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 |
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|
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 |
+ |
|