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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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