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/* Tridiagonal.java
* =========================================================================
* This file is part of the GrInvIn project - http://www.grinvin.org
*
* Copyright (C) 2005-2008 Universiteit Gent
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or (at
* your option) any later version.
*
* 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 GNU
* General Public License for more details.
*
* A copy of the GNU General Public License can be found in the file
* LICENSE.txt provided with the source distribution of this program (see
* the META-INF directory in the source jar). This license can also be
* found on the GNU website at http://www.gnu.org/licenses/gpl.html.
*
* If you did not receive a copy of the GNU General Public License along
* with this program, contact the lead developer, or write to the Free
* Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
* 02110-1301, USA.
*/
package org.grinvin.util;
/**
* This class represents a symmetric tridiagonal real matrix. The main purpose
* of this class is to provide information related to eigenvalues and
* eigenvectors of such matrices.
*/
public class Tridiagonal {
/** Order of this matrix. */
protected int order;
/**
* Diagonal elements of this matrix. Element d[i] contains the matrix
* element at position (i,i).
*/
protected double[] d;
/**
* Subdiagonal elements of this matrix. Element e[i] contains the matrix
* element at position (i,i-1). Element e[0] is set to zero.
*/
protected double[] e;
/**
* Create a tridiagonal matrix which is equivalent to the given symmetric
* real matrix a by some orthogonal transformation. Uses Householder
* reduction.
*
* <p>
* <b>Note:</b> This implementation is almost a literal copy of the
* algorithm <code>tred2</code> from <i>Numerical recipes in C</i>, ISBN
* 0-521-43108-5.
* </p>
*/
public Tridiagonal (double[][] a) {
// init variables
order = a.length;
d = new double[order];
e = new double[order];
// make copy of a
double[][] acopy = new double[order][];
for (int i = 0; i < order; i++)
acopy[i] = a[i].clone ();
// reduce to a tridiagonal matrix
householder (acopy);
e[0] = 0.0;
for (int i = 0; i < order; i++)
d[i] = acopy[i][i];
}
/**
* Auxiliary routine which performs the Householder reduction. Destroys the
* contents of array a.
*
* @param vectors If true, the orthogonal transformation Q such that
* Q<sup>T</sup>.A.Q is tridiagonal, is stored into parameter a as
* the result of this algorithm. Otherwise the elements of the
* matrix a are destroyed and contain no meaningful information.
*/
private void householder (double[][] a) {
for (int i = order - 1; i > 0; i--) {
int l = i - 1;
double h = 0.0;
double scale = 0.0;
if (l > 0) {
for (int k = 0; k <= l; k++)
if (a[i][k] < 0)
scale -= a[i][k];
else
scale += a[i][k];
if (scale == 0.0)
e[i] = a[i][l];
else {
for (int k = 0; k <= l; k++) {
a[i][k] /= scale;
h += a[i][k] * a[i][k];
}
double f = a[i][l];
double g = f >= 0.0 ? -Math.sqrt (h) : Math.sqrt (h);
e[i] = scale * g;
h -= f * g;
a[i][l] = f - g;
f = 0.0;
for (int j = 0; j <= l; j++) {
a[j][i] = a[i][j] / h; // [*]
g = 0.0;
for (int k = 0; k <= j; k++)
g += a[j][k] * a[i][k];
for (int k = j + 1; k <= l; k++)
g += a[k][j] * a[i][k];
e[j] = g / h;
f += e[j] * a[i][j];
}
double hh = f / (h + h);
for (int j = 0; j <= l; j++) {
f = a[i][j];
g = e[j] - hh * f;
e[j] = g;
for (int k = 0; k <= j; k++)
a[j][k] -= f * e[k] + g * a[i][k];
}
}
} else
e[i] = a[i][l];
d[i] = h;
}
}
/**
* Compute the eigenvalues of this symmetric tridiagonal matrix using the
* QL algorithm with implicit shifts.
*
* <p>
* <b>Note:</b> This implementation is almost a literal copy of the
* algorithm <code>tqli</code> from <i>Numerical recipes in C</i>, ISBN
* 0-521-43108-5.
* </p>
*/
public double[] eigenvalues () {
return eigen (null);
}
/**
* Auxiliary routine which performs the QL algorithm. Returns the
* eigenvalues of the tridiagonal matrix, and if z is not null, stores the
* corresponding eigenvectors into z, after applying the orthogonal
* transformation originally contained in z.
*/
private double[] eigen (double[][] z) {
double[] d = this.d.clone ();
double[] e = new double[order];
for (int i = 1; i < order; i++)
e[i - 1] = this.e[i];
e[order - 1] = 0.0;
for (int l = 0; l < order; l++) {
int iter = 0;
int m;
do {
for (m = l; m < order - 1; m++) {
double dd = Math.abs (d[m]) + Math.abs (d[m + 1]);
if (Math.abs (e[m]) + dd == dd)
break;
}
if (m != l) {
iter++;
if (iter == 30)
throw new RuntimeException(
"Too many iterations in QL algorithm ");
double g = (d[l + 1] - d[l]) / (2.0 * e[l]);
double r = Math.sqrt (1.0 + g * g); //[!] should use pythag?
if (g >= 0.0) //[!] equal sign correct ?
g = d[m] - d[l] + e[l] / (g + r);
else if (g < 0.0)
g = d[m] - d[l] + e[l] / (g - r);
double s = 1.0;
double c = 1.0;
double p = 0.0;
int i;
for (i = m - 1; i >= l; i--) {
double f = s * e[i];
double b = c * e[i];
r = Math.sqrt (f * f + g * g); //[!] should use pythag?
e[i + 1] = r;
if (r == 0.0) {
d[i + 1] -= p;
e[m] = 0.0;
break;
}
s = f / r;
c = g / r;
g = d[i + 1] - p;
r = (d[i] - g) * s + 2.0 * c * b;
p = s * r;
d[i + 1] = g + p;
g = c * r - b;
if (z != null)
for (int k = 0; k < order; k++) {
f = z[k][i + 1];
z[k][i + 1] = s * z[k][i] + c * f;
z[k][i] = c * z[k][i] - s * f;
}
}
if (r != 0.0 || i < l) {
d[l] -= p;
e[l] = g;
e[m] = 0.0;
}
}
} while (m != l);
}
return d;
}
}
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