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-// matrix/jama-eig.h
-
-// Copyright 2009-2011 Microsoft Corporation
-
-// See ../../COPYING for clarification regarding multiple authors
-//
-// Licensed under the Apache License, Version 2.0 (the "License");
-// you may not use this file except in compliance with the License.
-// You may obtain a copy of the License at
-
-// http://www.apache.org/licenses/LICENSE-2.0
-
-// THIS CODE IS PROVIDED *AS IS* BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
-// KIND, EITHER EXPRESS OR IMPLIED, INCLUDING WITHOUT LIMITATION ANY IMPLIED
-// WARRANTIES OR CONDITIONS OF TITLE, FITNESS FOR A PARTICULAR PURPOSE,
-// MERCHANTABLITY OR NON-INFRINGEMENT.
-// See the Apache 2 License for the specific language governing permissions and
-// limitations under the License.
-
-// This file consists of a port and modification of materials from
-// JAMA: A Java Matrix Package
-// under the following notice: This software is a cooperative product of
-// The MathWorks and the National Institute of Standards and Technology (NIST)
-// which has been released to the public. This notice and the original code are
-// available at http://math.nist.gov/javanumerics/jama/domain.notice
-
-
-
-#ifndef KALDI_MATRIX_JAMA_EIG_H_
-#define KALDI_MATRIX_JAMA_EIG_H_ 1
-
-#include "matrix/kaldi-matrix.h"
-
-namespace kaldi {
-
-// This class is not to be used externally. See the Eig function in the Matrix
-// class in kaldi-matrix.h. This is the external interface.
-
-template<typename Real> class EigenvalueDecomposition {
- // This class is based on the EigenvalueDecomposition class from the JAMA
- // library (version 1.0.2).
- public:
- EigenvalueDecomposition(const MatrixBase<Real> &A);
-
- ~EigenvalueDecomposition(); // free memory.
-
- void GetV(MatrixBase<Real> *V_out) { // V is what we call P externally; it's the matrix of
- // eigenvectors.
- KALDI_ASSERT(V_out->NumRows() == static_cast<MatrixIndexT>(n_)
- && V_out->NumCols() == static_cast<MatrixIndexT>(n_));
- for (int i = 0; i < n_; i++)
- for (int j = 0; j < n_; j++)
- (*V_out)(i, j) = V(i, j); // V(i, j) is member function.
- }
- void GetRealEigenvalues(VectorBase<Real> *r_out) {
- // returns real part of eigenvalues.
- KALDI_ASSERT(r_out->Dim() == static_cast<MatrixIndexT>(n_));
- for (int i = 0; i < n_; i++)
- (*r_out)(i) = d_[i];
- }
- void GetImagEigenvalues(VectorBase<Real> *i_out) {
- // returns imaginary part of eigenvalues.
- KALDI_ASSERT(i_out->Dim() == static_cast<MatrixIndexT>(n_));
- for (int i = 0; i < n_; i++)
- (*i_out)(i) = e_[i];
- }
- private:
-
- inline Real &H(int r, int c) { return H_[r*n_ + c]; }
- inline Real &V(int r, int c) { return V_[r*n_ + c]; }
-
- // complex division
- inline static void cdiv(Real xr, Real xi, Real yr, Real yi, Real *cdivr, Real *cdivi) {
- Real r, d;
- if (std::abs(yr) > std::abs(yi)) {
- r = yi/yr;
- d = yr + r*yi;
- *cdivr = (xr + r*xi)/d;
- *cdivi = (xi - r*xr)/d;
- } else {
- r = yr/yi;
- d = yi + r*yr;
- *cdivr = (r*xr + xi)/d;
- *cdivi = (r*xi - xr)/d;
- }
- }
-
- // Nonsymmetric reduction from Hessenberg to real Schur form.
- void Hqr2 ();
-
-
- int n_; // matrix dimension.
-
- Real *d_, *e_; // real and imaginary parts of eigenvalues.
- Real *V_; // the eigenvectors (P in our external notation)
- Real *H_; // the nonsymmetric Hessenberg form.
- Real *ort_; // working storage for nonsymmetric algorithm.
-
- // Symmetric Householder reduction to tridiagonal form.
- void Tred2 ();
-
- // Symmetric tridiagonal QL algorithm.
- void Tql2 ();
-
- // Nonsymmetric reduction to Hessenberg form.
- void Orthes ();
-
-};
-
-template class EigenvalueDecomposition<float>; // force instantiation.
-template class EigenvalueDecomposition<double>; // force instantiation.
-
-template<typename Real> void EigenvalueDecomposition<Real>::Tred2() {
- // This is derived from the Algol procedures tred2 by
- // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
- // Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
- // Fortran subroutine in EISPACK.
-
- for (int j = 0; j < n_; j++) {
- d_[j] = V(n_-1, j);
- }
-
- // Householder reduction to tridiagonal form.
-
- for (int i = n_-1; i > 0; i--) {
-
- // Scale to avoid under/overflow.
-
- Real scale = 0.0;
- Real h = 0.0;
- for (int k = 0; k < i; k++) {
- scale = scale + std::abs(d_[k]);
- }
- if (scale == 0.0) {
- e_[i] = d_[i-1];
- for (int j = 0; j < i; j++) {
- d_[j] = V(i-1, j);
- V(i, j) = 0.0;
- V(j, i) = 0.0;
- }
- } else {
-
- // Generate Householder vector.
-
- for (int k = 0; k < i; k++) {
- d_[k] /= scale;
- h += d_[k] * d_[k];
- }
- Real f = d_[i-1];
- Real g = std::sqrt(h);
- if (f > 0) {
- g = -g;
- }
- e_[i] = scale * g;
- h = h - f * g;
- d_[i-1] = f - g;
- for (int j = 0; j < i; j++) {
- e_[j] = 0.0;
- }
-
- // Apply similarity transformation to remaining columns.
-
- for (int j = 0; j < i; j++) {
- f = d_[j];
- V(j, i) = f;
- g =e_[j] + V(j, j) * f;
- for (int k = j+1; k <= i-1; k++) {
- g += V(k, j) * d_[k];
- e_[k] += V(k, j) * f;
- }
- e_[j] = g;
- }
- f = 0.0;
- for (int j = 0; j < i; j++) {
- e_[j] /= h;
- f += e_[j] * d_[j];
- }
- Real hh = f / (h + h);
- for (int j = 0; j < i; j++) {
- e_[j] -= hh * d_[j];
- }
- for (int j = 0; j < i; j++) {
- f = d_[j];
- g = e_[j];
- for (int k = j; k <= i-1; k++) {
- V(k, j) -= (f * e_[k] + g * d_[k]);
- }
- d_[j] = V(i-1, j);
- V(i, j) = 0.0;
- }
- }
- d_[i] = h;
- }
-
- // Accumulate transformations.
-
- for (int i = 0; i < n_-1; i++) {
- V(n_-1, i) = V(i, i);
- V(i, i) = 1.0;
- Real h = d_[i+1];
- if (h != 0.0) {
- for (int k = 0; k <= i; k++) {
- d_[k] = V(k, i+1) / h;
- }
- for (int j = 0; j <= i; j++) {
- Real g = 0.0;
- for (int k = 0; k <= i; k++) {
- g += V(k, i+1) * V(k, j);
- }
- for (int k = 0; k <= i; k++) {
- V(k, j) -= g * d_[k];
- }
- }
- }
- for (int k = 0; k <= i; k++) {
- V(k, i+1) = 0.0;
- }
- }
- for (int j = 0; j < n_; j++) {
- d_[j] = V(n_-1, j);
- V(n_-1, j) = 0.0;
- }
- V(n_-1, n_-1) = 1.0;
- e_[0] = 0.0;
-}
-
-template<typename Real> void EigenvalueDecomposition<Real>::Tql2() {
- // This is derived from the Algol procedures tql2, by
- // Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
- // Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
- // Fortran subroutine in EISPACK.
-
- for (int i = 1; i < n_; i++) {
- e_[i-1] = e_[i];
- }
- e_[n_-1] = 0.0;
-
- Real f = 0.0;
- Real tst1 = 0.0;
- Real eps = std::numeric_limits<Real>::epsilon();
- for (int l = 0; l < n_; l++) {
-
- // Find small subdiagonal element
-
- tst1 = std::max(tst1, std::abs(d_[l]) + std::abs(e_[l]));
- int m = l;
- while (m < n_) {
- if (std::abs(e_[m]) <= eps*tst1) {
- break;
- }
- m++;
- }
-
- // If m == l, d_[l] is an eigenvalue,
- // otherwise, iterate.
-
- if (m > l) {
- int iter = 0;
- do {
- iter = iter + 1; // (Could check iteration count here.)
-
- // Compute implicit shift
-
- Real g = d_[l];
- Real p = (d_[l+1] - g) / (2.0 *e_[l]);
- Real r = Hypot(p, static_cast<Real>(1.0)); // This is a Kaldi version of hypot that works with templates.
- if (p < 0) {
- r = -r;
- }
- d_[l] =e_[l] / (p + r);
- d_[l+1] =e_[l] * (p + r);
- Real dl1 = d_[l+1];
- Real h = g - d_[l];
- for (int i = l+2; i < n_; i++) {
- d_[i] -= h;
- }
- f = f + h;
-
- // Implicit QL transformation.
-
- p = d_[m];
- Real c = 1.0;
- Real c2 = c;
- Real c3 = c;
- Real el1 =e_[l+1];
- Real s = 0.0;
- Real s2 = 0.0;
- for (int i = m-1; i >= l; i--) {
- c3 = c2;
- c2 = c;
- s2 = s;
- g = c *e_[i];
- h = c * p;
- r = Hypot(p, e_[i]); // This is a Kaldi version of Hypot that works with templates.
- e_[i+1] = s * r;
- s =e_[i] / r;
- c = p / r;
- p = c * d_[i] - s * g;
- d_[i+1] = h + s * (c * g + s * d_[i]);
-
- // Accumulate transformation.
-
- for (int k = 0; k < n_; k++) {
- h = V(k, i+1);
- V(k, i+1) = s * V(k, i) + c * h;
- V(k, i) = c * V(k, i) - s * h;
- }
- }
- p = -s * s2 * c3 * el1 *e_[l] / dl1;
- e_[l] = s * p;
- d_[l] = c * p;
-
- // Check for convergence.
-
- } while (std::abs(e_[l]) > eps*tst1);
- }
- d_[l] = d_[l] + f;
- e_[l] = 0.0;
- }
-
- // Sort eigenvalues and corresponding vectors.
-
- for (int i = 0; i < n_-1; i++) {
- int k = i;
- Real p = d_[i];
- for (int j = i+1; j < n_; j++) {
- if (d_[j] < p) {
- k = j;
- p = d_[j];
- }
- }
- if (k != i) {
- d_[k] = d_[i];
- d_[i] = p;
- for (int j = 0; j < n_; j++) {
- p = V(j, i);
- V(j, i) = V(j, k);
- V(j, k) = p;
- }
- }
- }
-}
-
-template<typename Real>
-void EigenvalueDecomposition<Real>::Orthes() {
-
- // This is derived from the Algol procedures orthes and ortran,
- // by Martin and Wilkinson, Handbook for Auto. Comp.,
- // Vol.ii-Linear Algebra, and the corresponding
- // Fortran subroutines in EISPACK.
-
- int low = 0;
- int high = n_-1;
-
- for (int m = low+1; m <= high-1; m++) {
-
- // Scale column.
-
- Real scale = 0.0;
- for (int i = m; i <= high; i++) {
- scale = scale + std::abs(H(i, m-1));
- }
- if (scale != 0.0) {
-
- // Compute Householder transformation.
-
- Real h = 0.0;
- for (int i = high; i >= m; i--) {
- ort_[i] = H(i, m-1)/scale;
- h += ort_[i] * ort_[i];
- }
- Real g = std::sqrt(h);
- if (ort_[m] > 0) {
- g = -g;
- }
- h = h - ort_[m] * g;
- ort_[m] = ort_[m] - g;
-
- // Apply Householder similarity transformation
- // H = (I-u*u'/h)*H*(I-u*u')/h)
-
- for (int j = m; j < n_; j++) {
- Real f = 0.0;
- for (int i = high; i >= m; i--) {
- f += ort_[i]*H(i, j);
- }
- f = f/h;
- for (int i = m; i <= high; i++) {
- H(i, j) -= f*ort_[i];
- }
- }
-
- for (int i = 0; i <= high; i++) {
- Real f = 0.0;
- for (int j = high; j >= m; j--) {
- f += ort_[j]*H(i, j);
- }
- f = f/h;
- for (int j = m; j <= high; j++) {
- H(i, j) -= f*ort_[j];
- }
- }
- ort_[m] = scale*ort_[m];
- H(m, m-1) = scale*g;
- }
- }
-
- // Accumulate transformations (Algol's ortran).
-
- for (int i = 0; i < n_; i++) {
- for (int j = 0; j < n_; j++) {
- V(i, j) = (i == j ? 1.0 : 0.0);
- }
- }
-
- for (int m = high-1; m >= low+1; m--) {
- if (H(m, m-1) != 0.0) {
- for (int i = m+1; i <= high; i++) {
- ort_[i] = H(i, m-1);
- }
- for (int j = m; j <= high; j++) {
- Real g = 0.0;
- for (int i = m; i <= high; i++) {
- g += ort_[i] * V(i, j);
- }
- // Double division avoids possible underflow
- g = (g / ort_[m]) / H(m, m-1);
- for (int i = m; i <= high; i++) {
- V(i, j) += g * ort_[i];
- }
- }
- }
- }
-}
-
-template<typename Real> void EigenvalueDecomposition<Real>::Hqr2() {
- // This is derived from the Algol procedure hqr2,
- // by Martin and Wilkinson, Handbook for Auto. Comp.,
- // Vol.ii-Linear Algebra, and the corresponding
- // Fortran subroutine in EISPACK.
-
- int nn = n_;
- int n = nn-1;
- int low = 0;
- int high = nn-1;
- Real eps = std::numeric_limits<Real>::epsilon();
- Real exshift = 0.0;
- Real p = 0, q = 0, r = 0, s = 0, z=0, t, w, x, y;
-
- // Store roots isolated by balanc and compute matrix norm
-
- Real norm = 0.0;
- for (int i = 0; i < nn; i++) {
- if (i < low || i > high) {
- d_[i] = H(i, i);
- e_[i] = 0.0;
- }
- for (int j = std::max(i-1, 0); j < nn; j++) {
- norm = norm + std::abs(H(i, j));
- }
- }
-
- // Outer loop over eigenvalue index
-
- int iter = 0;
- while (n >= low) {
-
- // Look for single small sub-diagonal element
-
- int l = n;
- while (l > low) {
- s = std::abs(H(l-1, l-1)) + std::abs(H(l, l));
- if (s == 0.0) {
- s = norm;
- }
- if (std::abs(H(l, l-1)) < eps * s) {
- break;
- }
- l--;
- }
-
- // Check for convergence
- // One root found
-
- if (l == n) {
- H(n, n) = H(n, n) + exshift;
- d_[n] = H(n, n);
- e_[n] = 0.0;
- n--;
- iter = 0;
-
- // Two roots found
-
- } else if (l == n-1) {
- w = H(n, n-1) * H(n-1, n);
- p = (H(n-1, n-1) - H(n, n)) / 2.0;
- q = p * p + w;
- z = std::sqrt(std::abs(q));
- H(n, n) = H(n, n) + exshift;
- H(n-1, n-1) = H(n-1, n-1) + exshift;
- x = H(n, n);
-
- // Real pair
-
- if (q >= 0) {
- if (p >= 0) {
- z = p + z;
- } else {
- z = p - z;
- }
- d_[n-1] = x + z;
- d_[n] = d_[n-1];
- if (z != 0.0) {
- d_[n] = x - w / z;
- }
- e_[n-1] = 0.0;
- e_[n] = 0.0;
- x = H(n, n-1);
- s = std::abs(x) + std::abs(z);
- p = x / s;
- q = z / s;
- r = std::sqrt(p * p+q * q);
- 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 = std::abs(H(n, n-1)) + std::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 = std::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 = std::abs(p) + std::abs(q) + std::abs(r);
- p = p / s;
- q = q / s;
- r = r / s;
- if (m == l) {
- break;
- }
- if (std::abs(H(m, m-1)) * (std::abs(q) + std::abs(r)) <
- eps * (std::abs(p) * (std::abs(H(m-1, m-1)) + std::abs(z) +
- std::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++) {
- bool 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 = std::abs(p) + std::abs(q) + std::abs(r);
- if (x != 0.0) {
- p = p / x;
- q = q / x;
- r = r / x;
- }
- }
- if (x == 0.0) {
- break;
- }
- s = std::sqrt(p * p + q * q + r * r);
- if (p < 0) {
- s = -s;
- }
- if (s != 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 <= std::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 == 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] < 0.0) {
- z = w;
- s = r;
- } else {
- l = i;
- if (e_[i] == 0.0) {
- if (w != 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 (std::abs(x) > std::abs(z)) {
- H(i+1, n) = (-r - w * t) / x;
- } else {
- H(i+1, n) = (-s - y * t) / z;
- }
- }
-
- // Overflow control
-
- t = std::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 (std::abs(H(n, n-1)) > std::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 {
- Real cdivr, cdivi;
- cdiv(0.0, -H(n-1, n), H(n-1, n-1)-p, q, &cdivr, &cdivi);
- 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] < 0.0) {
- z = w;
- r = ra;
- s = sa;
- } else {
- l = i;
- if (e_[i] == 0) {
- Real cdivr, cdivi;
- cdiv(-ra, -sa, w, q, &cdivr, &cdivi);
- H(i, n-1) = cdivr;
- H(i, n) = cdivi;
- } else {
- Real cdivr, cdivi;
- // 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 == 0.0 && vi == 0.0) {
- vr = eps * norm * (std::abs(w) + std::abs(q) +
- std::abs(x) + std::abs(y) + std::abs(z));
- }
- cdiv(x*r-z*ra+q*sa, x*s-z*sa-q*ra, vr, vi, &cdivr, &cdivi);
- H(i, n-1) = cdivr;
- H(i, n) = cdivi;
- if (std::abs(x) > (std::abs(z) + std::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, &cdivr, &cdivi);
- H(i+1, n-1) = cdivr;
- H(i+1, n) = cdivi;
- }
- }
-
- // Overflow control
-
- t = std::max(std::abs(H(i, n-1)), std::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 <= std::min(j, high); k++) {
- z = z + V(i, k) * H(k, j);
- }
- V(i, j) = z;
- }
- }
-}
-
-template<typename Real>
-EigenvalueDecomposition<Real>::EigenvalueDecomposition(const MatrixBase<Real> &A) {
- KALDI_ASSERT(A.NumCols() == A.NumRows() && A.NumCols() >= 1);
- n_ = A.NumRows();
- V_ = new Real[n_*n_];
- d_ = new Real[n_];
- e_ = new Real[n_];
- H_ = NULL;
- ort_ = NULL;
- if (A.IsSymmetric(0.0)) {
-
- for (int i = 0; i < n_; i++)
- for (int j = 0; j < n_; j++)
- V(i, j) = A(i, j); // Note that V(i, j) is a member function; A(i, j) is an operator
- // of the matrix A.
- // Tridiagonalize.
- Tred2();
-
- // Diagonalize.
- Tql2();
- } else {
- H_ = new Real[n_*n_];
- ort_ = new Real[n_];
- for (int i = 0; i < n_; i++)
- for (int j = 0; j < n_; j++)
- H(i, j) = A(i, j); // as before: H is member function, A(i, j) is operator of matrix.
-
- // Reduce to Hessenberg form.
- Orthes();
-
- // Reduce Hessenberg to real Schur form.
- Hqr2();
- }
-}
-
-template<typename Real>
-EigenvalueDecomposition<Real>::~EigenvalueDecomposition() {
- delete [] d_;
- delete [] e_;
- delete [] V_;
- if (H_) delete [] H_;
- if (ort_) delete [] ort_;
-}
-
-// see function MatrixBase<Real>::Eig in kaldi-matrix.cc
-
-
-} // namespace kaldi
-
-#endif // KALDI_MATRIX_JAMA_EIG_H_
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