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Jeometry, a Mathematic and Geometry library for Java
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package org.jeometry.simple.math.decomposition;
import java.util.ArrayList;
import java.util.List;
import org.jeometry.Jeometry;
import org.jeometry.factory.JeometryFactory;
import org.jeometry.math.Matrix;
import org.jeometry.math.decomposition.EigenDecomposition;
/**
* A simple implementation of {@link EigenDecomposition Eigen decomposition}.
* This implantation is inspired by Jama Eigen Decomposition.
* @author Julien Seinturier - COMEX S.A. - [email protected] - https://github.com/jorigin/jeometry
* @version {@value Jeometry#version} b{@value Jeometry#BUILD}
* @since 1.0.0
*/
public class SimpleEigenDecomposition implements EigenDecomposition {
/** Row and column dimension (square matrix).
* @serial matrix dimension.
*/
private int n;
/** Symmetry flag.
* @serial internal symmetry flag.
*/
private boolean issymmetric;
/**
* Arrays for internal storage of eigenvalues.
*/
private double[] d;
/**
* Arrays for internal storage of eigenvalues.
*/
private double[] e;
/**
* Array for internal storage of eigenvectors.
*/
private Matrix V = null;
/**
* Array for internal storage of eigenvectors.
*/
private Matrix D = null;
/**
* Array for internal storage of nonsymmetric Hessenberg form.
* @serial internal storage of nonsymmetric Hessenberg form.
*/
private Matrix H;
/**
* Working storage for nonsymmetric algorithm.
* @serial working storage for nonsymmetric algorithm.
*/
private double[] ort;
/**
* Temporary variable.
*/
private transient double cdivr;
/**
* Temporary variable.
*/
private transient double cdivi;
/**
* Construct a new {@link EigenDecomposition Eigen decomposition} from the given {@link Matrix matrix}
* @param matrix the matrix to decompose
*/
public SimpleEigenDecomposition(Matrix matrix){
double[][] A = matrix.getDataArray2D();
n = matrix.getColumnsCount();
V = JeometryFactory.createMatrix(n, n);
d = new double[n];
e = new double[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++) {
V.setValue(i, j, A[i][j]);
}
}
// Tridiagonalize.
tred2();
// Diagonalize.
tql2();
} else {
H = JeometryFactory.createMatrix(n, n);
ort = new double[n];
for (int j = 0; j < n; j++) {
for (int i = 0; i < n; i++) {
H.setValue(i, j, A[i][j]);
}
}
// Reduce to Hessenberg form.
orthes();
// Reduce Hessenberg to real Schur form.
hqr2();
}
// Compute D matrix
D = JeometryFactory.createMatrix(n, n);
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
D.setValue(i, j, 0.0);
}
D.setValue(i, i, d[i]);
if (e[i] > 0) {
D.setValue(i, i+1, e[i]);
} else if (e[i] < 0) {
D.setValue(i, i-1, e[i]);
}
}
}
@Override
public Matrix getD() {
return D;
}
@Override
public Matrix getV() {
return V;
}
@Override
public List getComponents() {
List matrices = new ArrayList(2);
matrices.add(V);
matrices.add(D);
return matrices;
}
/** Compute sqrt(a^2 + b^2) without under/overflow.
* @param a the first
* @param b the second
* @return the result
**/
private double hypot(double a, double b) {
double r;
if (Math.abs(a) > Math.abs(b)) {
r = b/a;
r = Math.abs(a)*Math.sqrt(1+r*r);
} else if (b != 0) {
r = a/b;
r = Math.abs(b)*Math.sqrt(1+r*r);
} else {
r = 0.0;
}
return r;
}
/**
* Compute the complex scalar divisio x/y.
* @param xr real part of x
* @param xi imaginary part of x
* @param yr real part of y
* @param yi imaginary part of y
*/
private void cdiv(double xr, double xi, double yr, double yi) {
double r,d;
if (Math.abs(yr) > Math.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;
}
}
/**
* 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.
*/
private void tred2 () {
for (int j = 0; j < n; j++) {
d[j] = V.getValue(n-1, j);
}
// Householder reduction to tridiagonal form.
for (int i = n-1; i > 0; i--) {
// Scale to avoid under/overflow.
double scale = 0.0;
double h = 0.0;
for (int k = 0; k < i; k++) {
scale = scale + Math.abs(d[k]);
}
if (scale == 0.0) {
e[i] = d[i-1];
for (int j = 0; j < i; j++) {
d[j] = V.getValue(i-1, j);
V.setValue(i, j, 0.0);
V.setValue(j, i, 0.0);
}
} else {
// Generate Householder vector.
for (int k = 0; k < i; k++) {
d[k] /= scale;
h += d[k] * d[k];
}
double f = d[i-1];
double g = Math.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.setValue(j, i, f);
g = e[j] + V.getValue(j, j) * f;
for (int k = j+1; k <= i-1; k++) {
g += V.getValue(k, j) * d[k];
e[k] += V.getValue(k, j) * f;
}
e[j] = g;
}
f = 0.0;
for (int j = 0; j < i; j++) {
e[j] /= h;
f += e[j] * d[j];
}
double 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.setValue(k, j, V.getValue(k, j) - (f * e[k] + g * d[k]));
}
d[j] = V.getValue(i-1, j);
V.setValue(i, j, 0.0);
}
}
d[i] = h;
}
// Accumulate transformations.
for (int i = 0; i < n-1; i++) {
V.setValue(n-1, i, V.getValue(i, i));
V.setValue(i, i, 1.0);
double h = d[i+1];
if (h != 0.0) {
for (int k = 0; k <= i; k++) {
d[k] = V.getValue(k, i+1) / h;
}
for (int j = 0; j <= i; j++) {
double g = 0.0;
for (int k = 0; k <= i; k++) {
g += V.getValue(k, i+1) * V.getValue(k, j);
}
for (int k = 0; k <= i; k++) {
V.setValue(k, j, V.getValue(k, j) - g * d[k]);
}
}
}
for (int k = 0; k <= i; k++) {
V.setValue(k, i+1, 0.0);
}
}
for (int j = 0; j < n; j++) {
d[j] = V.getValue(n-1, j);
V.setValue(n-1, j, 0.0);
}
V.setValue(n-1, n-1, 1.0);
e[0] = 0.0;
}
/**
* Symmetric tridiagonal QL algorithm.
*/
private void 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;
double f = 0.0;
double tst1 = 0.0;
double eps = Math.pow(2.0,-52.0);
for (int l = 0; l < n; l++) {
// Find small subdiagonal element
tst1 = Math.max(tst1,Math.abs(d[l]) + Math.abs(e[l]));
int m = l;
while (m < n) {
if (Math.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
double g = d[l];
double p = (d[l+1] - g) / (2.0 * e[l]);
double r = hypot(p,1.0);
if (p < 0) {
r = -r;
}
d[l] = e[l] / (p + r);
d[l+1] = e[l] * (p + r);
double dl1 = d[l+1];
double h = g - d[l];
for (int i = l+2; i < n; i++) {
d[i] -= h;
}
f = f + h;
// Implicit QL transformation.
p = d[m];
double c = 1.0;
double c2 = c;
double c3 = c;
double el1 = e[l+1];
double s = 0.0;
double 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]);
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.getValue(k, i+1);
V.setValue(k, i+1, s * V.getValue(k, i) + c * h);
V.setValue(k, i, c * V.getValue(k, i) - s * h);
}
}
p = -s * s2 * c3 * el1 * e[l] / dl1;
e[l] = s * p;
d[l] = c * p;
// Check for convergence.
} while (Math.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;
double 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.getValue(j, i);
V.setValue(j, i, V.getValue(j, k));
V.setValue(j, k, p);
}
}
}
}
/**
* Nonsymmetric reduction to Hessenberg form.
*/
private void 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.
double scale = 0.0;
for (int i = m; i <= high; i++) {
scale = scale + Math.abs(H.getValue(i, m-1));
}
if (scale != 0.0) {
// Compute Householder transformation.
double h = 0.0;
for (int i = high; i >= m; i--) {
ort[i] = H.getValue(i, m-1)/scale;
h += ort[i] * ort[i];
}
double g = Math.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++) {
double f = 0.0;
for (int i = high; i >= m; i--) {
f += ort[i]*H.getValue(i, j);
}
f = f/h;
for (int i = m; i <= high; i++) {
H.setValue(i, j, H.getValue(i, j) - f*ort[i]);
}
}
for (int i = 0; i <= high; i++) {
double f = 0.0;
for (int j = high; j >= m; j--) {
f += ort[j]*H.getValue(i, j);
}
f = f/h;
for (int j = m; j <= high; j++) {
H.setValue(i, j, H.getValue(i, j) - f*ort[j]);
}
}
ort[m] = scale*ort[m];
H.setValue(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.setValue(i, j, (i == j ? 1.0 : 0.0));
}
}
for (int m = high-1; m >= low+1; m--) {
if (H.getValue(m, m-1) != 0.0) {
for (int i = m+1; i <= high; i++) {
ort[i] = H.getValue(i, m-1);
}
for (int j = m; j <= high; j++) {
double g = 0.0;
for (int i = m; i <= high; i++) {
g += ort[i] * V.getValue(i, j);
}
// Double division avoids possible underflow
g = (g / ort[m]) / H.getValue(m, m-1);
for (int i = m; i <= high; i++) {
V.setValue(i, j, V.getValue(i, j) + g * ort[i]);
}
}
}
}
}
/**
* Nonsymmetric reduction from Hessenberg to real Schur form.
*/
private void 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.
// Initialize
int nn = this.n;
int n = nn-1;
int low = 0;
int high = nn-1;
double eps = Math.pow(2.0,-52.0);
double exshift = 0.0;
double p=0,q=0,r=0,s=0,z=0,t,w,x,y;
// Store roots isolated by balanc and compute matrix norm
double norm = 0.0;
for (int i = 0; i < nn; i++) {
if (i < low | i > high) {
d[i] = H.getValue(i, i);
e[i] = 0.0;
}
for (int j = Math.max(i-1,0); j < nn; j++) {
norm = norm + Math.abs(H.getValue(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 = Math.abs(H.getValue(l-1, l-1)) + Math.abs(H.getValue(l, l));
if (s == 0.0) {
s = norm;
}
if (Math.abs(H.getValue(l, l-1)) < eps * s) {
break;
}
l--;
}
// Check for convergence
// One root found
if (l == n) {
H.setValue(n, n, H.getValue(n, n) + exshift);
d[n] = H.getValue(n, n);
e[n] = 0.0;
n--;
iter = 0;
// Two roots found
} else if (l == n-1) {
w = H.getValue(n, n-1) * H.getValue(n-1, n);
p = (H.getValue(n-1, n-1) - H.getValue(n, n)) / 2.0;
q = p * p + w;
z = Math.sqrt(Math.abs(q));
H.setValue(n, n, H.getValue(n, n) + exshift);
H.setValue(n-1, n-1, H.getValue(n-1, n-1) + exshift);
x = H.getValue(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.getValue(n, n-1);
s = Math.abs(x) + Math.abs(z);
p = x / s;
q = z / s;
r = Math.sqrt(p * p+q * q);
p = p / r;
q = q / r;
// Row modification
for (int j = n-1; j < nn; j++) {
z = H.getValue(n-1, j);
H.setValue(n-1, j, q * z + p * H.getValue(n, j));
H.setValue(n, j, q * H.getValue(n, j) - p * z);
}
// Column modification
for (int i = 0; i <= n; i++) {
z = H.getValue(i, n-1);
H.setValue(i, n-1, q * z + p * H.getValue(i, n));
H.setValue(i, n, q * H.getValue(i, n) - p * z);
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
z = V.getValue(i, n-1);
V.setValue(i, n-1, q * z + p * V.getValue(i, n));
V.setValue(i, n, q * V.getValue(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.getValue(n, n);
y = 0.0;
w = 0.0;
if (l < n) {
y = H.getValue(n-1, n-1);
w = H.getValue(n, n-1) * H.getValue(n-1, n);
}
// Wilkinson's original ad hoc shift
if (iter == 10) {
exshift += x;
for (int i = low; i <= n; i++) {
H.setValue(i, i, H.getValue(i, i)- x);
}
s = Math.abs(H.getValue(n, n-1)) + Math.abs(H.getValue(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 = Math.sqrt(s);
if (y < x) {
s = -s;
}
s = x - w / ((y - x) / 2.0 + s);
for (int i = low; i <= n; i++) {
H.setValue(i, i, H.getValue(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.getValue(m, m);
r = x - z;
s = y - z;
p = (r * s - w) / H.getValue(m+1, m) + H.getValue(m, m+1);
q = H.getValue(m+1, m+1) - z - r - s;
r = H.getValue(m+2, m+1);
s = Math.abs(p) + Math.abs(q) + Math.abs(r);
p = p / s;
q = q / s;
r = r / s;
if (m == l) {
break;
}
if (Math.abs(H.getValue(m, m-1)) * (Math.abs(q) + Math.abs(r)) <
eps * (Math.abs(p) * (Math.abs(H.getValue(m-1, m-1)) + Math.abs(z) +
Math.abs(H.getValue(m+1, m+1))))) {
break;
}
m--;
}
for (int i = m+2; i <= n; i++) {
H.setValue(i, i-2, 0.0);
if (i > m+2) {
H.setValue(i, i-3, 0.0);
}
}
// Double QR step involving rows l:n and columns m:n
for (int k = m; k <= n-1; k++) {
boolean notlast = (k != n-1);
if (k != m) {
p = H.getValue(k, k-1);
q = H.getValue(k+1, k-1);
r = (notlast ? H.getValue(k+2, k-1) : 0.0);
x = Math.abs(p) + Math.abs(q) + Math.abs(r);
if (x == 0.0) {
continue;
}
p = p / x;
q = q / x;
r = r / x;
}
s = Math.sqrt(p * p + q * q + r * r);
if (p < 0) {
s = -s;
}
if (s != 0) {
if (k != m) {
H.setValue(k, k-1, -s * x);
} else if (l != m) {
H.setValue(k, k-1, -H.getValue(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.getValue(k, j) + q * H.getValue(k+1, j);
if (notlast) {
p = p + r * H.getValue(k+2, j);
H.setValue(k+2, j, H.getValue(k+2, j) - p * z);
}
H.setValue(k, j, H.getValue(k, j) - p * x);
H.setValue(k+1,j, H.getValue(k+1, j) - p * y);
}
// Column modification
for (int i = 0; i <= Math.min(n,k+3); i++) {
p = x * H.getValue(i, k) + y * H.getValue(i, k+1);
if (notlast) {
p = p + z * H.getValue(i, k+2);
H.setValue(i, k+2, H.getValue(i, k+2) - p * r);
}
H.setValue(i, k, H.getValue(i, k) - p);
H.setValue(i, k+1, H.getValue(i, k+1) - p * q);
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
p = x * V.getValue(i, k) + y * V.getValue(i, k+1);
if (notlast) {
p = p + z * V.getValue(i, k+2);
V.setValue(i, k+2, V.getValue(i, k+2) - p * r);
}
V.setValue(i, k, V.getValue(i, k) - p);
V.setValue(i, k+1, V.getValue(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.setValue(n, n, 1.0);
for (int i = n-1; i >= 0; i--) {
w = H.getValue(i, i) - p;
r = 0.0;
for (int j = l; j <= n; j++) {
r = r + H.getValue(i, j) * H.getValue(j, n);
}
if (e[i] < 0.0) {
z = w;
s = r;
} else {
l = i;
if (e[i] == 0.0) {
if (w != 0.0) {
H.setValue(i, n, -r / w);
} else {
H.setValue(i, n, -r / (eps * norm));
}
// Solve real equations
} else {
x = H.getValue(i, i+1);
y = H.getValue(i+1, i);
q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
t = (x * s - z * r) / q;
H.setValue(i, n, t);
if (Math.abs(x) > Math.abs(z)) {
H.setValue(i+1, n, (-r - w * t) / x);
} else {
H.setValue(i+1, n, (-s - y * t) / z);
}
}
// Overflow control
t = Math.abs(H.getValue(i, n));
if ((eps * t) * t > 1) {
for (int j = i; j <= n; j++) {
H.setValue(j, n, H.getValue(j, n) / t);
}
}
}
}
// Complex vector
} else if (q < 0) {
int l = n-1;
// Last vector component imaginary so matrix is triangular
if (Math.abs(H.getValue(n, n-1)) > Math.abs(H.getValue(n-1, n))) {
H.setValue(n-1, n-1, q / H.getValue(n, n-1));
H.setValue(n-1, n, -(H.getValue(n, n) - p) / H.getValue(n, n-1));
} else {
cdiv(0.0,-H.getValue(n-1, n),H.getValue(n-1, n-1)-p,q);
H.setValue(n-1, n-1, cdivr);
H.setValue(n-1, n, cdivi);
}
H.setValue(n, n-1, 0.0);
H.setValue(n, n, 1.0);
for (int i = n-2; i >= 0; i--) {
double ra,sa,vr,vi;
ra = 0.0;
sa = 0.0;
for (int j = l; j <= n; j++) {
ra = ra + H.getValue(i, j) * H.getValue(j, n-1);
sa = sa + H.getValue(i, j)* H.getValue(j, n);
}
w = H.getValue(i, i) - p;
if (e[i] < 0.0) {
z = w;
r = ra;
s = sa;
} else {
l = i;
if (e[i] == 0) {
cdiv(-ra,-sa,w,q);
H.setValue(i, n-1, cdivr);
H.setValue(i, n, cdivi);
} else {
// Solve complex equations
x = H.getValue(i, i+1);
y = H.getValue(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 * (Math.abs(w) + Math.abs(q) +
Math.abs(x) + Math.abs(y) + Math.abs(z));
}
cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
H.setValue(i, n-1, cdivr);
H.setValue(i, n, cdivi);
if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {
H.setValue(i+1, n-1, (-ra - w * H.getValue(i, n-1) + q * H.getValue(i, n)) / x);
H.setValue(i+1, n, (-sa - w * H.getValue(i, n) - q * H.getValue(i, n-1)) / x);
} else {
cdiv(-r-y*H.getValue(i, n-1),-s-y*H.getValue(i, n),z,q);
H.setValue(i+1, n-1, cdivr);
H.setValue(i+1, n, cdivi);
}
}
// Overflow control
t = Math.max(Math.abs(H.getValue(i, n-1)),Math.abs(H.getValue(i, n)));
if ((eps * t) * t > 1) {
for (int j = i; j <= n; j++) {
H.setValue(j, n-1, H.getValue(j, n-1) / t);
H.setValue(j, n, H.getValue(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.setValue(i, j, H.getValue(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 <= Math.min(j,high); k++) {
z = z + V.getValue(i, k) * H.getValue(k, j);
}
V.setValue(i, j, z);
}
}
}
/** Return the real parts of the eigenvalues.
* @return the real parts of the eigenvalues
*/
public double[] getRealEigenvalues () {
return d;
}
/** Return the imaginary parts of the eigenvalues.
* @return the imaginary parts of the eigenvalues
*/
public double[] getImagEigenvalues () {
return e;
}
}
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