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blender/intern/smoke/intern/EIGENVALUE_HELPER.cpp

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/** \file smoke/intern/EIGENVALUE_HELPER.cpp
* \ingroup smoke
*/
#include "EIGENVALUE_HELPER.h"
void Eigentred2(sEigenvalue& eval) {
// 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.
int n=eval.n;
for (int j = 0; j < n; j++) {
eval.d[j] = eval.V[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(eval.d[k]);
}
if (scale == 0.0f) {
eval.e[i] = eval.d[i-1];
for (int j = 0; j < i; j++) {
eval.d[j] = eval.V[i-1][j];
eval.V[i][j] = 0.0;
eval.V[j][i] = 0.0;
}
} else {
// Generate Householder vector.
for (int k = 0; k < i; k++) {
eval.d[k] /= scale;
h += eval.d[k] * eval.d[k];
}
float f = eval.d[i-1];
float g = sqrt(h);
if (f > 0) {
g = -g;
}
eval.e[i] = scale * g;
h = h - f * g;
eval.d[i-1] = f - g;
for (int j = 0; j < i; j++) {
eval.e[j] = 0.0;
}
// Apply similarity transformation to remaining columns.
for (int j = 0; j < i; j++) {
f = eval.d[j];
eval.V[j][i] = f;
g = eval.e[j] + eval.V[j][j] * f;
for (int k = j+1; k <= i-1; k++) {
g += eval.V[k][j] * eval.d[k];
eval.e[k] += eval.V[k][j] * f;
}
eval.e[j] = g;
}
f = 0.0;
for (int j = 0; j < i; j++) {
eval.e[j] /= h;
f += eval.e[j] * eval.d[j];
}
float hh = f / (h + h);
for (int j = 0; j < i; j++) {
eval.e[j] -= hh * eval.d[j];
}
for (int j = 0; j < i; j++) {
f = eval.d[j];
g = eval.e[j];
for (int k = j; k <= i-1; k++) {
eval.V[k][j] -= (f * eval.e[k] + g * eval.d[k]);
}
eval.d[j] = eval.V[i-1][j];
eval.V[i][j] = 0.0;
}
}
eval.d[i] = h;
}
// Accumulate transformations.
for (int i = 0; i < n-1; i++) {
eval.V[n-1][i] = eval.V[i][i];
eval.V[i][i] = 1.0;
float h = eval.d[i+1];
if (h != 0.0f) {
for (int k = 0; k <= i; k++) {
eval.d[k] = eval.V[k][i+1] / h;
}
for (int j = 0; j <= i; j++) {
float g = 0.0;
for (int k = 0; k <= i; k++) {
g += eval.V[k][i+1] * eval.V[k][j];
}
for (int k = 0; k <= i; k++) {
eval.V[k][j] -= g * eval.d[k];
}
}
}
for (int k = 0; k <= i; k++) {
eval.V[k][i+1] = 0.0;
}
}
for (int j = 0; j < n; j++) {
eval.d[j] = eval.V[n-1][j];
eval.V[n-1][j] = 0.0;
}
eval.V[n-1][n-1] = 1.0;
eval.e[0] = 0.0;
}
void Eigencdiv(sEigenvalue& eval, float xr, float xi, float yr, float yi) {
float r,d;
if (fabs(yr) > fabs(yi)) {
r = yi/yr;
d = yr + r*yi;
eval.cdivr = (xr + r*xi)/d;
eval.cdivi = (xi - r*xr)/d;
} else {
r = yr/yi;
d = yi + r*yr;
eval.cdivr = (r*xr + xi)/d;
eval.cdivi = (r*xi - xr)/d;
}
}
void Eigentql2 (sEigenvalue& eval) {
// 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.
int n=eval.n;
for (int i = 1; i < n; i++) {
eval.e[i-1] = eval.e[i];
}
eval.e[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 = max(tst1,fabs(eval.d[l]) + fabs(eval.e[l]));
int m = l;
// Original while-loop from Java code
while (m < n) {
if (fabs(eval.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
float g = eval.d[l];
float p = (eval.d[l+1] - g) / (2.0f * eval.e[l]);
float r = hypot(p,1.0);
if (p < 0) {
r = -r;
}
eval.d[l] = eval.e[l] / (p + r);
eval.d[l+1] = eval.e[l] * (p + r);
float dl1 = eval.d[l+1];
float h = g - eval.d[l];
for (int i = l+2; i < n; i++) {
eval.d[i] -= h;
}
f = f + h;
// Implicit QL transformation.
p = eval.d[m];
float c = 1.0;
float c2 = c;
float c3 = c;
float el1 = eval.e[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 * eval.e[i];
h = c * p;
r = hypot(p,eval.e[i]);
eval.e[i+1] = s * r;
s = eval.e[i] / r;
c = p / r;
p = c * eval.d[i] - s * g;
eval.d[i+1] = h + s * (c * g + s * eval.d[i]);
// Accumulate transformation.
for (int k = 0; k < n; k++) {
h = eval.V[k][i+1];
eval.V[k][i+1] = s * eval.V[k][i] + c * h;
eval.V[k][i] = c * eval.V[k][i] - s * h;
}
}
p = -s * s2 * c3 * el1 * eval.e[l] / dl1;
eval.e[l] = s * p;
eval.d[l] = c * p;
// Check for convergence.
} while (fabs(eval.e[l]) > eps*tst1);
}
eval.d[l] = eval.d[l] + f;
eval.e[l] = 0.0;
}
// Sort eigenvalues and corresponding vectors.
for (int i = 0; i < n-1; i++) {
int k = i;
float p = eval.d[i];
for (int j = i+1; j < n; j++) {
if (eval.d[j] < p) {
k = j;
p = eval.d[j];
}
}
if (k != i) {
eval.d[k] = eval.d[i];
eval.d[i] = p;
for (int j = 0; j < n; j++) {
p = eval.V[j][i];
eval.V[j][i] = eval.V[j][k];
eval.V[j][k] = p;
}
}
}
}
void Eigenorthes (sEigenvalue& eval) {
// 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 n=eval.n;
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(eval.H[i][m-1]);
}
if (scale != 0.0f) {
// Compute Householder transformation.
float h = 0.0;
for (int i = high; i >= m; i--) {
eval.ort[i] = eval.H[i][m-1]/scale;
h += eval.ort[i] * eval.ort[i];
}
float g = sqrt(h);
if (eval.ort[m] > 0) {
g = -g;
}
h = h - eval.ort[m] * g;
eval.ort[m] = eval.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 += eval.ort[i]*eval.H[i][j];
}
f = f/h;
for (int i = m; i <= high; i++) {
eval.H[i][j] -= f*eval.ort[i];
}
}
for (int i = 0; i <= high; i++) {
float f = 0.0;
for (int j = high; j >= m; j--) {
f += eval.ort[j]*eval.H[i][j];
}
f = f/h;
for (int j = m; j <= high; j++) {
eval.H[i][j] -= f*eval.ort[j];
}
}
eval.ort[m] = scale*eval.ort[m];
eval.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++) {
eval.V[i][j] = (i == j ? 1.0 : 0.0);
}
}
for (int m = high-1; m >= low+1; m--) {
if (eval.H[m][m-1] != 0.0f) {
for (int i = m+1; i <= high; i++) {
eval.ort[i] = eval.H[i][m-1];
}
for (int j = m; j <= high; j++) {
float g = 0.0;
for (int i = m; i <= high; i++) {
g += eval.ort[i] * eval.V[i][j];
}
// Double division avoids possible underflow
g = (g / eval.ort[m]) / eval.H[m][m-1];
for (int i = m; i <= high; i++) {
eval.V[i][j] += g * eval.ort[i];
}
}
}
}
}
void Eigenhqr2 (sEigenvalue& eval) {
// 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 = eval.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)) {
eval.d[i] = eval.H[i][i];
eval.e[i] = 0.0;
}
for (int j = max(i-1,0); j < nn; j++) {
norm = norm + fabs(eval.H[i][j]);
}
}
// Outer loop over eigenvalue index
int iter = 0;
int totIter = 0;
while (n >= low) {
// NT limit no. of iterations
totIter++;
if(totIter>100) {
//if(totIter>15) std::cout<<"!!!!iter ABORT !!!!!!! "<<totIter<<"\n";
// NT hack/fix, return large eigenvalues
for (int i = 0; i < nn; i++) {
eval.d[i] = 10000.;
eval.e[i] = 10000.;
}
return;
}
// Look for single small sub-diagonal element
int l = n;
while (l > low) {
s = fabs(eval.H[l-1][l-1]) + fabs(eval.H[l][l]);
if (s == 0.0f) {
s = norm;
}
if (fabs(eval.H[l][l-1]) < eps * s) {
break;
}
l--;
}
// Check for convergence
// One root found
if (l == n) {
eval.H[n][n] = eval.H[n][n] + exshift;
eval.d[n] = eval.H[n][n];
eval.e[n] = 0.0;
n--;
iter = 0;
// Two roots found
} else if (l == n-1) {
w = eval.H[n][n-1] * eval.H[n-1][n];
p = (eval.H[n-1][n-1] - eval.H[n][n]) / 2.0f;
q = p * p + w;
z = sqrt(fabs(q));
eval.H[n][n] = eval.H[n][n] + exshift;
eval.H[n-1][n-1] = eval.H[n-1][n-1] + exshift;
x = eval.H[n][n];
// float pair
if (q >= 0) {
if (p >= 0) {
z = p + z;
} else {
z = p - z;
}
eval.d[n-1] = x + z;
eval.d[n] = eval.d[n-1];
if (z != 0.0f) {
eval.d[n] = x - w / z;
}
eval.e[n-1] = 0.0;
eval.e[n] = 0.0;
x = eval.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 = eval.H[n-1][j];
eval.H[n-1][j] = q * z + p * eval.H[n][j];
eval.H[n][j] = q * eval.H[n][j] - p * z;
}
// Column modification
for (int i = 0; i <= n; i++) {
z = eval.H[i][n-1];
eval.H[i][n-1] = q * z + p * eval.H[i][n];
eval.H[i][n] = q * eval.H[i][n] - p * z;
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
z = eval.V[i][n-1];
eval.V[i][n-1] = q * z + p * eval.V[i][n];
eval.V[i][n] = q * eval.V[i][n] - p * z;
}
// Complex pair
} else {
eval.d[n-1] = x + p;
eval.d[n] = x + p;
eval.e[n-1] = z;
eval.e[n] = -z;
}
n = n - 2;
iter = 0;
// No convergence yet
} else {
// Form shift
x = eval.H[n][n];
y = 0.0;
w = 0.0;
if (l < n) {
y = eval.H[n-1][n-1];
w = eval.H[n][n-1] * eval.H[n-1][n];
}
// Wilkinson's original ad hoc shift
if (iter == 10) {
exshift += x;
for (int i = low; i <= n; i++) {
eval.H[i][i] -= x;
}
s = fabs(eval.H[n][n-1]) + fabs(eval.H[n-1][n-2]);
x = y = 0.75f * s;
w = -0.4375f * s * s;
}
// MATLAB's new ad hoc shift
if (iter == 30) {
s = (y - x) / 2.0f;
s = s * s + w;
if (s > 0) {
s = sqrt(s);
if (y < x) {
s = -s;
}
s = x - w / ((y - x) / 2.0f + s);
for (int i = low; i <= n; i++) {
eval.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 = eval.H[m][m];
r = x - z;
s = y - z;
p = (r * s - w) / eval.H[m+1][m] + eval.H[m][m+1];
q = eval.H[m+1][m+1] - z - r - s;
r = eval.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(eval.H[m][m-1]) * (fabs(q) + fabs(r)) <
eps * (fabs(p) * (fabs(eval.H[m-1][m-1]) + fabs(z) +
fabs(eval.H[m+1][m+1])))) {
break;
}
m--;
}
for (int i = m+2; i <= n; i++) {
eval.H[i][i-2] = 0.0;
if (i > m+2) {
eval.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++) {
int notlast = (k != n-1);
if (k != m) {
p = eval.H[k][k-1];
q = eval.H[k+1][k-1];
r = (notlast ? eval.H[k+2][k-1] : 0.0f);
x = fabs(p) + fabs(q) + fabs(r);
if (x != 0.0f) {
p = p / x;
q = q / x;
r = r / x;
}
}
if (x == 0.0f) {
break;
}
s = sqrt(p * p + q * q + r * r);
if (p < 0) {
s = -s;
}
if (s != 0) {
if (k != m) {
eval.H[k][k-1] = -s * x;
} else if (l != m) {
eval.H[k][k-1] = -eval.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 = eval.H[k][j] + q * eval.H[k+1][j];
if (notlast) {
p = p + r * eval.H[k+2][j];
eval.H[k+2][j] = eval.H[k+2][j] - p * z;
}
eval.H[k][j] = eval.H[k][j] - p * x;
eval.H[k+1][j] = eval.H[k+1][j] - p * y;
}
// Column modification
for (int i = 0; i <= min(n,k+3); i++) {
p = x * eval.H[i][k] + y * eval.H[i][k+1];
if (notlast) {
p = p + z * eval.H[i][k+2];
eval.H[i][k+2] = eval.H[i][k+2] - p * r;
}
eval.H[i][k] = eval.H[i][k] - p;
eval.H[i][k+1] = eval.H[i][k+1] - p * q;
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
p = x * eval.V[i][k] + y * eval.V[i][k+1];
if (notlast) {
p = p + z * eval.V[i][k+2];
eval.V[i][k+2] = eval.V[i][k+2] - p * r;
}
eval.V[i][k] = eval.V[i][k] - p;
eval.V[i][k+1] = eval.V[i][k+1] - p * q;
}
} // (s != 0)
} // k loop
} // check convergence
} // while (n >= low)
//if(totIter>15) std::cout<<"!!!!iter "<<totIter<<"\n";
// Backsubstitute to find vectors of upper triangular form
if (norm == 0.0f) {
return;
}
for (n = nn-1; n >= 0; n--) {
p = eval.d[n];
q = eval.e[n];
// float vector
if (q == 0) {
int l = n;
eval.H[n][n] = 1.0;
for (int i = n-1; i >= 0; i--) {
w = eval.H[i][i] - p;
r = 0.0;
for (int j = l; j <= n; j++) {
r = r + eval.H[i][j] * eval.H[j][n];
}
if (eval.e[i] < 0.0f) {
z = w;
s = r;
} else {
l = i;
if (eval.e[i] == 0.0f) {
if (w != 0.0f) {
eval.H[i][n] = -r / w;
} else {
eval.H[i][n] = -r / (eps * norm);
}
// Solve real equations
} else {
x = eval.H[i][i+1];
y = eval.H[i+1][i];
q = (eval.d[i] - p) * (eval.d[i] - p) + eval.e[i] * eval.e[i];
t = (x * s - z * r) / q;
eval.H[i][n] = t;
if (fabs(x) > fabs(z)) {
eval.H[i+1][n] = (-r - w * t) / x;
} else {
eval.H[i+1][n] = (-s - y * t) / z;
}
}
// Overflow control
t = fabs(eval.H[i][n]);
if ((eps * t) * t > 1) {
for (int j = i; j <= n; j++) {
eval.H[j][n] = eval.H[j][n] / t;
}
}
}
}
// Complex vector
} else if (q < 0) {
int l = n-1;
// Last vector component imaginary so matrix is triangular
if (fabs(eval.H[n][n-1]) > fabs(eval.H[n-1][n])) {
eval.H[n-1][n-1] = q / eval.H[n][n-1];
eval.H[n-1][n] = -(eval.H[n][n] - p) / eval.H[n][n-1];
} else {
Eigencdiv(eval, 0.0,-eval.H[n-1][n],eval.H[n-1][n-1]-p,q);
eval.H[n-1][n-1] = eval.cdivr;
eval.H[n-1][n] = eval.cdivi;
}
eval.H[n][n-1] = 0.0;
eval.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 + eval.H[i][j] * eval.H[j][n-1];
sa = sa + eval.H[i][j] * eval.H[j][n];
}
w = eval.H[i][i] - p;
if (eval.e[i] < 0.0f) {
z = w;
r = ra;
s = sa;
} else {
l = i;
if (eval.e[i] == 0) {
Eigencdiv(eval,-ra,-sa,w,q);
eval.H[i][n-1] = eval.cdivr;
eval.H[i][n] = eval.cdivi;
} else {
// Solve complex equations
x = eval.H[i][i+1];
y = eval.H[i+1][i];
vr = (eval.d[i] - p) * (eval.d[i] - p) + eval.e[i] * eval.e[i] - q * q;
vi = (eval.d[i] - p) * 2.0f * q;
if ((vr == 0.0f) && (vi == 0.0f)) {
vr = eps * norm * (fabs(w) + fabs(q) +
fabs(x) + fabs(y) + fabs(z));
}
Eigencdiv(eval, x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
eval.H[i][n-1] = eval.cdivr;
eval.H[i][n] = eval.cdivi;
if (fabs(x) > (fabs(z) + fabs(q))) {
eval.H[i+1][n-1] = (-ra - w * eval.H[i][n-1] + q * eval.H[i][n]) / x;
eval.H[i+1][n] = (-sa - w * eval.H[i][n] - q * eval.H[i][n-1]) / x;
} else {
Eigencdiv(eval, -r-y*eval.H[i][n-1],-s-y*eval.H[i][n],z,q);
eval.H[i+1][n-1] = eval.cdivr;
eval.H[i+1][n] = eval.cdivi;
}
}
// Overflow control
t = max(fabs(eval.H[i][n-1]),fabs(eval.H[i][n]));
if ((eps * t) * t > 1) {
for (int j = i; j <= n; j++) {
eval.H[j][n-1] = eval.H[j][n-1] / t;
eval.H[j][n] = eval.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++) {
eval.V[i][j] = eval.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 <= min(j,high); k++) {
z = z + eval.V[i][k] * eval.H[k][j];
}
eval.V[i][j] = z;
}
}
}
int computeEigenvalues3x3(
float dout[3],
float a[3][3])
{
/*TNT::Array2D<float> A = TNT::Array2D<float>(3,3, &a[0][0]);
TNT::Array1D<float> eig = TNT::Array1D<float>(3);
TNT::Array1D<float> eigImag = TNT::Array1D<float>(3);
JAMA::Eigenvalue<float> jeig = JAMA::Eigenvalue<float>(A);*/
sEigenvalue jeig;
// Compute the values
{
jeig.n = 3;
int n=3;
//V = Array2D<float>(n,n);
//d = Array1D<float>(n);
//e = Array1D<float>(n);
for (int y=0; y<3; y++)
{
jeig.d[y]=0.0f;
jeig.e[y]=0.0f;
for (int t=0; t<3; t++) jeig.V[y][t]=0.0f;
}
jeig.issymmetric = 1;
for (int j = 0; (j < 3) && jeig.issymmetric; j++) {
for (int i = 0; (i < 3) && jeig.issymmetric; i++) {
jeig.issymmetric = (a[i][j] == a[j][i]);
}
}
if (jeig.issymmetric) {
for (int i = 0; i < 3; i++) {
for (int j = 0; j < 3; j++) {
jeig.V[i][j] = a[i][j];
}
}
// Tridiagonalize.
Eigentred2(jeig);
// Diagonalize.
Eigentql2(jeig);
} else {
//H = TNT::Array2D<float>(n,n);
for (int y=0; y<3; y++)
{
jeig.ort[y]=0.0f;
for (int t=0; t<3; t++) jeig.H[y][t]=0.0f;
}
//ort = TNT::Array1D<float>(n);
for (int j = 0; j < n; j++) {
for (int i = 0; i < n; i++) {
jeig.H[i][j] = a[i][j];
}
}
// Reduce to Hessenberg form.
Eigenorthes(jeig);
// Reduce Hessenberg to real Schur form.
Eigenhqr2(jeig);
}
}
//jeig.getfloatEigenvalues(eig);
// complex ones
//jeig.getImagEigenvalues(eigImag);
dout[0] = sqrt(jeig.d[0]*jeig.d[0] + jeig.e[0]*jeig.e[0]);
dout[1] = sqrt(jeig.d[1]*jeig.d[1] + jeig.e[1]*jeig.e[1]);
dout[2] = sqrt(jeig.d[2]*jeig.d[2] + jeig.e[2]*jeig.e[2]);
return 0;
}
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