| 29 |
|
* @date 10/11/2004 |
| 30 |
|
* @version 1.0 |
| 31 |
|
*/ |
| 32 |
< |
#ifndef MATH_SQUAREMATRIX_HPP |
| 32 |
> |
#ifndef MATH_SQUAREMATRIX_HPP |
| 33 |
|
#define MATH_SQUAREMATRIX_HPP |
| 34 |
|
|
| 35 |
|
#include "math/RectMatrix.hpp" |
| 80 |
|
|
| 81 |
|
/** |
| 82 |
|
* Retunrs the inversion of this matrix. |
| 83 |
< |
* @todo |
| 83 |
> |
* @todo need implementation |
| 84 |
|
*/ |
| 85 |
|
SquareMatrix<Real, Dim> inverse() { |
| 86 |
|
SquareMatrix<Real, Dim> result; |
| 90 |
|
|
| 91 |
|
/** |
| 92 |
|
* Returns the determinant of this matrix. |
| 93 |
< |
* @todo |
| 93 |
> |
* @todo need implementation |
| 94 |
|
*/ |
| 95 |
< |
double determinant() const { |
| 96 |
< |
double det; |
| 95 |
> |
Real determinant() const { |
| 96 |
> |
Real det; |
| 97 |
|
return det; |
| 98 |
|
} |
| 99 |
|
|
| 100 |
|
/** Returns the trace of this matrix. */ |
| 101 |
< |
double trace() const { |
| 102 |
< |
double tmp = 0; |
| 101 |
> |
Real trace() const { |
| 102 |
> |
Real tmp = 0; |
| 103 |
|
|
| 104 |
|
for (unsigned int i = 0; i < Dim ; i++) |
| 105 |
|
tmp += data_[i][i]; |
| 148 |
|
return true; |
| 149 |
|
} |
| 150 |
|
|
| 151 |
< |
/** @todo need implement */ |
| 151 |
> |
/** @todo need implementation */ |
| 152 |
|
void diagonalize() { |
| 153 |
|
//jacobi(m, eigenValues, ortMat); |
| 154 |
|
} |
| 155 |
|
|
| 156 |
|
/** |
| 157 |
– |
* Finds the eigenvalues and eigenvectors of a symmetric matrix |
| 158 |
– |
* @param eigenvals a reference to a vector3 where the |
| 159 |
– |
* eigenvalues will be stored. The eigenvalues are ordered so |
| 160 |
– |
* that eigenvals[0] <= eigenvals[1] <= eigenvals[2]. |
| 161 |
– |
* @return an orthogonal matrix whose ith column is an |
| 162 |
– |
* eigenvector for the eigenvalue eigenvals[i] |
| 163 |
– |
*/ |
| 164 |
– |
SquareMatrix<Real, Dim> findEigenvectors(Vector<Real, Dim>& eigenValues) { |
| 165 |
– |
SquareMatrix<Real, Dim> ortMat; |
| 166 |
– |
|
| 167 |
– |
if ( !isSymmetric()){ |
| 168 |
– |
throw(); |
| 169 |
– |
} |
| 170 |
– |
|
| 171 |
– |
SquareMatrix<Real, Dim> m(*this); |
| 172 |
– |
jacobi(m, eigenValues, ortMat); |
| 173 |
– |
|
| 174 |
– |
return ortMat; |
| 175 |
– |
} |
| 176 |
– |
/** |
| 157 |
|
* Jacobi iteration routines for computing eigenvalues/eigenvectors of |
| 158 |
|
* real symmetric matrix |
| 159 |
|
* |
| 160 |
|
* @return true if success, otherwise return false |
| 161 |
< |
* @param a source matrix |
| 162 |
< |
* @param w output eigenvalues |
| 163 |
< |
* @param v output eigenvectors |
| 161 |
> |
* @param a symmetric matrix whose eigenvectors are to be computed. On return, the matrix is |
| 162 |
> |
* overwritten |
| 163 |
> |
* @param w will contain the eigenvalues of the matrix On return of this function |
| 164 |
> |
* @param v the columns of this matrix will contain the eigenvectors. The eigenvectors are |
| 165 |
> |
* normalized and mutually orthogonal. |
| 166 |
|
*/ |
| 167 |
< |
bool jacobi(const SquareMatrix<Real, Dim>& a, Vector<Real, Dim>& w, |
| 167 |
> |
|
| 168 |
> |
static int jacobi(SquareMatrix<Real, Dim>& a, Vector<Real, Dim>& d, |
| 169 |
|
SquareMatrix<Real, Dim>& v); |
| 170 |
|
};//end SquareMatrix |
| 171 |
|
|
| 172 |
|
|
| 173 |
< |
#define ROT(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) |
| 191 |
< |
#define MAX_ROTATIONS 60 |
| 173 |
> |
/*========================================================================= |
| 174 |
|
|
| 175 |
< |
template<typename Real, int Dim> |
| 176 |
< |
bool SquareMatrix<Real, Dim>::jacobi(const SquareMatrix<Real, Dim>& a, Vector<Real, Dim>& w, |
| 195 |
< |
SquareMatrix<Real, Dim>& v) { |
| 196 |
< |
const int N = Dim; |
| 197 |
< |
int i, j, k, iq, ip; |
| 198 |
< |
double tresh, theta, tau, t, sm, s, h, g, c; |
| 199 |
< |
double tmp; |
| 200 |
< |
Vector<Real, Dim> b, z; |
| 175 |
> |
Program: Visualization Toolkit |
| 176 |
> |
Module: $RCSfile: SquareMatrix.hpp,v $ |
| 177 |
|
|
| 178 |
< |
// initialize |
| 179 |
< |
for (ip=0; ip<N; ip++) { |
| 180 |
< |
for (iq=0; iq<N; iq++) |
| 205 |
< |
v(ip, iq) = 0.0; |
| 206 |
< |
v(ip, ip) = 1.0; |
| 207 |
< |
} |
| 208 |
< |
|
| 209 |
< |
for (ip=0; ip<N; ip++) { |
| 210 |
< |
b(ip) = w(ip) = a(ip, ip); |
| 211 |
< |
z(ip) = 0.0; |
| 212 |
< |
} |
| 178 |
> |
Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen |
| 179 |
> |
All rights reserved. |
| 180 |
> |
See Copyright.txt or http://www.kitware.com/Copyright.htm for details. |
| 181 |
|
|
| 182 |
< |
// begin rotation sequence |
| 183 |
< |
for (i=0; i<MAX_ROTATIONS; i++) { |
| 184 |
< |
sm = 0.0; |
| 217 |
< |
for (ip=0; ip<2; ip++) { |
| 218 |
< |
for (iq=ip+1; iq<N; iq++) |
| 219 |
< |
sm += fabs(a(ip, iq)); |
| 220 |
< |
} |
| 221 |
< |
|
| 222 |
< |
if (sm == 0.0) |
| 223 |
< |
break; |
| 182 |
> |
This software is distributed WITHOUT ANY WARRANTY; without even |
| 183 |
> |
the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR |
| 184 |
> |
PURPOSE. See the above copyright notice for more information. |
| 185 |
|
|
| 186 |
< |
if (i < 4) |
| 226 |
< |
tresh = 0.2*sm/(9); |
| 227 |
< |
else |
| 228 |
< |
tresh = 0.0; |
| 186 |
> |
=========================================================================*/ |
| 187 |
|
|
| 188 |
< |
for (ip=0; ip<2; ip++) { |
| 189 |
< |
for (iq=ip+1; iq<N; iq++) { |
| 232 |
< |
g = 100.0*fabs(a(ip, iq)); |
| 233 |
< |
if (i > 4 && (fabs(w(ip))+g) == fabs(w(ip)) |
| 234 |
< |
&& (fabs(w(iq))+g) == fabs(w(iq))) { |
| 235 |
< |
a(ip, iq) = 0.0; |
| 236 |
< |
} else if (fabs(a(ip, iq)) > tresh) { |
| 237 |
< |
h = w(iq) - w(ip); |
| 238 |
< |
if ( (fabs(h)+g) == fabs(h)) { |
| 239 |
< |
t = (a(ip, iq)) / h; |
| 240 |
< |
} else { |
| 241 |
< |
theta = 0.5*h / (a(ip, iq)); |
| 242 |
< |
t = 1.0 / (fabs(theta)+sqrt(1.0+theta*theta)); |
| 188 |
> |
#define VTK_ROTATE(a,i,j,k,l) g=a(i, j);h=a(k, l);a(i, j)=g-s*(h+g*tau);\ |
| 189 |
> |
a(k, l)=h+s*(g-h*tau) |
| 190 |
|
|
| 191 |
< |
if (theta < 0.0) |
| 245 |
< |
t = -t; |
| 246 |
< |
} |
| 191 |
> |
#define VTK_MAX_ROTATIONS 20 |
| 192 |
|
|
| 193 |
< |
c = 1.0 / sqrt(1+t*t); |
| 194 |
< |
s = t*c; |
| 195 |
< |
tau = s/(1.0+c); |
| 196 |
< |
h = t*a(ip, iq); |
| 197 |
< |
z(ip) -= h; |
| 198 |
< |
z(iq) += h; |
| 199 |
< |
w(ip) -= h; |
| 200 |
< |
w(iq) += h; |
| 201 |
< |
a(ip, iq)=0.0; |
| 202 |
< |
|
| 203 |
< |
for (j=0;j<ip-1;j++) |
| 204 |
< |
ROT(a,j,ip,j,iq); |
| 193 |
> |
// Jacobi iteration for the solution of eigenvectors/eigenvalues of a nxn |
| 194 |
> |
// real symmetric matrix. Square nxn matrix a; size of matrix in n; |
| 195 |
> |
// output eigenvalues in w; and output eigenvectors in v. Resulting |
| 196 |
> |
// eigenvalues/vectors are sorted in decreasing order; eigenvectors are |
| 197 |
> |
// normalized. |
| 198 |
> |
template<typename Real, int Dim> |
| 199 |
> |
int SquareMatrix<Real, Dim>::jacobi(SquareMatrix<Real, Dim>& a, Vector<Real, Dim>& w, |
| 200 |
> |
SquareMatrix<Real, Dim>& v) { |
| 201 |
> |
const int n = Dim; |
| 202 |
> |
int i, j, k, iq, ip, numPos; |
| 203 |
> |
Real tresh, theta, tau, t, sm, s, h, g, c, tmp; |
| 204 |
> |
Real bspace[4], zspace[4]; |
| 205 |
> |
Real *b = bspace; |
| 206 |
> |
Real *z = zspace; |
| 207 |
|
|
| 208 |
< |
for (j=ip+1;j<iq-1;j++) |
| 209 |
< |
ROT(a,ip,j,j,iq); |
| 210 |
< |
|
| 211 |
< |
for (j=iq+1; j<N; j++) |
| 212 |
< |
ROT(a,ip,j,iq,j); |
| 266 |
< |
|
| 267 |
< |
for (j=0; j<N; j++) |
| 268 |
< |
ROT(v,j,ip,j,iq); |
| 269 |
< |
} |
| 270 |
< |
} |
| 271 |
< |
}//for (ip=0; ip<2; ip++) |
| 272 |
< |
|
| 273 |
< |
for (ip=0; ip<N; ip++) { |
| 274 |
< |
b(ip) += z(ip); |
| 275 |
< |
w(ip) = b(ip); |
| 276 |
< |
z(ip) = 0.0; |
| 208 |
> |
// only allocate memory if the matrix is large |
| 209 |
> |
if (n > 4) |
| 210 |
> |
{ |
| 211 |
> |
b = new Real[n]; |
| 212 |
> |
z = new Real[n]; |
| 213 |
|
} |
| 278 |
– |
|
| 279 |
– |
} // end for (i=0; i<MAX_ROTATIONS; i++) |
| 214 |
|
|
| 215 |
< |
if ( i >= MAX_ROTATIONS ) |
| 216 |
< |
return false; |
| 217 |
< |
|
| 218 |
< |
// sort eigenfunctions |
| 219 |
< |
for (j=0; j<N; j++) { |
| 220 |
< |
k = j; |
| 221 |
< |
tmp = w(k); |
| 222 |
< |
for (i=j; i<N; i++) { |
| 223 |
< |
if (w(i) >= tmp) { |
| 224 |
< |
k = i; |
| 225 |
< |
tmp = w(k); |
| 215 |
> |
// initialize |
| 216 |
> |
for (ip=0; ip<n; ip++) |
| 217 |
> |
{ |
| 218 |
> |
for (iq=0; iq<n; iq++) |
| 219 |
> |
{ |
| 220 |
> |
v(ip, iq) = 0.0; |
| 221 |
> |
} |
| 222 |
> |
v(ip, ip) = 1.0; |
| 223 |
> |
} |
| 224 |
> |
for (ip=0; ip<n; ip++) |
| 225 |
> |
{ |
| 226 |
> |
b[ip] = w[ip] = a(ip, ip); |
| 227 |
> |
z[ip] = 0.0; |
| 228 |
> |
} |
| 229 |
> |
|
| 230 |
> |
// begin rotation sequence |
| 231 |
> |
for (i=0; i<VTK_MAX_ROTATIONS; i++) |
| 232 |
> |
{ |
| 233 |
> |
sm = 0.0; |
| 234 |
> |
for (ip=0; ip<n-1; ip++) |
| 235 |
> |
{ |
| 236 |
> |
for (iq=ip+1; iq<n; iq++) |
| 237 |
> |
{ |
| 238 |
> |
sm += fabs(a(ip, iq)); |
| 239 |
|
} |
| 240 |
+ |
} |
| 241 |
+ |
if (sm == 0.0) |
| 242 |
+ |
{ |
| 243 |
+ |
break; |
| 244 |
+ |
} |
| 245 |
+ |
|
| 246 |
+ |
if (i < 3) // first 3 sweeps |
| 247 |
+ |
{ |
| 248 |
+ |
tresh = 0.2*sm/(n*n); |
| 249 |
+ |
} |
| 250 |
+ |
else |
| 251 |
+ |
{ |
| 252 |
+ |
tresh = 0.0; |
| 253 |
+ |
} |
| 254 |
+ |
|
| 255 |
+ |
for (ip=0; ip<n-1; ip++) |
| 256 |
+ |
{ |
| 257 |
+ |
for (iq=ip+1; iq<n; iq++) |
| 258 |
+ |
{ |
| 259 |
+ |
g = 100.0*fabs(a(ip, iq)); |
| 260 |
+ |
|
| 261 |
+ |
// after 4 sweeps |
| 262 |
+ |
if (i > 3 && (fabs(w[ip])+g) == fabs(w[ip]) |
| 263 |
+ |
&& (fabs(w[iq])+g) == fabs(w[iq])) |
| 264 |
+ |
{ |
| 265 |
+ |
a(ip, iq) = 0.0; |
| 266 |
+ |
} |
| 267 |
+ |
else if (fabs(a(ip, iq)) > tresh) |
| 268 |
+ |
{ |
| 269 |
+ |
h = w[iq] - w[ip]; |
| 270 |
+ |
if ( (fabs(h)+g) == fabs(h)) |
| 271 |
+ |
{ |
| 272 |
+ |
t = (a(ip, iq)) / h; |
| 273 |
+ |
} |
| 274 |
+ |
else |
| 275 |
+ |
{ |
| 276 |
+ |
theta = 0.5*h / (a(ip, iq)); |
| 277 |
+ |
t = 1.0 / (fabs(theta)+sqrt(1.0+theta*theta)); |
| 278 |
+ |
if (theta < 0.0) |
| 279 |
+ |
{ |
| 280 |
+ |
t = -t; |
| 281 |
+ |
} |
| 282 |
+ |
} |
| 283 |
+ |
c = 1.0 / sqrt(1+t*t); |
| 284 |
+ |
s = t*c; |
| 285 |
+ |
tau = s/(1.0+c); |
| 286 |
+ |
h = t*a(ip, iq); |
| 287 |
+ |
z[ip] -= h; |
| 288 |
+ |
z[iq] += h; |
| 289 |
+ |
w[ip] -= h; |
| 290 |
+ |
w[iq] += h; |
| 291 |
+ |
a(ip, iq)=0.0; |
| 292 |
+ |
|
| 293 |
+ |
// ip already shifted left by 1 unit |
| 294 |
+ |
for (j = 0;j <= ip-1;j++) |
| 295 |
+ |
{ |
| 296 |
+ |
VTK_ROTATE(a,j,ip,j,iq); |
| 297 |
+ |
} |
| 298 |
+ |
// ip and iq already shifted left by 1 unit |
| 299 |
+ |
for (j = ip+1;j <= iq-1;j++) |
| 300 |
+ |
{ |
| 301 |
+ |
VTK_ROTATE(a,ip,j,j,iq); |
| 302 |
+ |
} |
| 303 |
+ |
// iq already shifted left by 1 unit |
| 304 |
+ |
for (j=iq+1; j<n; j++) |
| 305 |
+ |
{ |
| 306 |
+ |
VTK_ROTATE(a,ip,j,iq,j); |
| 307 |
+ |
} |
| 308 |
+ |
for (j=0; j<n; j++) |
| 309 |
+ |
{ |
| 310 |
+ |
VTK_ROTATE(v,j,ip,j,iq); |
| 311 |
+ |
} |
| 312 |
+ |
} |
| 313 |
+ |
} |
| 314 |
+ |
} |
| 315 |
+ |
|
| 316 |
+ |
for (ip=0; ip<n; ip++) |
| 317 |
+ |
{ |
| 318 |
+ |
b[ip] += z[ip]; |
| 319 |
+ |
w[ip] = b[ip]; |
| 320 |
+ |
z[ip] = 0.0; |
| 321 |
+ |
} |
| 322 |
|
} |
| 323 |
< |
|
| 324 |
< |
if (k != j) { |
| 325 |
< |
w(k) = w(j); |
| 326 |
< |
w(j) = tmp; |
| 327 |
< |
for (i=0; i<N; i++) { |
| 328 |
< |
tmp = v(i, j); |
| 329 |
< |
v(i, j) = v(i, k); |
| 330 |
< |
v(i, k) = tmp; |
| 323 |
> |
|
| 324 |
> |
//// this is NEVER called |
| 325 |
> |
if ( i >= VTK_MAX_ROTATIONS ) |
| 326 |
> |
{ |
| 327 |
> |
std::cout << "vtkMath::Jacobi: Error extracting eigenfunctions" << std::endl; |
| 328 |
> |
return 0; |
| 329 |
> |
} |
| 330 |
> |
|
| 331 |
> |
// sort eigenfunctions these changes do not affect accuracy |
| 332 |
> |
for (j=0; j<n-1; j++) // boundary incorrect |
| 333 |
> |
{ |
| 334 |
> |
k = j; |
| 335 |
> |
tmp = w[k]; |
| 336 |
> |
for (i=j+1; i<n; i++) // boundary incorrect, shifted already |
| 337 |
> |
{ |
| 338 |
> |
if (w[i] >= tmp) // why exchage if same? |
| 339 |
> |
{ |
| 340 |
> |
k = i; |
| 341 |
> |
tmp = w[k]; |
| 342 |
|
} |
| 343 |
+ |
} |
| 344 |
+ |
if (k != j) |
| 345 |
+ |
{ |
| 346 |
+ |
w[k] = w[j]; |
| 347 |
+ |
w[j] = tmp; |
| 348 |
+ |
for (i=0; i<n; i++) |
| 349 |
+ |
{ |
| 350 |
+ |
tmp = v(i, j); |
| 351 |
+ |
v(i, j) = v(i, k); |
| 352 |
+ |
v(i, k) = tmp; |
| 353 |
+ |
} |
| 354 |
+ |
} |
| 355 |
|
} |
| 356 |
< |
} |
| 356 |
> |
// insure eigenvector consistency (i.e., Jacobi can compute vectors that |
| 357 |
> |
// are negative of one another (.707,.707,0) and (-.707,-.707,0). This can |
| 358 |
> |
// reek havoc in hyperstreamline/other stuff. We will select the most |
| 359 |
> |
// positive eigenvector. |
| 360 |
> |
int ceil_half_n = (n >> 1) + (n & 1); |
| 361 |
> |
for (j=0; j<n; j++) |
| 362 |
> |
{ |
| 363 |
> |
for (numPos=0, i=0; i<n; i++) |
| 364 |
> |
{ |
| 365 |
> |
if ( v(i, j) >= 0.0 ) |
| 366 |
> |
{ |
| 367 |
> |
numPos++; |
| 368 |
> |
} |
| 369 |
> |
} |
| 370 |
> |
// if ( numPos < ceil(double(n)/double(2.0)) ) |
| 371 |
> |
if ( numPos < ceil_half_n) |
| 372 |
> |
{ |
| 373 |
> |
for(i=0; i<n; i++) |
| 374 |
> |
{ |
| 375 |
> |
v(i, j) *= -1.0; |
| 376 |
> |
} |
| 377 |
> |
} |
| 378 |
> |
} |
| 379 |
|
|
| 380 |
< |
// insure eigenvector consistency (i.e., Jacobi can compute |
| 381 |
< |
// vectors that are negative of one another (.707,.707,0) and |
| 382 |
< |
// (-.707,-.707,0). This can reek havoc in |
| 383 |
< |
// hyperstreamline/other stuff. We will select the most |
| 384 |
< |
// positive eigenvector. |
| 385 |
< |
int numPos; |
| 312 |
< |
for (j=0; j<N; j++) { |
| 313 |
< |
for (numPos=0, i=0; i<N; i++) if ( v(i, j) >= 0.0 ) numPos++; |
| 314 |
< |
if ( numPos < 2 ) for(i=0; i<N; i++) v(i, j) *= -1.0; |
| 380 |
> |
if (n > 4) |
| 381 |
> |
{ |
| 382 |
> |
delete [] b; |
| 383 |
> |
delete [] z; |
| 384 |
> |
} |
| 385 |
> |
return 1; |
| 386 |
|
} |
| 387 |
|
|
| 317 |
– |
return true; |
| 318 |
– |
} |
| 388 |
|
|
| 320 |
– |
#undef ROT |
| 321 |
– |
#undef MAX_ROTATIONS |
| 322 |
– |
|
| 389 |
|
} |
| 324 |
– |
|
| 390 |
|
#endif //MATH_SQUAREMATRIX_HPP |
| 391 |
+ |
|
