EigenvalueDecomposition.cpp

/*
 This code is based on EigenvalueDecomposition.java from http://math.nist.gov/javanumerics/jama/
 The license below is from that file.
 */

/*
 * 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 domain. Neither The MathWorks nor NIST assumes any responsibility
 * whatsoever for its use by other parties, and makes no guarantees, expressed
 * or implied, about its quality, reliability, or any other characteristic.

 * EigenvalueDecomposition.java
 * Copyright (C) 1999 The Mathworks and NIST
 *
 */

#include "EigenvalueDecomposition.h"

GRT_BEGIN_NAMESPACE
 
EigenvalueDecomposition::EigenvalueDecomposition(){
 warningLog.setProceedingText("[WARNING EigenvalueDecomposition]");
}

EigenvalueDecomposition::~EigenvalueDecomposition(){

}
 
bool EigenvalueDecomposition::decompose(const MatrixFloat &a){
 
 n = a.getNumCols();
 eigenvectors.resize(n,n);
 realEigenvalues.resize(n);
 complexEigenvalues.resize(n);
 
 issymmetric = true;
 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++) {
 eigenvectors[i][j] = a[i][j];
 }
 }
 
 // Tridiagonalize.
 tred2();
 
 // Diagonalize.
 tql2();
 
 } else {
 h.resize(n,n);
 ort.resize(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 true;
}
 
void EigenvalueDecomposition::tred2(){
 
 for(int j = 0; j < n; j++) {
 realEigenvalues[j] = eigenvectors[n-1][j];
 }
 
 // Householder reduction to tridiagonal form.
 for(int i = n-1; i > 0; i--) {
 
 // Scale to avoid under/overflow.
 Float scale = 0.0;
 Float h = 0.0;
 for (int k = 0; k < i; k++) {
 scale = scale + fabs(realEigenvalues[k]);
 }
 if (scale == 0.0) {
 complexEigenvalues[i] = realEigenvalues[i-1];
 for (int j = 0; j < i; j++) {
 realEigenvalues[j] = eigenvectors[i-1][j];
 eigenvectors[i][j] = 0.0;
 eigenvectors[j][i] = 0.0;
 }
 } else {
 
 // Generate Householder vector.
 for (int k = 0; k < i; k++) {
 realEigenvalues[k] /= scale;
 h += realEigenvalues[k] * realEigenvalues[k];
 }
 Float f = realEigenvalues[i-1];
 Float g = sqrt(h);
 if (f > 0) {
 g = -g;
 }
 complexEigenvalues[i] = scale * g;
 h = h - f * g;
 realEigenvalues[i-1] = f - g;
 for (int j = 0; j < i; j++) {
 complexEigenvalues[j] = 0.0;
 }
 
 // Apply similarity transformation to remaining columns.
 for (int j = 0; j < i; j++) {
 f = realEigenvalues[j];
 eigenvectors[j][i] = f;
 g = complexEigenvalues[j] + eigenvectors[j][j] * f;
 for (int k = j+1; k <= i-1; k++) {
 g += eigenvectors[k][j] * realEigenvalues[k];
 complexEigenvalues[k] += eigenvectors[k][j] * f;
 }
 complexEigenvalues[j] = g;
 }
 f = 0.0;
 for (int j = 0; j < i; j++) {
 complexEigenvalues[j] /= h;
 f += complexEigenvalues[j] * realEigenvalues[j];
 }
 Float hh = f / (h + h);
 for (int j = 0; j < i; j++) {
 complexEigenvalues[j] -= hh * realEigenvalues[j];
 }
 for (int j = 0; j < i; j++) {
 f = realEigenvalues[j];
 g = complexEigenvalues[j];
 for (int k = j; k <= i-1; k++) {
 eigenvectors[k][j] -= (f * complexEigenvalues[k] + g * realEigenvalues[k]);
 }
 realEigenvalues[j] = eigenvectors[i-1][j];
 eigenvectors[i][j] = 0.0;
 }
 }
 realEigenvalues[i] = h;
 }
 
 // Accumulate transformations.
 for(int i = 0; i < n-1; i++) {
 eigenvectors[n-1][i] = eigenvectors[i][i];
 eigenvectors[i][i] = 1.0;
 Float h = realEigenvalues[i+1];
 if (h != 0.0) {
 for (int k = 0; k <= i; k++) {
 realEigenvalues[k] = eigenvectors[k][i+1] / h;
 }
 for (int j = 0; j <= i; j++) {
 Float g = 0.0;
 for (int k = 0; k <= i; k++) {
 g += eigenvectors[k][i+1] * eigenvectors[k][j];
 }
 for (int k = 0; k <= i; k++) {
 eigenvectors[k][j] -= g * realEigenvalues[k];
 }
 }
 }
 for(int k = 0; k <= i; k++) {
 eigenvectors[k][i+1] = 0.0;
 }
 }
 for(int j = 0; j < n; j++) {
 realEigenvalues[j] = eigenvectors[n-1][j];
 eigenvectors[n-1][j] = 0.0;
 }
 eigenvectors[n-1][n-1] = 1.0;
 complexEigenvalues[0] = 0.0;
 
 return;
}

void EigenvalueDecomposition::tql2(){
 
 for(int i = 1; i < n; i++) {
 complexEigenvalues[i-1] = complexEigenvalues[i];
 }
 complexEigenvalues[n-1] = 0.0;
 
 Float f = 0.0;
 Float tst1 = 0.0;
 Float eps = pow(2.0,-52.0);
 for (int l = 0; l < n; l++) {
 
 // Find small subdiagonal element
 tst1 = findMax(tst1,fabs(realEigenvalues[l]) + fabs(complexEigenvalues[l]));
 int m = l;
 while (m < n) {
 if(fabs(complexEigenvalues[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
 Float g = realEigenvalues[l];
 Float p = (realEigenvalues[l+1] - g) / (2.0 * complexEigenvalues[l]);
 Float r = hypot(p,1.0);
 if (p < 0) {
 r = -r;
 }
 realEigenvalues[l] = complexEigenvalues[l] / (p + r);
 realEigenvalues[l+1] = complexEigenvalues[l] * (p + r);
 Float dl1 = realEigenvalues[l+1];
 Float h = g - realEigenvalues[l];
 for (int i = l+2; i < n; i++) {
 realEigenvalues[i] -= h;
 }
 f = f + h;
 
 // Implicit QL transformation.
 p = realEigenvalues[m];
 Float c = 1.0;
 Float c2 = c;
 Float c3 = c;
 Float el1 = complexEigenvalues[l+1];
 Float s = 0.0;
 Float s2 = 0.0;
 for (int i = m-1; i >= l; i--) {
 c3 = c2;
 c2 = c;
 s2 = s;
 g = c * complexEigenvalues[i];
 h = c * p;
 r = hypot(p,complexEigenvalues[i]);
 complexEigenvalues[i+1] = s * r;
 s = complexEigenvalues[i] / r;
 c = p / r;
 p = c * realEigenvalues[i] - s * g;
 realEigenvalues[i+1] = h + s * (c * g + s * realEigenvalues[i]);
 
 // Accumulate transformation.
 for(int k = 0; k < n; k++) {
 h = eigenvectors[k][i+1];
 eigenvectors[k][i+1] = s * eigenvectors[k][i] + c * h;
 eigenvectors[k][i] = c * eigenvectors[k][i] - s * h;
 }
 }
 p = -s * s2 * c3 * el1 * complexEigenvalues[l] / dl1;
 complexEigenvalues[l] = s * p;
 realEigenvalues[l] = c * p;
 
 // Check for convergence.
 } while (fabs(complexEigenvalues[l]) > eps*tst1);
 }
 realEigenvalues[l] = realEigenvalues[l] + f;
 complexEigenvalues[l] = 0.0;
 }
 
 // Sort eigenvalues and corresponding vectors.
 for(int i = 0; i < n-1; i++) {
 int k = i;
 Float p = realEigenvalues[i];
 for (int j = i+1; j < n; j++) {
 if(realEigenvalues[j] < p) {
 k = j;
 p = realEigenvalues[j];
 }
 }
 if (k != i) {
 realEigenvalues[k] = realEigenvalues[i];
 realEigenvalues[i] = p;
 for (int j = 0; j < n; j++) {
 p = eigenvectors[j][i];
 eigenvectors[j][i] = eigenvectors[j][k];
 eigenvectors[j][k] = p;
 }
 }
 }
 return;
}

void EigenvalueDecomposition::orthes(){
 
 int low = 0;
 int high = n-1;
 
 for(int m = low+1; m <= high-1; m++) {
 
 // Scale column.
 Float scale = 0.0;
 for (int i = m; i <= high; i++) {
 scale = scale + fabs(h[i][m-1]);
 }
 if (scale != 0.0) {
 
 // Compute Householder transformation.
 Float ht = 0.0;
 for(int i = high; i >= m; i--) {
 ort[i] = h[i][m-1]/scale;
 ht += ort[i] * ort[i];
 }
 Float g = sqrt( ht );
 if (ort[m] > 0) {
 g = -g;
 }
 ht = ht - 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++) {
 Float f = 0.0;
 for (int i = high; i >= m; i--) {
 f += ort[i]*h[i][j];
 }
 f = f/ht;
 for (int i = m; i <= high; i++) {
 h[i][j] -= f*ort[i];
 }
 }
 
 for(int i = 0; i <= high; i++) {
 Float f = 0.0;
 for(int j = high; j >= m; j--) {
 f += ort[j]*h[i][j];
 }
 f = f/ht;
 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++) {
 eigenvectors[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++) {
 Float g = 0.0;
 for (int i = m; i <= high; i++) {
 g += ort[i] * eigenvectors[i][j];
 }
 // Double division avoids possible underflow
 g = (g / ort[m]) / h[m][m-1];
 for (int i = m; i <= high; i++) {
 eigenvectors[i][j] += g * ort[i];
 }
 }
 }
 }
 return;
}

void EigenvalueDecomposition::hqr2(){
 
 // Initialize
 int nn = this->n;
 int n = nn-1;
 int low = 0;
 int high = nn-1;
 Float eps = pow(2.0,-52.0);
 Float exshift = 0.0;
 Float p=0,q=0,r=0,s=0,z=0,t,w,x,y;
 
 // Store roots isolated by balanc and compute matrix norm
 Float norm = 0.0;
 for(int i = 0; i < nn; i++) {
 if( (i < low) | (i > high) ){
 realEigenvalues[i] = h[i][i];
 complexEigenvalues[i] = 0.0;
 }
 for (int j = findMax(i-1,0); j < nn; j++) {
 norm = norm + fabs(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 = fabs(h[l-1][l-1]) + fabs(h[l][l]);
 if (s == 0.0) {
 s = norm;
 }
 if(fabs(h[l][l-1]) < eps * s) {
 break;
 }
 l--;
 }
 
 // Check for convergence
 // One root found
 if(l == n) {
 h[n][n] = h[n][n] + exshift;
 realEigenvalues[n] = h[n][n];
 complexEigenvalues[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 = sqrt(fabs(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;
 }
 realEigenvalues[n-1] = x + z;
 realEigenvalues[n] = realEigenvalues[n-1];
 if (z != 0.0) {
 realEigenvalues[n] = x - w / z;
 }
 complexEigenvalues[n-1] = 0.0;
 complexEigenvalues[n] = 0.0;
 x = h[n][n-1];
 s = fabs(x) + fabs(z);
 p = x / s;
 q = z / s;
 r = 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 = eigenvectors[i][n-1];
 eigenvectors[i][n-1] = q * z + p * eigenvectors[i][n];
 eigenvectors[i][n] = q * eigenvectors[i][n] - p * z;
 }
 
 // Complex pair
 } else {
 realEigenvalues[n-1] = x + p;
 realEigenvalues[n] = x + p;
 complexEigenvalues[n-1] = z;
 complexEigenvalues[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 = fabs(h[n][n-1]) + fabs(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 = fabs(p) + fabs(q) + fabs(r);
 p = p / s;
 q = q / s;
 r = r / s;
 if(m == l){
 break;
 }
 if(fabs(h[m][m-1]) * (fabs(q) +fabs(r)) <
 eps * (fabs(p) * (fabs(h[m-1][m-1]) + fabs(z) +
 fabs(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 = fabs(p) + fabs(q) + fabs(r);
 if (x != 0.0) {
 p = p / x;
 q = q / x;
 r = r / x;
 }
 }
 if(x == 0.0){
 break;
 }
 s = 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 <= findMin(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 * eigenvectors[i][k] + y * eigenvectors[i][k+1];
 if(notlast){
 p = p + z * eigenvectors[i][k+2];
 eigenvectors[i][k+2] = eigenvectors[i][k+2] - p * r;
 }
 eigenvectors[i][k] = eigenvectors[i][k] - p;
 eigenvectors[i][k+1] = eigenvectors[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 = realEigenvalues[n];
 q = complexEigenvalues[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(complexEigenvalues[i] < 0.0) {
 z = w;
 s = r;
 } else {
 l = i;
 if (complexEigenvalues[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 = (realEigenvalues[i] - p) * (realEigenvalues[i] - p) + complexEigenvalues[i] * complexEigenvalues[i];
 t = (x * s - z * r) / q;
 h[i][n] = t;
 if(fabs(x) > fabs(z)) {
 h[i+1][n] = (-r - w * t) / x;
 } else {
 h[i+1][n] = (-s - y * t) / z;
 }
 }
 
 // Overflow control
 t = fabs(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 (fabs(h[n][n-1]) > fabs(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--) {
 Float 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(complexEigenvalues[i] < 0.0) {
 z = w;
 r = ra;
 s = sa;
 } else {
 l = i;
 if(complexEigenvalues[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 = (realEigenvalues[i] - p) * (realEigenvalues[i] - p) + complexEigenvalues[i] * complexEigenvalues[i] - q * q;
 vi = (realEigenvalues[i] - p) * 2.0 * q;
 if((vr == 0.0) & (vi == 0.0)){
 vr = eps * norm * (fabs(w) + fabs(q) + fabs(x) + fabs(y) + fabs(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 (fabs(x) > (fabs(z) + fabs(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 = findMax(fabs(h[i][n-1]),fabs(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++) {
 eigenvectors[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 <= findMin(j,high); k++) {
 z = z + eigenvectors[i][k] * h[k][j];
 }
 eigenvectors[i][j] = z;
 }
 }
 
 return;
}
 
void EigenvalueDecomposition::cdiv(Float xr, Float xi, Float yr, Float yi){
 Float r,d;
 if(fabs(yr) > fabs(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;
 }
 return;
}
 
MatrixFloat EigenvalueDecomposition::getDiagonalEigenvalueMatrix(){
 
 MatrixFloat x(n,n);
 for (int i = 0; i < n; i++) {
 for (int j = 0; j < n; j++) {
 x[i][j] = 0.0;
 }
 x[i][i] = realEigenvalues[i];
 if(complexEigenvalues[i] > 0) {
 x[i][i+1] = complexEigenvalues[i];
 } else if(complexEigenvalues[i] < 0) {
 x[i][i-1] = complexEigenvalues[i];
 }
 }
 return x;
}

VectorFloat EigenvalueDecomposition::getRealEigenvalues(){
 return realEigenvalues;
}
 
VectorFloat EigenvalueDecomposition::getComplexEigenvalues(){
 return complexEigenvalues;
}
 
GRT_END_NAMESPACE