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eig3.h
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eig3.h
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/* Eigen-decomposition for symmetric 3x3 real matrices.
Public domain, copied from the public domain Java library JAMA. */
#ifndef _eig_h
#ifdef MAX
#undef MAX
#endif
#define MAX(a, b) ((a)>(b)?(a):(b))
#define n 3
__device__
static double hypot2(double x, double y) {
return sqrt(x*x+y*y);
}
// Symmetric Householder reduction to tridiagonal form.
__device__
static void tred2(double V[n][n], double d[n], double e[n]) {
// 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 + fabs(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 = 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.
__device__
static void tql2(double V[n][n], double d[n], double e[n]) {
// 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 = pow(2.0,-52.0);
for (int l = 0; l < n; l++) {
// Find small subdiagonal element
tst1 = MAX(tst1,fabs(d[l]) + fabs(e[l]));
int m = l;
while (m < n) {
if (fabs(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 = hypot2(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 = hypot2(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 (fabs(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;
}
}
}
}
/* Symmetric matrix A => eigenvectors in columns of V, corresponding
eigenvalues in d. */
__device__
void eigen_decomposition(double A[n][n], double V[n][n], double d[n]) {
double e[n];
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
V[i][j] = A[i][j];
}
}
tred2(V, d, e);
tql2(V, d, e);
}
#endif