.addr.op(j, j)] = 1.0;
}
for (int k = nct - 1; k >= 0; k--) {
if (s.$[s.addr.op(k)] != 0.0) {
for (int j = k + 1; j < nu; j++) {
double t = 0;
for (int i = k; i < m; i++) {
t += U.$[U.addr.op(i, k)] * U.$[U.addr.op(i, j)];
}
t = -t / U.$[U.addr.op(k, k)];
for (int i = k; i < m; i++) {
U.$[U.addr.op(i, j)] += t * U.$[U.addr.op(i, k)];
}
}
for (int i = k; i < m; i++) {
U.$[U.addr.op(i, k)] = -U.$[U.addr.op(i, k)];
}
U.$[U.addr.op(k, k)] = 1.0 + U.$[U.addr.op(k, k)];
for (int i = 0; i < k - 1; i++) {
U.$[U.addr.op(i, k)] = 0.0;
}
} else {
for (int i = 0; i < m; i++) {
U.$[U.addr.op(i, k)] = 0.0;
}
U.$[U.addr.op(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)) { //FINDBUGS:: NS_DANGEROUS_NON_SHORT_CIRCUIT (solved)
for (int j = k + 1; j < nu; j++) {
double t = 0;
for (int i = k + 1; i < n; i++) {
t += V.$[V.addr.op(i, k)] * V.$[V.addr.op(i, j)];
}
t = -t / V.$[V.addr.op(k + 1, k)];
for (int i = k + 1; i < n; i++) {
V.$[V.addr.op(i, j)] += t * V.$[V.addr.op(i, k)];
}
}
}
for (int i = 0; i < n; i++) {
V.$[V.addr.op(i, k)] = 0.0;
}
V.$[V.addr.op(k, k)] = 1.0;
}
}
// Main iteration loop for the singular values.
final int pp = p - 1;
int iter = 0;
final double eps = Math.pow(2.0, -52.0);
while (p > 0) {
int k, kase;
// Here is where a test for too many iterations would go.
// This section of the program inspects for
// negligible elements in the s and e arrays. On
// completion the variables kase and k are set as follows.
// kase = 1 if s(p) and e[k-1] are negligible and k
= -1; k--) {
if (k == -1) {
break;
}
if (Math.abs(e[k]) <= eps * (Math.abs(s.$[s.addr.op(k)]) + Math.abs(s.$[s.addr.op(k + 1)]))) {
e[k] = 0.0;
break;
}
}
if (k == p - 2) {
kase = 4;
} else {
int ks;
for (ks = p - 1; ks >= k; ks--) {
if (ks == k) {
break;
}
final double t = (ks != p ? Math.abs(e[ks]) : 0.) + (ks != k + 1 ? Math.abs(e[ks - 1]) : 0.);
if (Math.abs(s.$[s.addr.op(ks)]) <= eps * t) {
s.$[s.addr.op(ks)] = 0.0;
break;
}
}
if (ks == k) {
kase = 3;
} else if (ks == p - 1) {
kase = 1;
} else {
kase = 2;
k = ks;
}
}
k++;
// Perform the task indicated by kase.
switch (kase) {
// Deflate negligible s(p).
case 1: {
double f = e[p - 2];
e[p - 2] = 0.0;
for (int j = p - 2; j >= k; j--) {
double t = Math.hypot(s.$[j], f);
final double cs = s.$[j] / t;
final double sn = f / t;
s.$[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.$[V.addr.op(i, j)] + sn * V.$[V.addr.op(i, p - 1)];
V.$[V.addr.op(i, p - 1)] = -sn * V.$[V.addr.op(i, j)] + cs * V.$[V.addr.op(i, p - 1)];
V.$[V.addr.op(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 = Math.hypot(s.$[j], f);
final double cs = s.$[j] / t;
final double sn = f / t;
s.$[j] = t;
f = -sn * e[j];
e[j] = cs * e[j];
if (wantu) {
for (int i = 0; i < m; i++) {
t = cs * U.$[U.addr.op(i, j)] + sn * U.$[U.addr.op(i, k - 1)];
U.$[U.addr.op(i, k - 1)] = -sn * U.$[U.addr.op(i, j)] + cs * U.$[U.addr.op(i, k - 1)];
U.$[U.addr.op(i, j)] = t;
}
}
}
}
break;
// Perform one qr step.
case 3: {
// Calculate the shift.
final double scale = Math.max(Math.max(Math.max(Math.max(Math.abs(s.$[p - 1]), Math.abs(s.$[p - 2])),
Math.abs(e[p - 2])), Math.abs(s.$[s.addr.op(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.$[s.addr.op(k)] / scale;
final double ek = e[k] / scale;
final double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;
final double c = (sp * epm1) * (sp * epm1);
double shift = 0.0;
if ((b != 0.0) | (c != 0.0)) {
shift = Math.sqrt(b * b + c);
if (b < 0.0) {
shift = -shift;
}
shift = c / (b + shift);
}
double f = (sk + sp) * (sk - sp) + shift;
double g = sk * ek;
// Chase zeros.
for (int j = k; j < p - 1; j++) {
double t = Math.hypot(f, g);
double cs = f / t;
double sn = g / t;
if (j != k) {
e[j - 1] = t;
}
f = cs * s.$[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.$[V.addr.op(i, j)] + sn * V.$[V.addr.op(i, j + 1)];
V.$[V.addr.op(i, j + 1)] = -sn * V.$[V.addr.op(i, j)] + cs * V.$[V.addr.op(i, j + 1)];
V.$[V.addr.op(i, j)] = t;
}
}
t = Math.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.$[U.addr.op(i, j)] + sn * U.$[U.addr.op(i, j + 1)];
U.$[U.addr.op(i, j + 1)] = -sn * U.$[U.addr.op(i, j)] + cs * U.$[U.addr.op(i, j + 1)];
U.$[U.addr.op(i, j)] = t;
}
}
}
e[p - 2] = f;
iter = iter + 1;
}
break;
// Convergence.
case 4: {
// Make the singular values positive.
if (s.$[s.addr.op(k)] <= 0.0) {
s.$[s.addr.op(k)] = (s.$[s.addr.op(k)] < 0.0 ? -s.$[s.addr.op(k)] : 0.0);
if (wantv) {
for (int i = 0; i <= pp; i++) {
V.$[V.addr.op(i, k)] = -V.$[V.addr.op(i, k)];
}
}
}
// Order the singular values.
while (k < pp) {
if (s.$[s.addr.op(k)] >= s.$[s.addr.op(k + 1)]) {
break;
}
double t = s.$[s.addr.op(k)];
s.$[s.addr.op(k)] = s.$[s.addr.op(k + 1)];
s.$[s.addr.op(k + 1)] = t;
if (wantv && (k < n - 1)) {
for (int i = 0; i < n; i++) {
t = V.$[V.addr.op(i, k + 1)];
V.$[V.addr.op(i, k + 1)] = V.$[V.addr.op(i, k)];
V.$[V.addr.op(i, k)] = t;
}
}
if (wantu && (k < m - 1)) {
for (int i = 0; i < m; i++) {
t = U.$[U.addr.op(i, k + 1)];
U.$[U.addr.op(i, k + 1)] = U.$[U.addr.op(i, k)];
U.$[U.addr.op(i, k)] = t;
}
}
k++;
}
iter = 0;
p--;
}
break;
}
}
// Populate S matrix
for (int i = 0; i < n; i++) {
// XXX: not needed
// for (int j = 0; j < n; j++) {
// S.data[S.addr.op(i,j)] = 0.0;
// }
S.$[S.addr.op(i, i)] = s.$[s.addr.op(i)];
}
}
//
// public methods
//
/**
* Return the left singular vectors
*
* @return U
*/
public Matrix U() {
return U.clone(); // new Matrix(U,m,Math.min(m+1,n));
}
/**
* Return the right singular vectors
*
* @return V
*/
public Matrix V() {
return V.clone(); // new Matrix(V,n,n);
}
/**
* Return the one-dimensional array of singular values
*
* @return diagonal of S.
*/
public Array singularValues() {
return s.clone();
}
/**
* Return the diagonal matrix of singular values
*
* @return S
*/
public Matrix S() {
return S.clone();
}
/**
* Two norm
*
* @return max(S)
*/
public double norm2() {
return s.$[s.addr.op(0)];
}
/**
* Two norm condition number
*
* @return max(S)/min(S)
*/
public double cond() {
return s.$[s.addr.op(0)] / s.$[Math.min(m, n) - 1];
}
/**
* Effective numerical matrix rank
*
* @return Number of nonnegligible singular values.
*/
public int rank() {
final double eps = Math.pow(2.0, -52.0);
final double tol = Math.max(m, n) * s.$[s.addr.op(0)] * eps;
int r = 0;
for (int i = 0; i < s.size(); i++) {
if (s.$[s.addr.op(i)] > tol) {
r++;
}
}
return r;
}
}
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