org.jeometry.simple.math.decomposition.SimpleSVDDecomposition Maven / Gradle / Ivy
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package org.jeometry.simple.math.decomposition;
import java.util.ArrayList;
import java.util.List;
import org.jeometry.Jeometry;
import org.jeometry.factory.JeometryFactory;
import org.jeometry.math.Matrix;
import org.jeometry.math.decomposition.SVDDecomposition;
/**
* A simple implementation of {@link SVDDecomposition SVDDecomposition}.
* This implantation is inspired by Jama LU Decomposition.
* @author Julien Seinturier - COMEX S.A. - [email protected] - https://github.com/jorigin/jeometry
* @version {@value Jeometry#version} b{@value Jeometry#BUILD}
* @since 1.0.0
*/
public class SimpleSVDDecomposition implements SVDDecomposition {
/**
* The U matrix;
*/
private Matrix U;
/**
* The D matrix.
*/
private Matrix S;
/**
* The V matrix.
*/
private Matrix V;
/**
* Array for internal storage of singular values.
*/
private double[] s;
/**
* The row dimension.
*/
private int inputRows;
/**
* The column dimension.
*/
private int inputColumns;
/**
* Construct the singular value decomposition Structure to access U, S and V.
* @param matrix the matrix to decompose
*/
public SimpleSVDDecomposition (Matrix matrix) {
// Derived from LINPACK code.
// Initialize.
double[][] A = matrix.getDataArray2D();
inputRows = matrix.getRowsCount();
inputColumns = matrix.getColumnsCount();
int nu = Math.min(inputRows,inputColumns);
s = new double [Math.min(inputRows+1,inputColumns)];
U = JeometryFactory.createMatrix(inputRows, nu);
V = JeometryFactory.createMatrix(inputColumns, inputColumns);
double[] e = new double [inputColumns];
double[] work = new double [inputRows];
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(inputRows-1,inputColumns);
int nrt = Math.max(0,Math.min(inputColumns-2,inputRows));
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 < inputRows; i++) {
s[k] = 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 < inputRows; i++) {
A[i][k] /= s[k];
}
A[k][k] += 1.0;
}
s[k] = -s[k];
}
for (int j = k+1; j < inputColumns; j++) {
if ((k < nct) & (s[k] != 0.0)) {
// Apply the transformation.
double t = 0;
for (int i = k; i < inputRows; i++) {
t += A[i][k]*A[i][j];
}
t = -t/A[k][k];
for (int i = k; i < inputRows; 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 < inputRows; i++) {
U.setValue(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 < inputColumns; i++) {
e[k] = 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 < inputColumns; i++) {
e[i] /= e[k];
}
e[k+1] += 1.0;
}
e[k] = -e[k];
if ((k+1 < inputRows) & (e[k] != 0.0)) {
// Apply the transformation.
for (int i = k+1; i < inputRows; i++) {
work[i] = 0.0;
}
for (int j = k+1; j < inputColumns; j++) {
for (int i = k+1; i < inputRows; i++) {
work[i] += e[j]*A[i][j];
}
}
for (int j = k+1; j < inputColumns; j++) {
double t = -e[j]/e[k+1];
for (int i = k+1; i < inputRows; i++) {
A[i][j] += t*work[i];
}
}
}
if (wantv) {
// Place the transformation in V for subsequent
// back multiplication.
for (int i = k+1; i < inputColumns; i++) {
V.setValue(i, k, e[i]);
}
}
}
}
// Set up the final bidiagonal matrix or order p.
int p = Math.min(inputColumns,inputRows+1);
if (nct < inputColumns) {
s[nct] = A[nct][nct];
}
if (inputRows < 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 < inputRows; i++) {
U.setValue(i, j, 0.0);
}
U.setValue(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 < inputRows; i++) {
t += U.getValue(i, k)*U.getValue(i, j);
}
t = -t/U.getValue(k, k);
for (int i = k; i < inputRows; i++) {
U.setValue(i, j, U.getValue(i, j) + t*U.getValue(i, k));
}
}
for (int i = k; i < inputRows; i++ ) {
U.setValue(i, k, -U.getValue(i, k));
}
U.setValue(k, k, 1.0 + U.getValue(k, k));
for (int i = 0; i < k-1; i++) {
U.setValue(i, k, 0.0);
}
} else {
for (int i = 0; i < inputRows; i++) {
U.setValue(i, k, 0.0);
}
U.setValue(k, k, 1.0);
}
}
}
// If required, generate V.
if (wantv) {
for (int k = inputColumns-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 < inputColumns; i++) {
t += V.getValue(i, k)*V.getValue(i, j);
}
t = -t/V.getValue(k+1, k);
for (int i = k+1; i < inputColumns; i++) {
V.setValue(i, j, V.getValue(i, j) + t*V.getValue(i, k));
}
}
}
for (int i = 0; i < inputColumns; i++) {
V.setValue(i, k, 0.0);
}
V.setValue(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= -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 = 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 < inputColumns; i++) {
t = cs*V.getValue(i, j) + sn*V.getValue(i, p-1);
V.setValue(i, p-1, -sn*V.getValue(i, j) + cs*V.getValue(i, p-1));
V.setValue(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 = 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 < inputRows; i++) {
t = cs*U.getValue(i, j) + sn*U.getValue(i, k-1);
U.setValue(i, k-1, -sn*U.getValue(i, j) + cs*U.getValue(i, k-1));
U.setValue(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 = 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 < inputColumns; i++) {
t = cs*V.getValue(i, j) + sn*V.getValue(i, j+1);
V.setValue(i, j+1, -sn*V.getValue(i, j) + cs*V.getValue(i, j+1));
V.setValue(i, j, t);
}
}
t = 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 < inputRows-1)) {
for (int i = 0; i < inputRows; i++) {
t = cs*U.getValue(i, j) + sn*U.getValue(i, j+1);
U.setValue(i, j+1, -sn*U.getValue(i, j) + cs*U.getValue(i, j+1));
U.setValue(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.setValue(i, k, -V.getValue(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 < inputColumns-1)) {
for (int i = 0; i < inputColumns; i++) {
t = V.getValue(i, k+1);
V.setValue(i, k+1, V.getValue(i, k));
V.setValue(i, k, t);
}
}
if (wantu && (k < inputRows-1)) {
for (int i = 0; i < inputRows; i++) {
t = U.getValue(i, k+1);
U.setValue(i, k+1, U.getValue(i, k));
U.setValue(i, k, t);
}
}
k++;
}
iter = 0;
p--;
}
break;
}
}
S = JeometryFactory.createMatrix(inputColumns,inputColumns);
for (int i = 0; i < inputColumns; i++) {
for (int j = 0; j < inputColumns; j++) {
S.setValue(i, j, 0.0);
}
S.setValue(i, i, s[i]);
}
}
/** 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(inputRows,inputColumns)-1];
}
/** Effective numerical matrix rank
* @return The number of non eligible singular values
*/
public int rank () {
double eps = Math.pow(2.0,-52.0);
double tol = Math.max(inputRows,inputColumns)*s[0]*eps;
int r = 0;
for (int i = 0; i < s.length; i++) {
if (s[i] > tol) {
r++;
}
}
return r;
}
@Override
public List getComponents() {
List components = new ArrayList(3);
components.add(U);
components.add(S);
components.add(V);
return components;
}
@Override
public Matrix getU() {
return U;
}
@Override
public Matrix getS() {
return S;
}
@Override
public Matrix getV() {
return V;
}
/** Compute sqrt(a^2 + b^2) without under/overflow.
* @param a the first
* @param b the second
* @return the result
**/
private double hypot(double a, double b) {
double r;
if (Math.abs(a) > Math.abs(b)) {
r = b/a;
r = Math.abs(a)*Math.sqrt(1+r*r);
} else if (b != 0) {
r = a/b;
r = Math.abs(b)*Math.sqrt(1+r*r);
} else {
r = 0.0;
}
return r;
}
}