mirror of https://github.com/CGAL/cgal
1027 lines
28 KiB
C++
1027 lines
28 KiB
C++
#ifndef CGAL_JAMA_EIG_H
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#define CGAL_JAMA_EIG_H
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#include <CGAL/PDB/basic.h>
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#include "tnt_array1d.h"
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#include "tnt_array2d.h"
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#include "tnt_math_utils.h"
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#include <algorithm>
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// for min(), max() below
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#include <cmath>
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// for abs() below
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CGAL_JAMA_BEGIN_NAMESPACE
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/**
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Computes eigenvalues and eigenvectors of a real (non-complex)
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matrix.
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<P>
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If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is
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diagonal and the eigenvector matrix V is orthogonal. That is,
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the diagonal values of D are the eigenvalues, and
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V*V' = I, where I is the identity matrix. The columns of V
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represent the eigenvectors in the sense that A*V = V*D.
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<P>
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If A is not symmetric, then the eigenvalue matrix D is block diagonal
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with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
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a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex
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eigenvalues look like
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<pre>
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u + iv . . . . .
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. u - iv . . . .
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. . a + ib . . .
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. . . a - ib . .
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. . . . x .
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. . . . . y
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</pre>
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then D looks like
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<pre>
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u v . . . .
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-v u . . . .
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. . a b . .
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. . -b a . .
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. . . . x .
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. . . . . y
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</pre>
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This keeps V a real matrix in both symmetric and non-symmetric
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cases, and A*V = V*D.
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<p>
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The matrix V may be badly
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conditioned, or even singular, so the validity of the equation
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A = V*D*inverse(V) depends upon the condition number of V.
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<p>
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(Adapted from JAMA, a Java Matrix Library, developed by jointly
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by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).
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**/
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template <class Real>
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class Eigenvalue
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{
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/** Row and column dimension (square matrix). */
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int n;
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int issymmetric; /* boolean*/
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/** Arrays for internal storage of eigenvalues. */
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TNT::Array1D<Real> d; /* real part */
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TNT::Array1D<Real> e; /* img part */
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/** Array for internal storage of eigenvectors. */
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TNT::Array2D<Real> V;
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/** Array for internal storage of nonsymmetric Hessenberg form.
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@serial internal storage of nonsymmetric Hessenberg form.
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*/
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TNT::Array2D<Real> H;
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/** Working storage for nonsymmetric algorithm.
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@serial working storage for nonsymmetric algorithm.
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*/
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TNT::Array1D<Real> ort;
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// Symmetric Householder reduction to tridiagonal form.
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void tred2() {
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// This is derived from the Algol procedures tred2 by
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// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
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// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
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// Fortran subroutine in EISPACK.
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for (int j = 0; j < n; j++) {
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d[j] = V[n-1][j];
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}
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// Householder reduction to tridiagonal form.
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for (int i = n-1; i > 0; i--) {
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// Scale to avoid under/overflow.
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Real scale = 0.0;
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Real h = 0.0;
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for (int k = 0; k < i; k++) {
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scale = scale + abs(d[k]);
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}
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if (scale == Real(0.0)) {
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e[i] = d[i-1];
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for (int j = 0; j < i; j++) {
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d[j] = V[i-1][j];
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V[i][j] = 0.0;
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V[j][i] = 0.0;
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}
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} else {
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// Generate Householder vector.
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for (int k = 0; k < i; k++) {
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d[k] /= scale;
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h += d[k] * d[k];
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}
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Real f = d[i-1];
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Real g = sqrt(h);
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if (f > 0) {
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g = -g;
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}
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e[i] = scale * g;
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h = h - f * g;
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d[i-1] = f - g;
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for (int j = 0; j < i; j++) {
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e[j] = 0.0;
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}
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// Apply similarity transformation to remaining columns.
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for (int j = 0; j < i; j++) {
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f = d[j];
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V[j][i] = f;
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g = e[j] + V[j][j] * f;
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for (int k = j+1; k <= i-1; k++) {
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g += V[k][j] * d[k];
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e[k] += V[k][j] * f;
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}
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e[j] = g;
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}
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f = 0.0;
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for (int j = 0; j < i; j++) {
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e[j] /= h;
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f += e[j] * d[j];
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}
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Real hh = f / (h + h);
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for (int j = 0; j < i; j++) {
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e[j] -= hh * d[j];
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}
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for (int j = 0; j < i; j++) {
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f = d[j];
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g = e[j];
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for (int k = j; k <= i-1; k++) {
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V[k][j] -= (f * e[k] + g * d[k]);
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}
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d[j] = V[i-1][j];
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V[i][j] = 0.0;
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}
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}
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d[i] = h;
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}
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// Accumulate transformations.
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for (int i = 0; i < n-1; i++) {
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V[n-1][i] = V[i][i];
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V[i][i] = 1.0;
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Real h = d[i+1];
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if (h != Real(0.0)) {
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for (int k = 0; k <= i; k++) {
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d[k] = V[k][i+1] / h;
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}
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for (int j = 0; j <= i; j++) {
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Real g = 0.0;
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for (int k = 0; k <= i; k++) {
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g += V[k][i+1] * V[k][j];
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}
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for (int k = 0; k <= i; k++) {
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V[k][j] -= g * d[k];
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}
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}
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}
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for (int k = 0; k <= i; k++) {
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V[k][i+1] = 0.0;
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}
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}
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for (int j = 0; j < n; j++) {
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d[j] = V[n-1][j];
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V[n-1][j] = 0.0;
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}
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V[n-1][n-1] = 1.0;
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e[0] = 0.0;
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}
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// Symmetric tridiagonal QL algorithm.
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void tql2 () {
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// This is derived from the Algol procedures tql2, by
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// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
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// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
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// Fortran subroutine in EISPACK.
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for (int i = 1; i < n; i++) {
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e[i-1] = e[i];
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}
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e[n-1] = 0.0;
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Real f = 0.0;
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Real tst1 = 0.0;
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Real eps = pow(2.0,-52.0);
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for (int l = 0; l < n; l++) {
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// Find small subdiagonal element
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tst1 = max(tst1,abs(d[l]) + abs(e[l]));
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int m = l;
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// Original while-loop from Java code
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while (m < n) {
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if (abs(e[m]) <= eps*tst1) {
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break;
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}
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m++;
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}
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// If m == l, d[l] is an eigenvalue,
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// otherwise, iterate.
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if (m > l) {
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int iter = 0;
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do {
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iter = iter + 1; // (Could check iteration count here.)
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// Compute implicit shift
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Real g = d[l];
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Real p = (d[l+1] - g) / (2.0 * e[l]);
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Real r = hypot(p,Real(1.0));
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if (p < 0) {
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r = -r;
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}
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d[l] = e[l] / (p + r);
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d[l+1] = e[l] * (p + r);
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Real dl1 = d[l+1];
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Real h = g - d[l];
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for (int i = l+2; i < n; i++) {
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d[i] -= h;
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}
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f = f + h;
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// Implicit QL transformation.
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p = d[m];
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Real c = 1.0;
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Real c2 = c;
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Real c3 = c;
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Real el1 = e[l+1];
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Real s = 0.0;
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Real s2 = 0.0;
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for (int i = m-1; i >= l; i--) {
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c3 = c2;
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c2 = c;
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s2 = s;
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g = c * e[i];
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h = c * p;
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r = hypot(p,e[i]);
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e[i+1] = s * r;
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s = e[i] / r;
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c = p / r;
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p = c * d[i] - s * g;
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d[i+1] = h + s * (c * g + s * d[i]);
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// Accumulate transformation.
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for (int k = 0; k < n; k++) {
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h = V[k][i+1];
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V[k][i+1] = s * V[k][i] + c * h;
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V[k][i] = c * V[k][i] - s * h;
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}
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}
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p = -s * s2 * c3 * el1 * e[l] / dl1;
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e[l] = s * p;
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d[l] = c * p;
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// Check for convergence.
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} while (abs(e[l]) > eps*tst1);
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}
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d[l] = d[l] + f;
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e[l] = 0.0;
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}
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// Sort eigenvalues and corresponding vectors.
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for (int i = 0; i < n-1; i++) {
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int k = i;
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Real p = d[i];
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for (int j = i+1; j < n; j++) {
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if (d[j] < p) {
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k = j;
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p = d[j];
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}
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}
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if (k != i) {
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d[k] = d[i];
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d[i] = p;
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for (int j = 0; j < n; j++) {
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p = V[j][i];
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V[j][i] = V[j][k];
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V[j][k] = p;
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}
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}
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}
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}
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// Nonsymmetric reduction to Hessenberg form.
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void orthes () {
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// This is derived from the Algol procedures orthes and ortran,
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// by Martin and Wilkinson, Handbook for Auto. Comp.,
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// Vol.ii-Linear Algebra, and the corresponding
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// Fortran subroutines in EISPACK.
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int low = 0;
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int high = n-1;
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for (int m = low+1; m <= high-1; m++) {
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// Scale column.
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Real scale = 0.0;
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for (int i = m; i <= high; i++) {
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scale = scale + abs(H[i][m-1]);
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}
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if (scale != Real(0.0)) {
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// Compute Householder transformation.
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Real h = 0.0;
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for (int i = high; i >= m; i--) {
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ort[i] = H[i][m-1]/scale;
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h += ort[i] * ort[i];
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}
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Real g = sqrt(h);
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if (ort[m] > 0) {
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g = -g;
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}
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h = h - ort[m] * g;
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ort[m] = ort[m] - g;
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// Apply Householder similarity transformation
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// H = (I-u*u'/h)*H*(I-u*u')/h)
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for (int j = m; j < n; j++) {
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Real f = 0.0;
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for (int i = high; i >= m; i--) {
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f += ort[i]*H[i][j];
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}
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f = f/h;
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for (int i = m; i <= high; i++) {
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H[i][j] -= f*ort[i];
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}
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}
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for (int i = 0; i <= high; i++) {
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Real f = 0.0;
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for (int j = high; j >= m; j--) {
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f += ort[j]*H[i][j];
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}
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f = f/h;
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for (int j = m; j <= high; j++) {
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H[i][j] -= f*ort[j];
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}
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}
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ort[m] = scale*ort[m];
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H[m][m-1] = scale*g;
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}
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}
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// Accumulate transformations (Algol's ortran).
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for (int i = 0; i < n; i++) {
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for (int j = 0; j < n; j++) {
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V[i][j] = (i == j ? 1.0 : 0.0);
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}
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}
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for (int m = high-1; m >= low+1; m--) {
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if (H[m][m-1] != Real(0.0)) {
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for (int i = m+1; i <= high; i++) {
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ort[i] = H[i][m-1];
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}
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for (int j = m; j <= high; j++) {
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Real g = 0.0;
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for (int i = m; i <= high; i++) {
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g += ort[i] * V[i][j];
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}
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// Double division avoids possible underflow
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g = (g / ort[m]) / H[m][m-1];
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for (int i = m; i <= high; i++) {
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V[i][j] += g * ort[i];
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}
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}
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}
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}
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}
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// Complex scalar division.
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Real cdivr, cdivi;
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void cdiv(Real xr, Real xi, Real yr, Real yi) {
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Real r,d;
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if (abs(yr) > abs(yi)) {
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r = yi/yr;
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d = yr + r*yi;
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cdivr = (xr + r*xi)/d;
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cdivi = (xi - r*xr)/d;
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} else {
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r = yr/yi;
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d = yi + r*yr;
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cdivr = (r*xr + xi)/d;
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cdivi = (r*xi - xr)/d;
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}
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}
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// Nonsymmetric reduction from Hessenberg to real Schur form.
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void hqr2 () {
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// This is derived from the Algol procedure hqr2,
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// by Martin and Wilkinson, Handbook for Auto. Comp.,
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// Vol.ii-Linear Algebra, and the corresponding
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// Fortran subroutine in EISPACK.
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// Initialize
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int nn = this->n;
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int n = nn-1;
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int low = 0;
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int high = nn-1;
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Real eps = pow(2.0,-52.0);
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Real exshift = 0.0;
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Real p=0,q=0,r=0,s=0,z=0,t,w,x,y;
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// Store roots isolated by balanc and compute matrix norm
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Real norm = 0.0;
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for (int i = 0; i < nn; i++) {
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if ((i < low) || (i > high)) {
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d[i] = H[i][i];
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e[i] = 0.0;
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}
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for (int j = max(i-1,0); j < nn; j++) {
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norm = norm + abs(H[i][j]);
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}
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}
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// Outer loop over eigenvalue index
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int iter = 0;
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while (n >= low) {
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// Look for single small sub-diagonal element
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int l = n;
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while (l > low) {
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s = abs(H[l-1][l-1]) + abs(H[l][l]);
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if (s == Real(0.0)) {
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s = norm;
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}
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if (abs(H[l][l-1]) < eps * s) {
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break;
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}
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l--;
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}
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// Check for convergence
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// One root found
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if (l == n) {
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H[n][n] = H[n][n] + exshift;
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d[n] = H[n][n];
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e[n] = 0.0;
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n--;
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iter = 0;
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// Two roots found
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} else if (l == n-1) {
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w = H[n][n-1] * H[n-1][n];
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p = (H[n-1][n-1] - H[n][n]) / 2.0;
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q = p * p + w;
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z = sqrt(abs(q));
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H[n][n] = H[n][n] + exshift;
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H[n-1][n-1] = H[n-1][n-1] + exshift;
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x = H[n][n];
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// Real pair
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if (q >= 0) {
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if (p >= 0) {
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z = p + z;
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} else {
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z = p - z;
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}
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d[n-1] = x + z;
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d[n] = d[n-1];
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if (z != Real(0.0)) {
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d[n] = x - w / z;
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}
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e[n-1] = 0.0;
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e[n] = 0.0;
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x = H[n][n-1];
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s = abs(x) + abs(z);
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p = x / s;
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q = z / s;
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r = sqrt(p * p+q * q);
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|
p = p / r;
|
|
q = q / r;
|
|
|
|
// Row modification
|
|
|
|
for (int j = n-1; j < nn; j++) {
|
|
z = H[n-1][j];
|
|
H[n-1][j] = q * z + p * H[n][j];
|
|
H[n][j] = q * H[n][j] - p * z;
|
|
}
|
|
|
|
// Column modification
|
|
|
|
for (int i = 0; i <= n; i++) {
|
|
z = H[i][n-1];
|
|
H[i][n-1] = q * z + p * H[i][n];
|
|
H[i][n] = q * H[i][n] - p * z;
|
|
}
|
|
|
|
// Accumulate transformations
|
|
|
|
for (int i = low; i <= high; i++) {
|
|
z = V[i][n-1];
|
|
V[i][n-1] = q * z + p * V[i][n];
|
|
V[i][n] = q * V[i][n] - p * z;
|
|
}
|
|
|
|
// Complex pair
|
|
|
|
} else {
|
|
d[n-1] = x + p;
|
|
d[n] = x + p;
|
|
e[n-1] = z;
|
|
e[n] = -z;
|
|
}
|
|
n = n - 2;
|
|
iter = 0;
|
|
|
|
// No convergence yet
|
|
|
|
} else {
|
|
|
|
// Form shift
|
|
|
|
x = H[n][n];
|
|
y = 0.0;
|
|
w = 0.0;
|
|
if (l < n) {
|
|
y = H[n-1][n-1];
|
|
w = H[n][n-1] * H[n-1][n];
|
|
}
|
|
|
|
// Wilkinson's original ad hoc shift
|
|
|
|
if (iter == 10) {
|
|
exshift += x;
|
|
for (int i = low; i <= n; i++) {
|
|
H[i][i] -= x;
|
|
}
|
|
s = abs(H[n][n-1]) + abs(H[n-1][n-2]);
|
|
x = y = 0.75 * s;
|
|
w = -0.4375 * s * s;
|
|
}
|
|
|
|
// MATLAB's new ad hoc shift
|
|
|
|
if (iter == 30) {
|
|
s = (y - x) / 2.0;
|
|
s = s * s + w;
|
|
if (s > 0) {
|
|
s = sqrt(s);
|
|
if (y < x) {
|
|
s = -s;
|
|
}
|
|
s = x - w / ((y - x) / 2.0 + s);
|
|
for (int i = low; i <= n; i++) {
|
|
H[i][i] -= s;
|
|
}
|
|
exshift += s;
|
|
x = y = w = 0.964;
|
|
}
|
|
}
|
|
|
|
iter = iter + 1; // (Could check iteration count here.)
|
|
|
|
// Look for two consecutive small sub-diagonal elements
|
|
|
|
int m = n-2;
|
|
while (m >= l) {
|
|
z = H[m][m];
|
|
r = x - z;
|
|
s = y - z;
|
|
p = (r * s - w) / H[m+1][m] + H[m][m+1];
|
|
q = H[m+1][m+1] - z - r - s;
|
|
r = H[m+2][m+1];
|
|
s = abs(p) + abs(q) + abs(r);
|
|
p = p / s;
|
|
q = q / s;
|
|
r = r / s;
|
|
if (m == l) {
|
|
break;
|
|
}
|
|
if (abs(H[m][m-1]) * (abs(q) + abs(r)) <
|
|
eps * (abs(p) * (abs(H[m-1][m-1]) + abs(z) +
|
|
abs(H[m+1][m+1])))) {
|
|
break;
|
|
}
|
|
m--;
|
|
}
|
|
|
|
for (int i = m+2; i <= n; i++) {
|
|
H[i][i-2] = 0.0;
|
|
if (i > m+2) {
|
|
H[i][i-3] = 0.0;
|
|
}
|
|
}
|
|
|
|
// Double QR step involving rows l:n and columns m:n
|
|
|
|
for (int k = m; k <= n-1; k++) {
|
|
int notlast = (k != n-1);
|
|
if (k != m) {
|
|
p = H[k][k-1];
|
|
q = H[k+1][k-1];
|
|
r = (notlast ? H[k+2][k-1] : 0.0);
|
|
x = abs(p) + abs(q) + abs(r);
|
|
if (x != Real(0.0)) {
|
|
p = p / x;
|
|
q = q / x;
|
|
r = r / x;
|
|
}
|
|
}
|
|
if (x == Real(0.0)) {
|
|
break;
|
|
}
|
|
s = sqrt(p * p + q * q + r * r);
|
|
if (p < 0) {
|
|
s = -s;
|
|
}
|
|
if (s != Real(0)) {
|
|
if (k != m) {
|
|
H[k][k-1] = -s * x;
|
|
} else if (l != m) {
|
|
H[k][k-1] = -H[k][k-1];
|
|
}
|
|
p = p + s;
|
|
x = p / s;
|
|
y = q / s;
|
|
z = r / s;
|
|
q = q / p;
|
|
r = r / p;
|
|
|
|
// Row modification
|
|
|
|
for (int j = k; j < nn; j++) {
|
|
p = H[k][j] + q * H[k+1][j];
|
|
if (notlast) {
|
|
p = p + r * H[k+2][j];
|
|
H[k+2][j] = H[k+2][j] - p * z;
|
|
}
|
|
H[k][j] = H[k][j] - p * x;
|
|
H[k+1][j] = H[k+1][j] - p * y;
|
|
}
|
|
|
|
// Column modification
|
|
|
|
for (int i = 0; i <= min(n,k+3); i++) {
|
|
p = x * H[i][k] + y * H[i][k+1];
|
|
if (notlast) {
|
|
p = p + z * H[i][k+2];
|
|
H[i][k+2] = H[i][k+2] - p * r;
|
|
}
|
|
H[i][k] = H[i][k] - p;
|
|
H[i][k+1] = H[i][k+1] - p * q;
|
|
}
|
|
|
|
// Accumulate transformations
|
|
|
|
for (int i = low; i <= high; i++) {
|
|
p = x * V[i][k] + y * V[i][k+1];
|
|
if (notlast) {
|
|
p = p + z * V[i][k+2];
|
|
V[i][k+2] = V[i][k+2] - p * r;
|
|
}
|
|
V[i][k] = V[i][k] - p;
|
|
V[i][k+1] = V[i][k+1] - p * q;
|
|
}
|
|
} // (s != 0)
|
|
} // k loop
|
|
} // check convergence
|
|
} // while (n >= low)
|
|
|
|
// Backsubstitute to find vectors of upper triangular form
|
|
|
|
if (norm == Real(0.0)) {
|
|
return;
|
|
}
|
|
|
|
for (n = nn-1; n >= 0; n--) {
|
|
p = d[n];
|
|
q = e[n];
|
|
|
|
// Real vector
|
|
|
|
if (q == 0) {
|
|
int l = n;
|
|
H[n][n] = 1.0;
|
|
for (int i = n-1; i >= 0; i--) {
|
|
w = H[i][i] - p;
|
|
r = 0.0;
|
|
for (int j = l; j <= n; j++) {
|
|
r = r + H[i][j] * H[j][n];
|
|
}
|
|
if (e[i] < Real(0.0)) {
|
|
z = w;
|
|
s = r;
|
|
} else {
|
|
l = i;
|
|
if (e[i] == Real(0.0)) {
|
|
if (w != Real(0.0)) {
|
|
H[i][n] = -r / w;
|
|
} else {
|
|
H[i][n] = -r / (eps * norm);
|
|
}
|
|
|
|
// Solve real equations
|
|
|
|
} else {
|
|
x = H[i][i+1];
|
|
y = H[i+1][i];
|
|
q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
|
|
t = (x * s - z * r) / q;
|
|
H[i][n] = t;
|
|
if (abs(x) > abs(z)) {
|
|
H[i+1][n] = (-r - w * t) / x;
|
|
} else {
|
|
H[i+1][n] = (-s - y * t) / z;
|
|
}
|
|
}
|
|
|
|
// Overflow control
|
|
|
|
t = abs(H[i][n]);
|
|
if ((eps * t) * t > 1) {
|
|
for (int j = i; j <= n; j++) {
|
|
H[j][n] = H[j][n] / t;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// Complex vector
|
|
|
|
} else if (q < 0) {
|
|
int l = n-1;
|
|
|
|
// Last vector component imaginary so matrix is triangular
|
|
|
|
if (abs(H[n][n-1]) > abs(H[n-1][n])) {
|
|
H[n-1][n-1] = q / H[n][n-1];
|
|
H[n-1][n] = -(H[n][n] - p) / H[n][n-1];
|
|
} else {
|
|
cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);
|
|
H[n-1][n-1] = cdivr;
|
|
H[n-1][n] = cdivi;
|
|
}
|
|
H[n][n-1] = 0.0;
|
|
H[n][n] = 1.0;
|
|
for (int i = n-2; i >= 0; i--) {
|
|
Real ra,sa,vr,vi;
|
|
ra = 0.0;
|
|
sa = 0.0;
|
|
for (int j = l; j <= n; j++) {
|
|
ra = ra + H[i][j] * H[j][n-1];
|
|
sa = sa + H[i][j] * H[j][n];
|
|
}
|
|
w = H[i][i] - p;
|
|
|
|
if (e[i] < Real(0.0)) {
|
|
z = w;
|
|
r = ra;
|
|
s = sa;
|
|
} else {
|
|
l = i;
|
|
if (e[i] == 0) {
|
|
cdiv(-ra,-sa,w,q);
|
|
H[i][n-1] = cdivr;
|
|
H[i][n] = cdivi;
|
|
} else {
|
|
|
|
// Solve complex equations
|
|
|
|
x = H[i][i+1];
|
|
y = H[i+1][i];
|
|
vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
|
|
vi = (d[i] - p) * 2.0 * q;
|
|
if ((vr == Real(0.0)) && (vi == Real(0.0))) {
|
|
vr = eps * norm * (abs(w) + abs(q) +
|
|
abs(x) + abs(y) + abs(z));
|
|
}
|
|
cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
|
|
H[i][n-1] = cdivr;
|
|
H[i][n] = cdivi;
|
|
if (abs(x) > (abs(z) + abs(q))) {
|
|
H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;
|
|
H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;
|
|
} else {
|
|
cdiv(-r-y*H[i][n-1],-s-y*H[i][n],z,q);
|
|
H[i+1][n-1] = cdivr;
|
|
H[i+1][n] = cdivi;
|
|
}
|
|
}
|
|
|
|
// Overflow control
|
|
|
|
t = max(abs(H[i][n-1]),abs(H[i][n]));
|
|
if ((eps * t) * t > 1) {
|
|
for (int j = i; j <= n; j++) {
|
|
H[j][n-1] = H[j][n-1] / t;
|
|
H[j][n] = H[j][n] / t;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// Vectors of isolated roots
|
|
|
|
for (int i = 0; i < nn; i++) {
|
|
if (i < low || i > high) {
|
|
for (int j = i; j < nn; j++) {
|
|
V[i][j] = H[i][j];
|
|
}
|
|
}
|
|
}
|
|
|
|
// Back transformation to get eigenvectors of original matrix
|
|
|
|
for (int j = nn-1; j >= low; j--) {
|
|
for (int i = low; i <= high; i++) {
|
|
z = 0.0;
|
|
for (int k = low; k <= min(j,high); k++) {
|
|
z = z + V[i][k] * H[k][j];
|
|
}
|
|
V[i][j] = z;
|
|
}
|
|
}
|
|
}
|
|
|
|
public:
|
|
|
|
|
|
/** Check for symmetry, then construct the eigenvalue decomposition
|
|
@param A Square real (non-complex) matrix
|
|
*/
|
|
|
|
Eigenvalue(const TNT::Array2D<Real> &A) {
|
|
n = A.dim2();
|
|
V = Array2D<Real>(n,n);
|
|
d = Array1D<Real>(n);
|
|
e = Array1D<Real>(n);
|
|
|
|
issymmetric = 1;
|
|
for (int j = 0; (j < n) && issymmetric; j++) {
|
|
for (int i = 0; (i < n) && issymmetric; i++) {
|
|
issymmetric = (A[i][j] == A[j][i]);
|
|
}
|
|
}
|
|
|
|
if (issymmetric) {
|
|
for (int i = 0; i < n; i++) {
|
|
for (int j = 0; j < n; j++) {
|
|
V[i][j] = A[i][j];
|
|
}
|
|
}
|
|
|
|
// Tridiagonalize.
|
|
tred2();
|
|
|
|
// Diagonalize.
|
|
tql2();
|
|
|
|
} else {
|
|
H = TNT::Array2D<Real>(n,n);
|
|
ort = TNT::Array1D<Real>(n);
|
|
|
|
for (int j = 0; j < n; j++) {
|
|
for (int i = 0; i < n; i++) {
|
|
H[i][j] = A[i][j];
|
|
}
|
|
}
|
|
|
|
// Reduce to Hessenberg form.
|
|
orthes();
|
|
|
|
// Reduce Hessenberg to real Schur form.
|
|
hqr2();
|
|
}
|
|
}
|
|
|
|
|
|
/** Return the eigenvector matrix
|
|
@return V
|
|
*/
|
|
|
|
void getV (TNT::Array2D<Real> &V_) {
|
|
V_ = V;
|
|
return;
|
|
}
|
|
|
|
/** Return the real parts of the eigenvalues
|
|
@return real(diag(D))
|
|
*/
|
|
|
|
void getRealEigenvalues (TNT::Array1D<Real> &d_) {
|
|
d_ = d;
|
|
return ;
|
|
}
|
|
|
|
/** Return the imaginary parts of the eigenvalues
|
|
in parameter e_.
|
|
|
|
@pararm e_: new matrix with imaginary parts of the eigenvalues.
|
|
*/
|
|
void getImagEigenvalues (TNT::Array1D<Real> &e_) {
|
|
e_ = e;
|
|
return;
|
|
}
|
|
|
|
|
|
/**
|
|
Computes the block diagonal eigenvalue matrix.
|
|
If the original matrix A is not symmetric, then the eigenvalue
|
|
matrix D is block diagonal with the real eigenvalues in 1-by-1
|
|
blocks and any complex eigenvalues,
|
|
a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex
|
|
eigenvalues look like
|
|
<pre>
|
|
|
|
u + iv . . . . .
|
|
. u - iv . . . .
|
|
. . a + ib . . .
|
|
. . . a - ib . .
|
|
. . . . x .
|
|
. . . . . y
|
|
</pre>
|
|
then D looks like
|
|
<pre>
|
|
|
|
u v . . . .
|
|
-v u . . . .
|
|
. . a b . .
|
|
. . -b a . .
|
|
. . . . x .
|
|
. . . . . y
|
|
</pre>
|
|
This keeps V a real matrix in both symmetric and non-symmetric
|
|
cases, and A*V = V*D.
|
|
|
|
@param D: upon return, the matrix is filled with the block diagonal
|
|
eigenvalue matrix.
|
|
|
|
*/
|
|
void getD (TNT::Array2D<Real> &D) {
|
|
D = Array2D<Real>(n,n);
|
|
for (int i = 0; i < n; i++) {
|
|
for (int j = 0; j < n; j++) {
|
|
D[i][j] = 0.0;
|
|
}
|
|
D[i][i] = d[i];
|
|
if (e[i] > 0) {
|
|
D[i][i+1] = e[i];
|
|
} else if (e[i] < 0) {
|
|
D[i][i-1] = e[i];
|
|
}
|
|
}
|
|
}
|
|
};
|
|
|
|
CGAL_JAMA_END_NAMESPACE
|
|
|
|
#endif
|
|
// JAMA_EIG_H
|