SingularValueDecomposition.java
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* Unless required by applicable law or agreed to in writing, software
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/*
* This is not the original file distributed by the Apache Software Foundation
* It has been modified by the Hipparchus project
*/
package org.hipparchus.linear;
import org.hipparchus.exception.LocalizedCoreFormats;
import org.hipparchus.exception.MathIllegalArgumentException;
import org.hipparchus.util.FastMath;
import org.hipparchus.util.Precision;
/**
* Calculates the compact Singular Value Decomposition of a matrix.
* <p>
* The Singular Value Decomposition of matrix A is a set of three matrices: U,
* Σ and V such that A = U × Σ × V<sup>T</sup>. Let A be
* a m × n matrix, then U is a m × p orthogonal matrix, Σ is a
* p × p diagonal matrix with positive or null elements, V is a p ×
* n orthogonal matrix (hence V<sup>T</sup> is also orthogonal) where
* p=min(m,n).
* </p>
* <p>This class is similar to the class with similar name from the
* <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library, with the
* following changes:</p>
* <ul>
* <li>the {@code norm2} method which has been renamed as {@link #getNorm()
* getNorm},</li>
* <li>the {@code cond} method which has been renamed as {@link
* #getConditionNumber() getConditionNumber},</li>
* <li>the {@code rank} method which has been renamed as {@link #getRank()
* getRank},</li>
* <li>a {@link #getUT() getUT} method has been added,</li>
* <li>a {@link #getVT() getVT} method has been added,</li>
* <li>a {@link #getSolver() getSolver} method has been added,</li>
* <li>a {@link #getCovariance(double) getCovariance} method has been added.</li>
* </ul>
* @see <a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">MathWorld</a>
* @see <a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia</a>
*/
public class SingularValueDecomposition {
/** Relative threshold for small singular values. */
private static final double EPS = 0x1.0p-52;
/** Absolute threshold for small singular values. */
private static final double TINY = 0x1.0p-966;
/** Computed singular values. */
private final double[] singularValues;
/** max(row dimension, column dimension). */
private final int m;
/** min(row dimension, column dimension). */
private final int n;
/** Cached value of U matrix. */
private final RealMatrix cachedU;
/** Cached value of transposed U matrix. */
private RealMatrix cachedUt;
/** Cached value of S (diagonal) matrix. */
private RealMatrix cachedS;
/** Cached value of V matrix. */
private final RealMatrix cachedV;
/** Cached value of transposed V matrix. */
private RealMatrix cachedVt;
/**
* Tolerance value for small singular values, calculated once we have
* populated "singularValues".
**/
private final double tol;
/**
* Calculates the compact Singular Value Decomposition of the given matrix.
*
* @param matrix Matrix to decompose.
*/
public SingularValueDecomposition(final RealMatrix matrix) {
final double[][] A;
// "m" is always the largest dimension.
final boolean transposed;
if (matrix.getRowDimension() < matrix.getColumnDimension()) {
transposed = true;
A = matrix.transpose().getData();
m = matrix.getColumnDimension();
n = matrix.getRowDimension();
} else {
transposed = false;
A = matrix.getData();
m = matrix.getRowDimension();
n = matrix.getColumnDimension();
}
singularValues = new double[n];
final double[][] U = new double[m][n];
final double[][] 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 = FastMath.min(m - 1, n);
final int nrt = FastMath.max(0, n - 2);
for (int k = 0; k < FastMath.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.
singularValues[k] = 0;
for (int i = k; i < m; i++) {
singularValues[k] = FastMath.hypot(singularValues[k], A[i][k]);
}
if (singularValues[k] != 0) {
if (A[k][k] < 0) {
singularValues[k] = -singularValues[k];
}
for (int i = k; i < m; i++) {
A[i][k] /= singularValues[k];
}
A[k][k] += 1;
}
singularValues[k] = -singularValues[k];
}
for (int j = k + 1; j < n; j++) {
if (k < nct &&
singularValues[k] != 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 (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] = FastMath.hypot(e[k], e[i]);
}
if (e[k] != 0) {
if (e[k + 1] < 0) {
e[k] = -e[k];
}
for (int i = k + 1; i < n; i++) {
e[i] /= e[k];
}
e[k + 1] += 1;
}
e[k] = -e[k];
if (k + 1 < m &&
e[k] != 0) {
// Apply the transformation.
for (int i = k + 1; i < m; i++) {
work[i] = 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++) {
final double t = -e[j] / e[k + 1];
for (int i = k + 1; i < m; i++) {
A[i][j] += t * work[i];
}
}
}
// 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 = n;
if (nct < n) {
singularValues[nct] = A[nct][nct];
}
if (m < p) {
singularValues[p - 1] = 0;
}
if (nrt + 1 < p) {
e[nrt] = A[nrt][p - 1];
}
e[p - 1] = 0;
// Generate U.
for (int j = nct; j < n; j++) {
for (int i = 0; i < m; i++) {
U[i][j] = 0;
}
U[j][j] = 1;
}
for (int k = nct - 1; k >= 0; k--) {
if (singularValues[k] != 0) {
for (int j = k + 1; j < n; 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 + U[k][k];
for (int i = 0; i < k - 1; i++) {
U[i][k] = 0;
}
} else {
for (int i = 0; i < m; i++) {
U[i][k] = 0;
}
U[k][k] = 1;
}
}
// Generate V.
for (int k = n - 1; k >= 0; k--) {
if (k < nrt &&
e[k] != 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;
}
V[k][k] = 1;
}
// Main iteration loop for the singular values.
final int pp = p - 1;
while (p > 0) {
int k;
int 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 >= 0; k--) {
final double threshold
= TINY + EPS * (FastMath.abs(singularValues[k]) +
FastMath.abs(singularValues[k + 1]));
// the following condition is written this way in order
// to break out of the loop when NaN occurs, writing it
// as "if (FastMath.abs(e[k]) <= threshold)" would loop
// indefinitely in case of NaNs because comparison on NaNs
// always return false, regardless of what is checked
// see issue MATH-947
if (!(FastMath.abs(e[k]) > threshold)) { // NOPMD - as explained above, the way this test is written is correct
e[k] = 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 ? FastMath.abs(e[ks]) : 0) +
(ks != k + 1 ? FastMath.abs(e[ks - 1]) : 0);
if (FastMath.abs(singularValues[ks]) <= TINY + EPS * t) {
singularValues[ks] = 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) { // NOPMD - breaking this complex algorithm into functions just to keep PMD happy would be artificial
// Deflate negligible s(p).
case 1: {
double f = e[p - 2];
e[p - 2] = 0;
for (int j = p - 2; j >= k; j--) {
double t = FastMath.hypot(singularValues[j], f);
final double cs = singularValues[j] / t;
final double sn = f / t;
singularValues[j] = t;
if (j != k) {
f = -sn * e[j - 1];
e[j - 1] = cs * e[j - 1];
}
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;
for (int j = k; j < p; j++) {
double t = FastMath.hypot(singularValues[j], f);
final double cs = singularValues[j] / t;
final double sn = f / t;
singularValues[j] = t;
f = -sn * e[j];
e[j] = cs * e[j];
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 maxPm1Pm2 = FastMath.max(FastMath.abs(singularValues[p - 1]),
FastMath.abs(singularValues[p - 2]));
final double scale = FastMath.max(FastMath.max(FastMath.max(maxPm1Pm2,
FastMath.abs(e[p - 2])),
FastMath.abs(singularValues[k])),
FastMath.abs(e[k]));
final double sp = singularValues[p - 1] / scale;
final double spm1 = singularValues[p - 2] / scale;
final double epm1 = e[p - 2] / scale;
final double sk = singularValues[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;
if (b != 0 ||
c != 0) {
shift = FastMath.sqrt(b * b + c);
if (b < 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 = FastMath.hypot(f, g);
double cs = f / t;
double sn = g / t;
if (j != k) {
e[j - 1] = t;
}
f = cs * singularValues[j] + sn * e[j];
e[j] = cs * e[j] - sn * singularValues[j];
g = sn * singularValues[j + 1];
singularValues[j + 1] = cs * singularValues[j + 1];
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 = FastMath.hypot(f, g);
cs = f / t;
sn = g / t;
singularValues[j] = t;
f = cs * e[j] + sn * singularValues[j + 1];
singularValues[j + 1] = -sn * e[j] + cs * singularValues[j + 1];
g = sn * e[j + 1];
e[j + 1] = cs * e[j + 1];
if (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.
default: {
// Make the singular values positive.
if (singularValues[k] <= 0) {
singularValues[k] = singularValues[k] < 0 ? -singularValues[k] : 0;
for (int i = 0; i <= pp; i++) {
V[i][k] = -V[i][k];
}
}
// Order the singular values.
while (k < pp) {
if (singularValues[k] >= singularValues[k + 1]) {
break;
}
double t = singularValues[k];
singularValues[k] = singularValues[k + 1];
singularValues[k + 1] = t;
if (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 (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;
}
}
// Set the small value tolerance used to calculate rank and pseudo-inverse
tol = FastMath.max(m * singularValues[0] * EPS,
FastMath.sqrt(Precision.SAFE_MIN));
if (!transposed) {
cachedU = MatrixUtils.createRealMatrix(U);
cachedV = MatrixUtils.createRealMatrix(V);
} else {
cachedU = MatrixUtils.createRealMatrix(V);
cachedV = MatrixUtils.createRealMatrix(U);
}
}
/**
* Returns the matrix U of the decomposition.
* <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
* @return the U matrix
* @see #getUT()
*/
public RealMatrix getU() {
// return the cached matrix
return cachedU;
}
/**
* Returns the transpose of the matrix U of the decomposition.
* <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
* @return the U matrix (or null if decomposed matrix is singular)
* @see #getU()
*/
public RealMatrix getUT() {
if (cachedUt == null) {
cachedUt = getU().transpose();
}
// return the cached matrix
return cachedUt;
}
/**
* Returns the diagonal matrix Σ of the decomposition.
* <p>Σ is a diagonal matrix. The singular values are provided in
* non-increasing order, for compatibility with Jama.</p>
* @return the Σ matrix
*/
public RealMatrix getS() {
if (cachedS == null) {
// cache the matrix for subsequent calls
cachedS = MatrixUtils.createRealDiagonalMatrix(singularValues);
}
return cachedS;
}
/**
* Returns the diagonal elements of the matrix Σ of the decomposition.
* <p>The singular values are provided in non-increasing order, for
* compatibility with Jama.</p>
* @return the diagonal elements of the Σ matrix
*/
public double[] getSingularValues() {
return singularValues.clone();
}
/**
* Returns the matrix V of the decomposition.
* <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
* @return the V matrix (or null if decomposed matrix is singular)
* @see #getVT()
*/
public RealMatrix getV() {
// return the cached matrix
return cachedV;
}
/**
* Returns the transpose of the matrix V of the decomposition.
* <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
* @return the V matrix (or null if decomposed matrix is singular)
* @see #getV()
*/
public RealMatrix getVT() {
if (cachedVt == null) {
cachedVt = getV().transpose();
}
// return the cached matrix
return cachedVt;
}
/**
* Returns the n × n covariance matrix.
* <p>The covariance matrix is V × J × V<sup>T</sup>
* where J is the diagonal matrix of the inverse of the squares of
* the singular values.</p>
* @param minSingularValue value below which singular values are ignored
* (a 0 or negative value implies all singular value will be used)
* @return covariance matrix
* @exception IllegalArgumentException if minSingularValue is larger than
* the largest singular value, meaning all singular values are ignored
*/
public RealMatrix getCovariance(final double minSingularValue) {
// get the number of singular values to consider
final int p = singularValues.length;
int dimension = 0;
while (dimension < p &&
singularValues[dimension] >= minSingularValue) {
++dimension;
}
if (dimension == 0) {
throw new MathIllegalArgumentException(LocalizedCoreFormats.TOO_LARGE_CUTOFF_SINGULAR_VALUE,
minSingularValue, singularValues[0], true);
}
final double[][] data = new double[dimension][p];
getVT().walkInOptimizedOrder(new DefaultRealMatrixPreservingVisitor() {
/** {@inheritDoc} */
@Override
public void visit(final int row, final int column,
final double value) {
data[row][column] = value / singularValues[row];
}
}, 0, dimension - 1, 0, p - 1);
RealMatrix jv = new Array2DRowRealMatrix(data, false);
return jv.transposeMultiply(jv);
}
/**
* Returns the L<sub>2</sub> norm of the matrix.
* <p>The L<sub>2</sub> norm is max(|A × u|<sub>2</sub> /
* |u|<sub>2</sub>), where |.|<sub>2</sub> denotes the vectorial 2-norm
* (i.e. the traditional euclidian norm).</p>
* @return norm
*/
public double getNorm() {
return singularValues[0];
}
/**
* Return the condition number of the matrix.
* @return condition number of the matrix
*/
public double getConditionNumber() {
return singularValues[0] / singularValues[n - 1];
}
/**
* Computes the inverse of the condition number.
* In cases of rank deficiency, the {@link #getConditionNumber() condition
* number} will become undefined.
*
* @return the inverse of the condition number.
*/
public double getInverseConditionNumber() {
return singularValues[n - 1] / singularValues[0];
}
/**
* Return the effective numerical matrix rank.
* <p>The effective numerical rank is the number of non-negligible
* singular values. The threshold used to identify non-negligible
* terms is max(m,n) × ulp(s<sub>1</sub>) where ulp(s<sub>1</sub>)
* is the least significant bit of the largest singular value.</p>
* @return effective numerical matrix rank
*/
public int getRank() {
int r = 0;
for (int i = 0; i < singularValues.length; i++) {
if (singularValues[i] > tol) {
r++;
}
}
return r;
}
/**
* Get a solver for finding the A × X = B solution in least square sense.
* @return a solver
*/
public DecompositionSolver getSolver() {
return new Solver(singularValues, getUT(), getV(), getRank() == m, tol);
}
/** Specialized solver. */
private static class Solver implements DecompositionSolver {
/** Pseudo-inverse of the initial matrix. */
private final RealMatrix pseudoInverse;
/** Singularity indicator. */
private final boolean nonSingular;
/**
* Build a solver from decomposed matrix.
*
* @param singularValues Singular values.
* @param uT U<sup>T</sup> matrix of the decomposition.
* @param v V matrix of the decomposition.
* @param nonSingular Singularity indicator.
* @param tol tolerance for singular values
*/
private Solver(final double[] singularValues, final RealMatrix uT,
final RealMatrix v, final boolean nonSingular, final double tol) {
final double[][] suT = uT.getData();
for (int i = 0; i < singularValues.length; ++i) {
final double a;
if (singularValues[i] > tol) {
a = 1 / singularValues[i];
} else {
a = 0;
}
final double[] suTi = suT[i];
for (int j = 0; j < suTi.length; ++j) {
suTi[j] *= a;
}
}
pseudoInverse = v.multiply(new Array2DRowRealMatrix(suT, false));
this.nonSingular = nonSingular;
}
/**
* Solve the linear equation A × X = B in least square sense.
* <p>
* The m×n matrix A may not be square, the solution X is such that
* ||A × X - B|| is minimal.
* </p>
* @param b Right-hand side of the equation A × X = B
* @return a vector X that minimizes the two norm of A × X - B
* @throws org.hipparchus.exception.MathIllegalArgumentException
* if the matrices dimensions do not match.
*/
@Override
public RealVector solve(final RealVector b) {
return pseudoInverse.operate(b);
}
/**
* Solve the linear equation A × X = B in least square sense.
* <p>
* The m×n matrix A may not be square, the solution X is such that
* ||A × X - B|| is minimal.
* </p>
*
* @param b Right-hand side of the equation A × X = B
* @return a matrix X that minimizes the two norm of A × X - B
* @throws org.hipparchus.exception.MathIllegalArgumentException
* if the matrices dimensions do not match.
*/
@Override
public RealMatrix solve(final RealMatrix b) {
return pseudoInverse.multiply(b);
}
/**
* Check if the decomposed matrix is non-singular.
*
* @return {@code true} if the decomposed matrix is non-singular.
*/
@Override
public boolean isNonSingular() {
return nonSingular;
}
/**
* Get the pseudo-inverse of the decomposed matrix.
*
* @return the inverse matrix.
*/
@Override
public RealMatrix getInverse() {
return pseudoInverse;
}
/** {@inheritDoc} */
@Override
public int getRowDimension() {
return pseudoInverse.getColumnDimension();
}
/** {@inheritDoc} */
@Override
public int getColumnDimension() {
return pseudoInverse.getRowDimension();
}
}
}