| 1 |
#include <stdio.h> |
| 2 |
#include <math.h> |
| 3 |
#include <stdlib.h> |
| 4 |
#include "MatVec3.h" |
| 5 |
|
| 6 |
/* |
| 7 |
* Contains various utilities for dealing with 3x3 matrices and |
| 8 |
* length 3 vectors |
| 9 |
*/ |
| 10 |
|
| 11 |
void identityMat3(double A[3][3]) { |
| 12 |
int i; |
| 13 |
for (i = 0; i < 3; i++) { |
| 14 |
A[i][0] = A[i][1] = A[i][2] = 0.0; |
| 15 |
A[i][i] = 1.0; |
| 16 |
} |
| 17 |
} |
| 18 |
|
| 19 |
void swapVectors3(double v1[3], double v2[3]) { |
| 20 |
int i; |
| 21 |
for (i = 0; i < 3; i++) { |
| 22 |
double tmp = v1[i]; |
| 23 |
v1[i] = v2[i]; |
| 24 |
v2[i] = tmp; |
| 25 |
} |
| 26 |
} |
| 27 |
|
| 28 |
double normalize3(double x[3]) { |
| 29 |
double den; |
| 30 |
int i; |
| 31 |
if ( (den = norm3(x)) != 0.0 ) { |
| 32 |
for (i=0; i < 3; i++) |
| 33 |
{ |
| 34 |
x[i] /= den; |
| 35 |
} |
| 36 |
} |
| 37 |
return den; |
| 38 |
} |
| 39 |
|
| 40 |
void matMul3(double a[3][3], double b[3][3], double c[3][3]) { |
| 41 |
double r00, r01, r02, r10, r11, r12, r20, r21, r22; |
| 42 |
|
| 43 |
r00 = a[0][0]*b[0][0] + a[0][1]*b[1][0] + a[0][2]*b[2][0]; |
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r01 = a[0][0]*b[0][1] + a[0][1]*b[1][1] + a[0][2]*b[2][1]; |
| 45 |
r02 = a[0][0]*b[0][2] + a[0][1]*b[1][2] + a[0][2]*b[2][2]; |
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|
| 47 |
r10 = a[1][0]*b[0][0] + a[1][1]*b[1][0] + a[1][2]*b[2][0]; |
| 48 |
r11 = a[1][0]*b[0][1] + a[1][1]*b[1][1] + a[1][2]*b[2][1]; |
| 49 |
r12 = a[1][0]*b[0][2] + a[1][1]*b[1][2] + a[1][2]*b[2][2]; |
| 50 |
|
| 51 |
r20 = a[2][0]*b[0][0] + a[2][1]*b[1][0] + a[2][2]*b[2][0]; |
| 52 |
r21 = a[2][0]*b[0][1] + a[2][1]*b[1][1] + a[2][2]*b[2][1]; |
| 53 |
r22 = a[2][0]*b[0][2] + a[2][1]*b[1][2] + a[2][2]*b[2][2]; |
| 54 |
|
| 55 |
c[0][0] = r00; c[0][1] = r01; c[0][2] = r02; |
| 56 |
c[1][0] = r10; c[1][1] = r11; c[1][2] = r12; |
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c[2][0] = r20; c[2][1] = r21; c[2][2] = r22; |
| 58 |
} |
| 59 |
|
| 60 |
void matVecMul3(double m[3][3], double inVec[3], double outVec[3]) { |
| 61 |
double a0, a1, a2; |
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|
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a0 = inVec[0]; a1 = inVec[1]; a2 = inVec[2]; |
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|
| 65 |
outVec[0] = m[0][0]*a0 + m[0][1]*a1 + m[0][2]*a2; |
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outVec[1] = m[1][0]*a0 + m[1][1]*a1 + m[1][2]*a2; |
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outVec[2] = m[2][0]*a0 + m[2][1]*a1 + m[2][2]*a2; |
| 68 |
} |
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|
| 70 |
double matDet3(double a[3][3]) { |
| 71 |
int i, j, k; |
| 72 |
double determinant; |
| 73 |
|
| 74 |
determinant = 0.0; |
| 75 |
|
| 76 |
for(i = 0; i < 3; i++) { |
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j = (i+1)%3; |
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k = (i+2)%3; |
| 79 |
|
| 80 |
determinant += a[0][i] * (a[1][j]*a[2][k] - a[1][k]*a[2][j]); |
| 81 |
} |
| 82 |
|
| 83 |
return determinant; |
| 84 |
} |
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|
| 86 |
void invertMat3(double a[3][3], double b[3][3]) { |
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|
| 88 |
int i, j, k, l, m, n; |
| 89 |
double determinant; |
| 90 |
|
| 91 |
determinant = matDet3( a ); |
| 92 |
|
| 93 |
if (determinant == 0.0) { |
| 94 |
printf( |
| 95 |
"Can't invert a matrix with a zero determinant!\n"); |
| 96 |
exit(-1); |
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} |
| 98 |
|
| 99 |
for (i=0; i < 3; i++) { |
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j = (i+1)%3; |
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k = (i+2)%3; |
| 102 |
for(l = 0; l < 3; l++) { |
| 103 |
m = (l+1)%3; |
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n = (l+2)%3; |
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|
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b[l][i] = (a[j][m]*a[k][n] - a[j][n]*a[k][m]) / determinant; |
| 107 |
} |
| 108 |
} |
| 109 |
} |
| 110 |
|
| 111 |
void transposeMat3(double in[3][3], double out[3][3]) { |
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double temp[3][3]; |
| 113 |
int i, j; |
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|
| 115 |
for (i = 0; i < 3; i++) { |
| 116 |
for (j = 0; j < 3; j++) { |
| 117 |
temp[j][i] = in[i][j]; |
| 118 |
} |
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} |
| 120 |
for (i = 0; i < 3; i++) { |
| 121 |
for (j = 0; j < 3; j++) { |
| 122 |
out[i][j] = temp[i][j]; |
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} |
| 124 |
} |
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} |
| 126 |
|
| 127 |
void printMat3(double A[3][3] ){ |
| 128 |
|
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fprintf(stderr, "[ %g, %g, %g ]\n[ %g, %g, %g ]\n[ %g, %g, %g ]\n", |
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A[0][0] , A[0][1] , A[0][2], |
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A[1][0] , A[1][1] , A[1][2], |
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A[2][0] , A[2][1] , A[2][2]) ; |
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} |
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|
| 135 |
void printMat9(double A[9] ){ |
| 136 |
|
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fprintf(stderr, "[ %g, %g, %g ]\n[ %g, %g, %g ]\n[ %g, %g, %g ]\n", |
| 138 |
A[0], A[1], A[2], |
| 139 |
A[3], A[4], A[5], |
| 140 |
A[6], A[7], A[8]); |
| 141 |
} |
| 142 |
|
| 143 |
double matTrace3(double m[3][3]){ |
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double trace; |
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trace = m[0][0] + m[1][1] + m[2][2]; |
| 146 |
|
| 147 |
return trace; |
| 148 |
} |
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|
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void crossProduct3(double a[3],double b[3], double out[3]){ |
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|
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out[0] = a[1] * b[2] - a[2] * b[1]; |
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out[1] = a[2] * b[0] - a[0] * b[2] ; |
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out[2] = a[0] * b[1] - a[1] * b[0]; |
| 155 |
|
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} |
| 157 |
|
| 158 |
double dotProduct3(double a[3], double b[3]){ |
| 159 |
return a[0]*b[0] + a[1]*b[1]+ a[2]*b[2]; |
| 160 |
} |
| 161 |
|
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//---------------------------------------------------------------------------- |
| 163 |
// Extract the eigenvalues and eigenvectors from a 3x3 matrix. |
| 164 |
// The eigenvectors (the columns of V) will be normalized. |
| 165 |
// The eigenvectors are aligned optimally with the x, y, and z |
| 166 |
// axes respectively. |
| 167 |
|
| 168 |
void diagonalize3x3(const double A[3][3], double w[3], double V[3][3]) { |
| 169 |
int i,j,k,maxI; |
| 170 |
double tmp, maxVal; |
| 171 |
|
| 172 |
// do the matrix[3][3] to **matrix conversion for Jacobi |
| 173 |
double C[3][3]; |
| 174 |
double *ATemp[3],*VTemp[3]; |
| 175 |
for (i = 0; i < 3; i++) |
| 176 |
{ |
| 177 |
C[i][0] = A[i][0]; |
| 178 |
C[i][1] = A[i][1]; |
| 179 |
C[i][2] = A[i][2]; |
| 180 |
ATemp[i] = C[i]; |
| 181 |
VTemp[i] = V[i]; |
| 182 |
} |
| 183 |
|
| 184 |
// diagonalize using Jacobi |
| 185 |
JacobiN(ATemp,3,w,VTemp); |
| 186 |
|
| 187 |
// if all the eigenvalues are the same, return identity matrix |
| 188 |
if (w[0] == w[1] && w[0] == w[2]) |
| 189 |
{ |
| 190 |
identityMat3(V); |
| 191 |
return; |
| 192 |
} |
| 193 |
|
| 194 |
// transpose temporarily, it makes it easier to sort the eigenvectors |
| 195 |
transposeMat3(V,V); |
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|
| 197 |
// if two eigenvalues are the same, re-orthogonalize to optimally line |
| 198 |
// up the eigenvectors with the x, y, and z axes |
| 199 |
for (i = 0; i < 3; i++) |
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{ |
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if (w[(i+1)%3] == w[(i+2)%3]) // two eigenvalues are the same |
| 202 |
{ |
| 203 |
// find maximum element of the independant eigenvector |
| 204 |
maxVal = fabs(V[i][0]); |
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maxI = 0; |
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for (j = 1; j < 3; j++) |
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{ |
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if (maxVal < (tmp = fabs(V[i][j]))) |
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{ |
| 210 |
maxVal = tmp; |
| 211 |
maxI = j; |
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} |
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} |
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// swap the eigenvector into its proper position |
| 215 |
if (maxI != i) |
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{ |
| 217 |
tmp = w[maxI]; |
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w[maxI] = w[i]; |
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w[i] = tmp; |
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swapVectors3(V[i],V[maxI]); |
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} |
| 222 |
// maximum element of eigenvector should be positive |
| 223 |
if (V[maxI][maxI] < 0) |
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{ |
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V[maxI][0] = -V[maxI][0]; |
| 226 |
V[maxI][1] = -V[maxI][1]; |
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V[maxI][2] = -V[maxI][2]; |
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} |
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|
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// re-orthogonalize the other two eigenvectors |
| 231 |
j = (maxI+1)%3; |
| 232 |
k = (maxI+2)%3; |
| 233 |
|
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V[j][0] = 0.0; |
| 235 |
V[j][1] = 0.0; |
| 236 |
V[j][2] = 0.0; |
| 237 |
V[j][j] = 1.0; |
| 238 |
crossProduct3(V[maxI],V[j],V[k]); |
| 239 |
normalize3(V[k]); |
| 240 |
crossProduct3(V[k],V[maxI],V[j]); |
| 241 |
|
| 242 |
// transpose vectors back to columns |
| 243 |
transposeMat3(V,V); |
| 244 |
return; |
| 245 |
} |
| 246 |
} |
| 247 |
|
| 248 |
// the three eigenvalues are different, just sort the eigenvectors |
| 249 |
// to align them with the x, y, and z axes |
| 250 |
|
| 251 |
// find the vector with the largest x element, make that vector |
| 252 |
// the first vector |
| 253 |
maxVal = fabs(V[0][0]); |
| 254 |
maxI = 0; |
| 255 |
for (i = 1; i < 3; i++) |
| 256 |
{ |
| 257 |
if (maxVal < (tmp = fabs(V[i][0]))) |
| 258 |
{ |
| 259 |
maxVal = tmp; |
| 260 |
maxI = i; |
| 261 |
} |
| 262 |
} |
| 263 |
// swap eigenvalue and eigenvector |
| 264 |
if (maxI != 0) |
| 265 |
{ |
| 266 |
tmp = w[maxI]; |
| 267 |
w[maxI] = w[0]; |
| 268 |
w[0] = tmp; |
| 269 |
swapVectors3(V[maxI],V[0]); |
| 270 |
} |
| 271 |
// do the same for the y element |
| 272 |
if (fabs(V[1][1]) < fabs(V[2][1])) |
| 273 |
{ |
| 274 |
tmp = w[2]; |
| 275 |
w[2] = w[1]; |
| 276 |
w[1] = tmp; |
| 277 |
swapVectors3(V[2],V[1]); |
| 278 |
} |
| 279 |
|
| 280 |
// ensure that the sign of the eigenvectors is correct |
| 281 |
for (i = 0; i < 2; i++) |
| 282 |
{ |
| 283 |
if (V[i][i] < 0) |
| 284 |
{ |
| 285 |
V[i][0] = -V[i][0]; |
| 286 |
V[i][1] = -V[i][1]; |
| 287 |
V[i][2] = -V[i][2]; |
| 288 |
} |
| 289 |
} |
| 290 |
// set sign of final eigenvector to ensure that determinant is positive |
| 291 |
if (matDet3(V) < 0) |
| 292 |
{ |
| 293 |
V[2][0] = -V[2][0]; |
| 294 |
V[2][1] = -V[2][1]; |
| 295 |
V[2][2] = -V[2][2]; |
| 296 |
} |
| 297 |
|
| 298 |
// transpose the eigenvectors back again |
| 299 |
transposeMat3(V,V); |
| 300 |
} |
| 301 |
|
| 302 |
|
| 303 |
#define MAT_ROTATE(a,i,j,k,l) g=a[i][j];h=a[k][l];a[i][j]=g-s*(h+g*tau); a[k][l]=h+s*(g-h*tau); |
| 304 |
|
| 305 |
#define MAX_ROTATIONS 20 |
| 306 |
|
| 307 |
// Jacobi iteration for the solution of eigenvectors/eigenvalues of a nxn |
| 308 |
// real symmetric matrix. Square nxn matrix a; size of matrix in n; |
| 309 |
// output eigenvalues in w; and output eigenvectors in v. Resulting |
| 310 |
// eigenvalues/vectors are sorted in decreasing order; eigenvectors are |
| 311 |
// normalized. |
| 312 |
int JacobiN(double **a, int n, double *w, double **v) { |
| 313 |
|
| 314 |
int i, j, k, iq, ip, numPos; |
| 315 |
double tresh, theta, tau, t, sm, s, h, g, c, tmp; |
| 316 |
double bspace[4], zspace[4]; |
| 317 |
double *b = bspace; |
| 318 |
double *z = zspace; |
| 319 |
|
| 320 |
// only allocate memory if the matrix is large |
| 321 |
if (n > 4) |
| 322 |
{ |
| 323 |
b = (double *) calloc(n, sizeof(double)); |
| 324 |
z = (double *) calloc(n, sizeof(double)); |
| 325 |
} |
| 326 |
|
| 327 |
// initialize |
| 328 |
for (ip=0; ip<n; ip++) |
| 329 |
{ |
| 330 |
for (iq=0; iq<n; iq++) |
| 331 |
{ |
| 332 |
v[ip][iq] = 0.0; |
| 333 |
} |
| 334 |
v[ip][ip] = 1.0; |
| 335 |
} |
| 336 |
for (ip=0; ip<n; ip++) |
| 337 |
{ |
| 338 |
b[ip] = w[ip] = a[ip][ip]; |
| 339 |
z[ip] = 0.0; |
| 340 |
} |
| 341 |
|
| 342 |
// begin rotation sequence |
| 343 |
for (i=0; i<MAX_ROTATIONS; i++) |
| 344 |
{ |
| 345 |
sm = 0.0; |
| 346 |
for (ip=0; ip<n-1; ip++) |
| 347 |
{ |
| 348 |
for (iq=ip+1; iq<n; iq++) |
| 349 |
{ |
| 350 |
sm += fabs(a[ip][iq]); |
| 351 |
} |
| 352 |
} |
| 353 |
if (sm == 0.0) |
| 354 |
{ |
| 355 |
break; |
| 356 |
} |
| 357 |
|
| 358 |
if (i < 3) // first 3 sweeps |
| 359 |
{ |
| 360 |
tresh = 0.2*sm/(n*n); |
| 361 |
} |
| 362 |
else |
| 363 |
{ |
| 364 |
tresh = 0.0; |
| 365 |
} |
| 366 |
|
| 367 |
for (ip=0; ip<n-1; ip++) |
| 368 |
{ |
| 369 |
for (iq=ip+1; iq<n; iq++) |
| 370 |
{ |
| 371 |
g = 100.0*fabs(a[ip][iq]); |
| 372 |
|
| 373 |
// after 4 sweeps |
| 374 |
if (i > 3 && (fabs(w[ip])+g) == fabs(w[ip]) |
| 375 |
&& (fabs(w[iq])+g) == fabs(w[iq])) |
| 376 |
{ |
| 377 |
a[ip][iq] = 0.0; |
| 378 |
} |
| 379 |
else if (fabs(a[ip][iq]) > tresh) |
| 380 |
{ |
| 381 |
h = w[iq] - w[ip]; |
| 382 |
if ( (fabs(h)+g) == fabs(h)) |
| 383 |
{ |
| 384 |
t = (a[ip][iq]) / h; |
| 385 |
} |
| 386 |
else |
| 387 |
{ |
| 388 |
theta = 0.5*h / (a[ip][iq]); |
| 389 |
t = 1.0 / (fabs(theta)+sqrt(1.0+theta*theta)); |
| 390 |
if (theta < 0.0) |
| 391 |
{ |
| 392 |
t = -t; |
| 393 |
} |
| 394 |
} |
| 395 |
c = 1.0 / sqrt(1+t*t); |
| 396 |
s = t*c; |
| 397 |
tau = s/(1.0+c); |
| 398 |
h = t*a[ip][iq]; |
| 399 |
z[ip] -= h; |
| 400 |
z[iq] += h; |
| 401 |
w[ip] -= h; |
| 402 |
w[iq] += h; |
| 403 |
a[ip][iq]=0.0; |
| 404 |
|
| 405 |
// ip already shifted left by 1 unit |
| 406 |
for (j = 0;j <= ip-1;j++) |
| 407 |
{ |
| 408 |
MAT_ROTATE(a,j,ip,j,iq) |
| 409 |
} |
| 410 |
// ip and iq already shifted left by 1 unit |
| 411 |
for (j = ip+1;j <= iq-1;j++) |
| 412 |
{ |
| 413 |
MAT_ROTATE(a,ip,j,j,iq) |
| 414 |
} |
| 415 |
// iq already shifted left by 1 unit |
| 416 |
for (j=iq+1; j<n; j++) |
| 417 |
{ |
| 418 |
MAT_ROTATE(a,ip,j,iq,j) |
| 419 |
} |
| 420 |
for (j=0; j<n; j++) |
| 421 |
{ |
| 422 |
MAT_ROTATE(v,j,ip,j,iq) |
| 423 |
} |
| 424 |
} |
| 425 |
} |
| 426 |
} |
| 427 |
|
| 428 |
for (ip=0; ip<n; ip++) |
| 429 |
{ |
| 430 |
b[ip] += z[ip]; |
| 431 |
w[ip] = b[ip]; |
| 432 |
z[ip] = 0.0; |
| 433 |
} |
| 434 |
} |
| 435 |
|
| 436 |
//// this is NEVER called |
| 437 |
if ( i >= MAX_ROTATIONS ) |
| 438 |
{ |
| 439 |
printf( "Jacobi: Error extracting eigenfunctions!\n"); |
| 440 |
exit(-1); |
| 441 |
} |
| 442 |
|
| 443 |
// sort eigenfunctions these changes do not affect accuracy |
| 444 |
for (j=0; j<n-1; j++) // boundary incorrect |
| 445 |
{ |
| 446 |
k = j; |
| 447 |
tmp = w[k]; |
| 448 |
for (i=j+1; i<n; i++) // boundary incorrect, shifted already |
| 449 |
{ |
| 450 |
if (w[i] >= tmp) // why exchage if same? |
| 451 |
{ |
| 452 |
k = i; |
| 453 |
tmp = w[k]; |
| 454 |
} |
| 455 |
} |
| 456 |
if (k != j) |
| 457 |
{ |
| 458 |
w[k] = w[j]; |
| 459 |
w[j] = tmp; |
| 460 |
for (i=0; i<n; i++) |
| 461 |
{ |
| 462 |
tmp = v[i][j]; |
| 463 |
v[i][j] = v[i][k]; |
| 464 |
v[i][k] = tmp; |
| 465 |
} |
| 466 |
} |
| 467 |
} |
| 468 |
// insure eigenvector consistency (i.e., Jacobi can compute vectors that |
| 469 |
// are negative of one another (.707,.707,0) and (-.707,-.707,0). This can |
| 470 |
// reek havoc in hyperstreamline/other stuff. We will select the most |
| 471 |
// positive eigenvector. |
| 472 |
int ceil_half_n = (n >> 1) + (n & 1); |
| 473 |
for (j=0; j<n; j++) |
| 474 |
{ |
| 475 |
for (numPos=0, i=0; i<n; i++) |
| 476 |
{ |
| 477 |
if ( v[i][j] >= 0.0 ) |
| 478 |
{ |
| 479 |
numPos++; |
| 480 |
} |
| 481 |
} |
| 482 |
// if ( numPos < ceil(double(n)/double(2.0)) ) |
| 483 |
if ( numPos < ceil_half_n) |
| 484 |
{ |
| 485 |
for(i=0; i<n; i++) |
| 486 |
{ |
| 487 |
v[i][j] *= -1.0; |
| 488 |
} |
| 489 |
} |
| 490 |
} |
| 491 |
|
| 492 |
if (n > 4) |
| 493 |
{ |
| 494 |
free(b); |
| 495 |
free(z); |
| 496 |
} |
| 497 |
return 1; |
| 498 |
} |
| 499 |
|
| 500 |
#undef MAT_ROTATE |
| 501 |
#undef MAX_ROTATIONS |
