/*
* Open Source Physics software is free software as described near the bottom of this code file.
*
* For additional information and documentation on Open Source Physics please see:
* <http://www.opensourcephysics.org/>
*/
/*
* Code adapted from JAMA library to use double[][] arrays rather than JAMA Matrix class.
*/
package org.opensourcephysics.numerics;
/**
* Eigenvalues and eigenvectors of a real matrix.
* <P>
* If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is
* diagonal and the eigenvector matrix V is orthogonal.
* I.e. A = V.times(D.times(V.transpose())) and
* V.times(V.transpose()) equals the identity matrix.
* <P>
* If 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,
* lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The
* columns of V represent the eigenvectors in the sense that A*V = V*D,
* i.e. A.times(V) equals V.times(D). The matrix V may be badly
* conditioned, or even singular, so the validity of the equation
* A = V*D*inverse(V) depends upon V.cond().
*/
public class EigenvalueDecomposition implements java.io.Serializable {
/*
* ------------------------
* Class variables
* ------------------------
*/
/**
* 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.
* @serial internal storage of eigenvalues.
*/
private double[] d, e;
/**
* Array for internal storage of eigenvectors.
* @serial internal storage of eigenvectors.
*/
private double[][] V;
/**
* Array for internal storage of nonsymmetric Hessenberg form.
* @serial internal storage of nonsymmetric Hessenberg form.
*/
private double[][] H;
/**
* Working storage for nonsymmetric algorithm.
* @serial working storage for nonsymmetric algorithm.
*/
private double[] ort;
/*
* ------------------------
* Private Methods
* ------------------------
*/
// Symmetric Householder reduction to tridiagonal form.
private void 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.
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[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];
}
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[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];
}
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[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;
double 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++) {
double 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;
}
// 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[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(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[j][i];
V[j][i] = V[j][k];
V[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[i][m-1]);
}
if(scale!=0.0) {
// Compute Householder transformation.
double h = 0.0;
for(int i = high; i>=m; i--) {
ort[i] = H[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[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++) {
double 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++) {
double 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];
}
}
}
}
}
// Complex scalar division.
private transient double cdivr, cdivi;
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;
}
}
// 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[i][i];
e[i] = 0.0;
}
for(int j = Math.max(i-1, 0); j<nn; j++) {
norm = norm+Math.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 = Math.abs(H[l-1][l-1])+Math.abs(H[l][l]);
if(s==0.0) {
s = norm;
}
if(Math.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 = Math.sqrt(Math.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 = 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[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 = Math.abs(H[n][n-1])+Math.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 = Math.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 = 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[m][m-1])*(Math.abs(q)+Math.abs(r))<eps*(Math.abs(p)*(Math.abs(H[m-1][m-1])+Math.abs(z)+Math.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++) {
boolean 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 = Math.abs(p)+Math.abs(q)+Math.abs(r);
if(x!=0.0) {
p = p/x;
q = q/x;
r = r/x;
}
}
if(x==0.0) {
break;
}
s = Math.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<=Math.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(Math.abs(x)>Math.abs(z)) {
H[i+1][n] = (-r-w*t)/x;
} else {
H[i+1][n] = (-s-y*t)/z;
}
}
// Overflow control
t = Math.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(Math.abs(H[n][n-1])>Math.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--) {
double 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) {
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==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[i][n-1] = cdivr;
H[i][n] = cdivi;
if(Math.abs(x)>(Math.abs(z)+Math.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 = Math.max(Math.abs(H[i][n-1]), Math.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<=Math.min(j, high); k++) {
z = z+V[i][k]*H[k][j];
}
V[i][j] = z;
}
}
}
/*
* ------------------------
* Constructor
* ------------------------
*/
/**
* Constructs the EigenvalueDecomposition.
*
* Allocates arrays, checks for symmetry and then constructs the eigenvalue decomposition
* @param A square matrix
*/
public EigenvalueDecomposition(double[][] A) {
if((A==null)||(A.length!=A[0].length)) {
throw new IllegalArgumentException("Matrix must be not-null and square for eigenvalue decompositon."); //$NON-NLS-1$
}
n = A.length;
V = new double[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[i][j] = A[i][j];
}
}
// Tridiagonalize.
tred2();
// Diagonalize.
tql2();
} else {
H = new double[n][n];
ort = new double[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();
}
}
/**
* Check for symmetry, then recompute the eigenvalue decomposition
* @param A square matrix
*/
public void recompute(double[][] A) {
if((A==null)||(A.length!=n)||(A[0].length!=n)) {
throw new IllegalArgumentException("Matrix must dimension must match that of original matrix."); //$NON-NLS-1$
}
for(int i = 0; i<n; i++) {
d[i] = 0;
e[i] = 0;
}
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[i][j] = A[i][j];
}
}
// Tridiagonalize.
tred2();
// Diagonalize.
tql2();
} else {
if((H==null)||(ort==null)) {
H = new double[n][n];
ort = new double[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();
}
}
/*
* ------------------------
* Public Methods
* ------------------------
*/
/**
* Return the eigenvector matrix
* @return V
*/
public double[][] getEigenvectors() {
return V;
}
/**
* Return the real parts of the eigenvalues
* @return real(diag(D))
*/
public double[] getRealEigenvalues() {
return d;
}
/**
* Return the imaginary parts of the eigenvalues
* @return imag(diag(D))
*/
public double[] getImagEigenvalues() {
return e;
}
/**
* Return the block diagonal eigenvalue matrix
* @return D
*/
public double[][] getEigenvalueMatrix() {
double[][] D = new double[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];
}
}
return D;
}
/** sqrt(a^2 + b^2) without under/overflow. */
public static 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;
}
}
/*
* EigenvalueDecompostion class adapted from Jama library for use in the OSP project.
*
* Open Source Physics software is free software; you can redistribute
* it and/or modify it under the terms of the GNU General Public License (GPL) as
* published by the Free Software Foundation; either version 2 of the License,
* or(at your option) any later version.
* Code that uses any portion of the code in the org.opensourcephysics package
* or any subpackage (subdirectory) of this package must must also be be released
* under the GNU GPL license.
*
* This software is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston MA 02111-1307 USA
* or view the license online at http://www.gnu.org/copyleft/gpl.html
*
* Copyright (c) 2007 The Open Source Physics project
* http://www.opensourcephysics.org
*/