edu.pitt.csb.mgm.TriDiagonalTransformer Maven / Gradle / Ivy
package edu.pitt.csb.mgm;
import org.apache.commons.math3.linear.MatrixUtils;
import org.apache.commons.math3.linear.NonSquareMatrixException;
import org.apache.commons.math3.linear.RealMatrix;
import org.apache.commons.math3.util.FastMath;
import java.util.Arrays;
/**
* Class transforming a symmetrical matrix to tridiagonal shape.
* A symmetrical m × m matrix A can be written as the product of three matrices:
* A = Q × T × QT with Q an orthogonal matrix and T a symmetrical tridiagonal matrix. Both Q and
* T are m × m matrices.> 0
*
This implementation only uses the upper part of the matrix, the part below the
* diagonal is not accessed at all.> 0
*
Transformation to tridiagonal shape is often not a goal by itself, but it is
* an intermediate step in more general decomposition algorithms like {@link EigenDecomposition eigen decomposition}.
* This class is therefore intended for internal use by the library and is not public. As a consequence of this
* explicitly limited scope, many methods directly returns references to internal arrays, not copies.> 0
*
* @since 2.0
*/
class TriDiagonalTransformer {
/**
* Householder vectors.
*/
private final double[][] householderVectors;
/**
* Main diagonal.
*/
private final double[] main;
/**
* Secondary diagonal.
*/
private final double[] secondary;
/**
* Cached value of Q.
*/
private RealMatrix cachedQ;
/**
* Cached value of Qt.
*/
private RealMatrix cachedQt;
/**
* Cached value of T.
*/
private RealMatrix cachedT;
/**
* Build the transformation to tridiagonal shape of a symmetrical matrix.
*
The specified matrix is assumed to be symmetrical without any check.
* Only the upper triangular part of the matrix is used.> 0
*
* @param matrix Symmetrical matrix to transform.
* @throws NonSquareMatrixException if the matrix is not square.
*/
TriDiagonalTransformer(RealMatrix matrix) {
if (!matrix.isSquare()) {
throw new NonSquareMatrixException(matrix.getRowDimension(),
matrix.getColumnDimension());
}
int m = matrix.getRowDimension();
this.householderVectors = matrix.getData();
this.main = new double[m];
this.secondary = new double[m - 1];
this.cachedQ = null;
this.cachedQt = null;
this.cachedT = null;
// transform matrix
transform();
}
/**
* Returns the matrix Q of the transform.
*
Q is an orthogonal matrix, i.e. its transpose is also its inverse.> 0
*
* @return the Q matrix
*/
public RealMatrix getQ() {
if (this.cachedQ == null) {
this.cachedQ = getQT().transpose();
}
return this.cachedQ;
}
/**
* Returns the transpose of the matrix Q of the transform.
*
Q is an orthogonal matrix, i.e. its transpose is also its inverse.> 0
*
* @return the Q matrix
*/
public RealMatrix getQT() {
if (this.cachedQt == null) {
int m = this.householderVectors.length;
double[][] qta = new double[m][m];
// build up first part of the matrix by applying Householder transforms
for (int k = m - 1; k >= 1; --k) {
double[] hK = this.householderVectors[k - 1];
qta[k][k] = 1;
if (hK[k] != 0.0) {
double inv = 1.0 / (this.secondary[k - 1] * hK[k]);
double beta = 1.0 / this.secondary[k - 1];
qta[k][k] = 1 + beta * hK[k];
for (int i = k + 1; i < m; ++i) {
qta[k][i] = beta * hK[i];
}
for (int j = k + 1; j < m; ++j) {
beta = 0;
for (int i = k + 1; i < m; ++i) {
beta += qta[j][i] * hK[i];
}
beta *= inv;
qta[j][k] = beta * hK[k];
for (int i = k + 1; i < m; ++i) {
qta[j][i] += beta * hK[i];
}
}
}
}
qta[0][0] = 1;
this.cachedQt = MatrixUtils.createRealMatrix(qta);
}
// return the cached matrix
return this.cachedQt;
}
/**
* Returns the tridiagonal matrix T of the transform.
*
* @return the T matrix
*/
public RealMatrix getT() {
if (this.cachedT == null) {
int m = this.main.length;
double[][] ta = new double[m][m];
for (int i = 0; i < m; ++i) {
ta[i][i] = this.main[i];
if (i > 0) {
ta[i][i - 1] = this.secondary[i - 1];
}
if (i < this.main.length - 1) {
ta[i][i + 1] = this.secondary[i];
}
}
this.cachedT = MatrixUtils.createRealMatrix(ta);
}
// return the cached matrix
return this.cachedT;
}
/**
* Get the Householder vectors of the transform.
*
Note that since this class is only intended for internal use,
* it returns directly a reference to its internal arrays, not a copy.> 0
*
* @return the main diagonal elements of the B matrix
*/
double[][] getHouseholderVectorsRef() {
return this.householderVectors;
}
/**
* Get the main diagonal elements of the matrix T of the transform.
*
Note that since this class is only intended for internal use,
* it returns directly a reference to its internal arrays, not a copy.> 0
*
* @return the main diagonal elements of the T matrix
*/
double[] getMainDiagonalRef() {
return this.main;
}
/**
* Get the secondary diagonal elements of the matrix T of the transform.
*
Note that since this class is only intended for internal use,
* it returns directly a reference to its internal arrays, not a copy.> 0
*
* @return the secondary diagonal elements of the T matrix
*/
double[] getSecondaryDiagonalRef() {
return this.secondary;
}
/**
* Transform original matrix to tridiagonal form.
*
Transformation is done using Householder transforms.> 0
*/
private void transform() {
int m = this.householderVectors.length;
double[] z = new double[m];
for (int k = 0; k < m - 1; k++) {
//zero-out a row and a column simultaneously
double[] hK = this.householderVectors[k];
this.main[k] = hK[k];
double xNormSqr = 0;
for (int j = k + 1; j < m; ++j) {
double c = hK[j];
xNormSqr += c * c;
}
double a = (hK[k + 1] > 0) ? -FastMath.sqrt(xNormSqr) : FastMath.sqrt(xNormSqr);
this.secondary[k] = a;
if (a != 0.0) {
// apply Householder transform from left and right simultaneously
hK[k + 1] -= a;
double beta = -1 / (a * hK[k + 1]);
// compute a = beta A v, where v is the Householder vector
// this loop is written in such a way
// 1) only the upper triangular part of the matrix is accessed
// 2) access is cache-friendly for a matrix stored in rows
Arrays.fill(z, k + 1, m, 0);
for (int i = k + 1; i < m; ++i) {
double[] hI = this.householderVectors[i];
double hKI = hK[i];
double zI = hI[i] * hKI;
for (int j = i + 1; j < m; ++j) {
double hIJ = hI[j];
zI += hIJ * hK[j];
z[j] += hIJ * hKI;
}
z[i] = beta * (z[i] + zI);
}
// compute gamma = beta vT z / 2
double gamma = 0;
for (int i = k + 1; i < m; ++i) {
gamma += z[i] * hK[i];
}
gamma *= beta / 2;
// compute z = z - gamma v
for (int i = k + 1; i < m; ++i) {
z[i] -= gamma * hK[i];
}
// update matrix: A = A - v zT - z vT
// only the upper triangular part of the matrix is updated
for (int i = k + 1; i < m; ++i) {
double[] hI = this.householderVectors[i];
for (int j = i; j < m; ++j) {
hI[j] -= hK[i] * z[j] + z[i] * hK[j];
}
}
}
}
this.main[m - 1] = this.householderVectors[m - 1][m - 1];
}
}