83 |
|
SquareMatrix<Real, Dim> result; |
84 |
|
|
85 |
|
return result; |
86 |
< |
} |
86 |
> |
} |
87 |
|
|
88 |
– |
|
89 |
– |
|
88 |
|
/** Returns the determinant of this matrix. */ |
89 |
|
double determinant() const { |
90 |
|
double det; |
111 |
|
return true; |
112 |
|
} |
113 |
|
|
114 |
< |
/** Tests if this matrix is orthogona. */ |
114 |
> |
/** Tests if this matrix is orthogonal. */ |
115 |
|
bool isOrthogonal() { |
116 |
|
SquareMatrix<Real, Dim> tmp; |
117 |
|
|
118 |
|
tmp = *this * transpose(); |
119 |
|
|
120 |
< |
return tmp.isUnitMatrix(); |
120 |
> |
return tmp.isDiagonal(); |
121 |
|
} |
122 |
|
|
123 |
|
/** Tests if this matrix is diagonal. */ |
142 |
|
return true; |
143 |
|
} |
144 |
|
|
145 |
+ |
void diagonalize() { |
146 |
+ |
jacobi(m, eigenValues, ortMat); |
147 |
+ |
} |
148 |
+ |
|
149 |
+ |
/** |
150 |
+ |
* Finds the eigenvalues and eigenvectors of a symmetric matrix |
151 |
+ |
* @param eigenvals a reference to a vector3 where the |
152 |
+ |
* eigenvalues will be stored. The eigenvalues are ordered so |
153 |
+ |
* that eigenvals[0] <= eigenvals[1] <= eigenvals[2]. |
154 |
+ |
* @return an orthogonal matrix whose ith column is an |
155 |
+ |
* eigenvector for the eigenvalue eigenvals[i] |
156 |
+ |
*/ |
157 |
+ |
SquareMatrix<Real, Dim> findEigenvectors(Vector<Real, Dim>& eigenValues) { |
158 |
+ |
SquareMatrix<Real, Dim> ortMat; |
159 |
+ |
|
160 |
+ |
if ( !isSymmetric()){ |
161 |
+ |
throw(); |
162 |
+ |
} |
163 |
+ |
|
164 |
+ |
SquareMatrix<Real, Dim> m(*this); |
165 |
+ |
jacobi(m, eigenValues, ortMat); |
166 |
+ |
|
167 |
+ |
return ortMat; |
168 |
+ |
} |
169 |
+ |
/** |
170 |
+ |
* Jacobi iteration routines for computing eigenvalues/eigenvectors of |
171 |
+ |
* real symmetric matrix |
172 |
+ |
* |
173 |
+ |
* @return true if success, otherwise return false |
174 |
+ |
* @param a source matrix |
175 |
+ |
* @param w output eigenvalues |
176 |
+ |
* @param v output eigenvectors |
177 |
+ |
*/ |
178 |
+ |
void jacobi(const SquareMatrix<Real, Dim>& a, |
179 |
+ |
Vector<Real, Dim>& w, |
180 |
+ |
SquareMatrix<Real, Dim>& v); |
181 |
|
};//end SquareMatrix |
182 |
|
|
183 |
+ |
|
184 |
+ |
#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) |
185 |
+ |
#define MAX_ROTATIONS 60 |
186 |
+ |
|
187 |
+ |
template<Real, int Dim> |
188 |
+ |
void SquareMatrix<Real, int Dim>::jacobi(SquareMatrix<Real, Dim>& a, |
189 |
+ |
Vector<Real, Dim>& w, |
190 |
+ |
SquareMatrix<Real, Dim>& v) { |
191 |
+ |
const int N = Dim; |
192 |
+ |
int i, j, k, iq, ip; |
193 |
+ |
double tresh, theta, tau, t, sm, s, h, g, c; |
194 |
+ |
double tmp; |
195 |
+ |
Vector<Real, Dim> b, z; |
196 |
+ |
|
197 |
+ |
// initialize |
198 |
+ |
for (ip=0; ip<N; ip++) |
199 |
+ |
{ |
200 |
+ |
for (iq=0; iq<N; iq++) v(ip, iq) = 0.0; |
201 |
+ |
v(ip, ip) = 1.0; |
202 |
+ |
} |
203 |
+ |
for (ip=0; ip<N; ip++) |
204 |
+ |
{ |
205 |
+ |
b(ip) = w(ip) = a(ip, ip); |
206 |
+ |
z(ip) = 0.0; |
207 |
+ |
} |
208 |
+ |
|
209 |
+ |
// begin rotation sequence |
210 |
+ |
for (i=0; i<MAX_ROTATIONS; i++) |
211 |
+ |
{ |
212 |
+ |
sm = 0.0; |
213 |
+ |
for (ip=0; ip<2; ip++) |
214 |
+ |
{ |
215 |
+ |
for (iq=ip+1; iq<N; iq++) sm += fabs(a(ip, iq)); |
216 |
+ |
} |
217 |
+ |
if (sm == 0.0) break; |
218 |
+ |
|
219 |
+ |
if (i < 4) tresh = 0.2*sm/(9); |
220 |
+ |
else tresh = 0.0; |
221 |
+ |
|
222 |
+ |
for (ip=0; ip<2; ip++) |
223 |
+ |
{ |
224 |
+ |
for (iq=ip+1; iq<N; iq++) |
225 |
+ |
{ |
226 |
+ |
g = 100.0*fabs(a(ip, iq)); |
227 |
+ |
if (i > 4 && (fabs(w(ip))+g) == fabs(w(ip)) |
228 |
+ |
&& (fabs(w(iq))+g) == fabs(w(iq))) |
229 |
+ |
{ |
230 |
+ |
a(ip, iq) = 0.0; |
231 |
+ |
} |
232 |
+ |
else if (fabs(a(ip, iq)) > tresh) |
233 |
+ |
{ |
234 |
+ |
h = w(iq) - w(ip); |
235 |
+ |
if ( (fabs(h)+g) == fabs(h)) t = (a(ip, iq)) / h; |
236 |
+ |
else |
237 |
+ |
{ |
238 |
+ |
theta = 0.5*h / (a(ip, iq)); |
239 |
+ |
t = 1.0 / (fabs(theta)+sqrt(1.0+theta*theta)); |
240 |
+ |
if (theta < 0.0) t = -t; |
241 |
+ |
} |
242 |
+ |
c = 1.0 / sqrt(1+t*t); |
243 |
+ |
s = t*c; |
244 |
+ |
tau = s/(1.0+c); |
245 |
+ |
h = t*a(ip, iq); |
246 |
+ |
z(ip) -= h; |
247 |
+ |
z(iq) += h; |
248 |
+ |
w(ip) -= h; |
249 |
+ |
w(iq) += h; |
250 |
+ |
a(ip, iq)=0.0; |
251 |
+ |
for (j=0;j<ip-1;j++) |
252 |
+ |
{ |
253 |
+ |
ROT(a,j,ip,j,iq); |
254 |
+ |
} |
255 |
+ |
for (j=ip+1;j<iq-1;j++) |
256 |
+ |
{ |
257 |
+ |
ROT(a,ip,j,j,iq); |
258 |
+ |
} |
259 |
+ |
for (j=iq+1; j<N; j++) |
260 |
+ |
{ |
261 |
+ |
ROT(a,ip,j,iq,j); |
262 |
+ |
} |
263 |
+ |
for (j=0; j<N; j++) |
264 |
+ |
{ |
265 |
+ |
ROT(v,j,ip,j,iq); |
266 |
+ |
} |
267 |
+ |
} |
268 |
+ |
} |
269 |
+ |
} |
270 |
+ |
|
271 |
+ |
for (ip=0; ip<N; ip++) |
272 |
+ |
{ |
273 |
+ |
b(ip) += z(ip); |
274 |
+ |
w(ip) = b(ip); |
275 |
+ |
z(ip) = 0.0; |
276 |
+ |
} |
277 |
+ |
} |
278 |
+ |
|
279 |
+ |
if ( i >= MAX_ROTATIONS ) |
280 |
+ |
return false; |
281 |
+ |
|
282 |
+ |
// sort eigenfunctions |
283 |
+ |
for (j=0; j<N; j++) |
284 |
+ |
{ |
285 |
+ |
k = j; |
286 |
+ |
tmp = w(k); |
287 |
+ |
for (i=j; i<N; i++) |
288 |
+ |
{ |
289 |
+ |
if (w(i) >= tmp) |
290 |
+ |
{ |
291 |
+ |
k = i; |
292 |
+ |
tmp = w(k); |
293 |
+ |
} |
294 |
+ |
} |
295 |
+ |
if (k != j) |
296 |
+ |
{ |
297 |
+ |
w(k) = w(j); |
298 |
+ |
w(j) = tmp; |
299 |
+ |
for (i=0; i<N; i++) |
300 |
+ |
{ |
301 |
+ |
tmp = v(i, j); |
302 |
+ |
v(i, j) = v(i, k); |
303 |
+ |
v(i, k) = tmp; |
304 |
+ |
} |
305 |
+ |
} |
306 |
+ |
} |
307 |
+ |
|
308 |
+ |
// insure eigenvector consistency (i.e., Jacobi can compute |
309 |
+ |
// vectors that are negative of one another (.707,.707,0) and |
310 |
+ |
// (-.707,-.707,0). This can reek havoc in |
311 |
+ |
// hyperstreamline/other stuff. We will select the most |
312 |
+ |
// positive eigenvector. |
313 |
+ |
int numPos; |
314 |
+ |
for (j=0; j<N; j++) |
315 |
+ |
{ |
316 |
+ |
for (numPos=0, i=0; i<N; i++) if ( v(i, j) >= 0.0 ) numPos++; |
317 |
+ |
if ( numPos < 2 ) for(i=0; i<N; i++) v(i, j) *= -1.0; |
318 |
+ |
} |
319 |
+ |
|
320 |
+ |
return true; |
321 |
|
} |
322 |
+ |
|
323 |
+ |
#undef ROT |
324 |
+ |
#undef MAX_ROTATIONS |
325 |
+ |
|
326 |
+ |
} |
327 |
+ |
|
328 |
+ |
|
329 |
+ |
} |
330 |
|
#endif //MATH_SQUAREMATRIX_HPP |