/[escript]/trunk/escript/src/LocalOps.h
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Contents of /trunk/escript/src/LocalOps.h

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Revision 583 - (show annotations)
Wed Mar 8 08:15:34 2006 UTC (13 years, 7 months ago) by gross
File MIME type: text/plain
File size: 13882 byte(s)
_eigenvalues_and_eigenvector method added of data object. the algorithm has been tested on floats in python but not on data objects.
1 // $Id$
2 /*
3 ******************************************************************************
4 * *
5 * COPYRIGHT ACcESS 2004 - All Rights Reserved *
6 * *
7 * This software is the property of ACcESS. No part of this code *
8 * may be copied in any form or by any means without the expressed written *
9 * consent of ACcESS. Copying, use or modification of this software *
10 * by any unauthorised person is illegal unless that person has a software *
11 * license agreement with ACcESS. *
12 * *
13 ******************************************************************************
14 */
15
16 #if !defined escript_LocalOps_H
17 #define escript_LocalOps_H
18 #include <math.h>
19 namespace escript {
20
21
22 /**
23 \brief
24 solves a 1x1 eigenvalue A*V=ev*V problem
25
26 \param A00 Input - A_00
27 \param ev0 Output - eigenvalue
28 */
29 inline
30 void eigenvalues1(const double A00,double* ev0) {
31
32 *ev0=A00;
33
34 }
35 /**
36 \brief
37 solves a 2x2 eigenvalue A*V=ev*V problem for symmetric A
38
39 \param A00 Input - A_00
40 \param A01 Input - A_01
41 \param A11 Input - A_11
42 \param ev0 Output - smallest eigenvalue
43 \param ev1 Output - largest eigenvalue
44 */
45 inline
46 void eigenvalues2(const double A00,const double A01,const double A11,
47 double* ev0, double* ev1) {
48 const register double trA=(A00+A11)/2.;
49 const register double A_00=A00-trA;
50 const register double A_11=A11-trA;
51 const register double s=sqrt(A01*A01-A_00*A_11);
52 *ev0=trA-s;
53 *ev1=trA+s;
54 }
55 /**
56 \brief
57 solves a 3x3 eigenvalue A*V=ev*V problem for symmetric A
58
59 \param A00 Input - A_00
60 \param A01 Input - A_01
61 \param A02 Input - A_02
62 \param A11 Input - A_11
63 \param A12 Input - A_12
64 \param A22 Input - A_22
65 \param ev0 Output - smallest eigenvalue
66 \param ev1 Output - eigenvalue
67 \param ev2 Output - largest eigenvalue
68 */
69 inline
70 void eigenvalues3(const double A00, const double A01, const double A02,
71 const double A11, const double A12,
72 const double A22,
73 double* ev0, double* ev1,double* ev2) {
74
75 const register double trA=(A00+A11+A22)/3.;
76 const register double A_00=A00-trA;
77 const register double A_11=A11-trA;
78 const register double A_22=A22-trA;
79 const register double A01_2=A01*A01;
80 const register double A02_2=A02*A02;
81 const register double A12_2=A12*A12;
82 const register double p=A02_2+A12_2+A01_2+(A_00*A_00+A_11*A_11+A_22*A_22)/2.;
83 const register double q=(A02_2*A_11+A12_2*A_00+A01_2*A_22)-(A_00*A_11*A_22+2*A01*A12*A02);
84 const register double sq_p=sqrt(p/3.);
85 register double z=-q/(2*pow(sq_p,3));
86 if (z<-1.) {
87 z=-1.;
88 } else if (z>1.) {
89 z=1.;
90 }
91 const register double alpha_3=acos(z)/3.;
92 *ev2=trA+2.*sq_p*cos(alpha_3);
93 *ev1=trA-2.*sq_p*cos(alpha_3+M_PI/3.);
94 *ev0=trA-2.*sq_p*cos(alpha_3-M_PI/3.);
95 }
96 /**
97 \brief
98 solves a 1x1 eigenvalue A*V=ev*V problem for symmetric A
99
100 \param A00 Input - A_00
101 \param ev0 Output - eigenvalue
102 \param V00 Output - eigenvector
103 \param tol Input - tolerance to identify to eigenvalues
104 */
105 inline
106 void eigenvalues_and_eigenvectors1(const double A00,double* ev0,double* V00,const double tol)
107 {
108 eigenvalues1(A00,ev0);
109 *V00=1.;
110 return;
111 }
112 /**
113 \brief
114 returns a non-zero vector in the kernel of [[A00,A01],[A01,A11]] assuming that the kernel dimension is at least 1.
115
116 \param A00 Input - matrix component
117 \param A10 Input - matrix component
118 \param A01 Input - matrix component
119 \param A11 Input - matrix component
120 \param V0 Output - vector component
121 \param V1 Output - vector component
122 */
123 inline
124 void vectorInKernel2(const double A00,const double A10,const double A01,const double A11,
125 double* V0, double*V1)
126 {
127 register double absA00=fabs(A00);
128 register double absA01=fabs(A01);
129 register double absA10=fabs(A10);
130 register double absA11=fabs(A11);
131 register double m=absA11>absA01 ? absA11 : absA01;
132 if (absA00>m || absA10>m) {
133 *V0=-A10;
134 *V1=A00;
135 } else {
136 if (m<=0) {
137 *V0=1.;
138 *V1=0.;
139 } else {
140 *V0=A11;
141 *V1=-A01;
142 }
143 }
144 }
145 /**
146 \brief
147 returns a non-zero vector in the kernel of [[A00,A01,A02],[A10,A11,A12],[A20,A21,A22]]
148 assuming that the kernel dimension is at least 1 and A00 is non zero.
149
150 \param A00 Input - matrix component
151 \param A10 Input - matrix component
152 \param A20 Input - matrix component
153 \param A01 Input - matrix component
154 \param A11 Input - matrix component
155 \param A21 Input - matrix component
156 \param A02 Input - matrix component
157 \param A12 Input - matrix component
158 \param A22 Input - matrix component
159 \param V0 Output - vector component
160 \param V1 Output - vector component
161 \param V2 Output - vector component
162 */
163 inline
164 void vectorInKernel3__nonZeroA00(const double A00,const double A10,const double A20,
165 const double A01,const double A11,const double A21,
166 const double A02,const double A12,const double A22,
167 double* V0,double* V1,double* V2)
168 {
169 double TEMP0,TEMP1;
170 register const double I00=1./A00;
171 register const double IA10=I00*A10;
172 register const double IA20=I00*A20;
173 vectorInKernel2(A11-IA10*A01,A21-IA20*A01,A12-IA10*A02,A22-IA20*A02,&TEMP0,&TEMP1);
174 *V0=-(A10*TEMP0+A20*TEMP1);
175 *V1=A00*TEMP0;
176 *V2=A00*TEMP1;
177 }
178
179 /**
180 \brief
181 solves a 2x2 eigenvalue A*V=ev*V problem for symmetric A. Eigenvectors are
182 ordered by increasing value and eigen vectors are normalizeVector3d such that
183 length is zero and first non-zero component is positive.
184
185 \param A00 Input - A_00
186 \param A01 Input - A_01
187 \param A11 Input - A_11
188 \param ev0 Output - smallest eigenvalue
189 \param ev1 Output - eigenvalue
190 \param V00 Output - eigenvector componenent coresponding to ev0
191 \param V10 Output - eigenvector componenent coresponding to ev0
192 \param V01 Output - eigenvector componenent coresponding to ev1
193 \param V11 Output - eigenvector componenent coresponding to ev1
194 \param tol Input - tolerance to identify to eigenvalues
195 */
196 inline
197 void eigenvalues_and_eigenvectors2(const double A00,const double A01,const double A11,
198 double* ev0, double* ev1,
199 double* V00, double* V10, double* V01, double* V11,
200 const double tol)
201 {
202 double TEMP0,TEMP1;
203 eigenvalues2(A00,A01,A11,ev0,ev1);
204 const register double absev0=fabs(*ev0);
205 const register double absev1=fabs(*ev1);
206 register double max_ev=absev0>absev1 ? absev0 : absev1;
207 if (fabs((*ev0)-(*ev1))<tol*max_ev) {
208 *V00=1.;
209 *V10=0.;
210 *V01=0.;
211 *V11=1.;
212 } else {
213 vectorInKernel2(A00-(*ev0),A01,A01,A11-(*ev0),&TEMP0,&TEMP1);
214 const register double scale=1./sqrt(TEMP0*TEMP0+TEMP1*TEMP1);
215 if (TEMP0<0.) {
216 *V00=-TEMP0*scale;
217 *V10=-TEMP1*scale;
218 if (TEMP1<0.) {
219 *V01= *V10;
220 *V11=-(*V00);
221 } else {
222 *V01=-(*V10);
223 *V11= (*V00);
224 }
225 } else if (TEMP0>0.) {
226 *V00=TEMP0*scale;
227 *V10=TEMP1*scale;
228 if (TEMP1<0.) {
229 *V01=-(*V10);
230 *V11= (*V00);
231 } else {
232 *V01= (*V10);
233 *V11=-(*V00);
234 }
235 } else {
236 *V00=0.;
237 *V10=1;
238 *V11=0.;
239 *V01=1.;
240 }
241 }
242 }
243 /**
244 \brief
245 nomalizes a 3-d vector such that length is one and first non-zero component is positive.
246
247 \param V0 - vector componenent
248 \param V1 - vector componenent
249 \param V2 - vector componenent
250 */
251 inline
252 void normalizeVector3(double* V0,double* V1,double* V2)
253 {
254 register double s;
255 if (*V0>0) {
256 s=1./sqrt((*V0)*(*V0)+(*V1)*(*V1)+(*V2)*(*V2));
257 *V0*=s;
258 *V1*=s;
259 *V2*=s;
260 } else if (*V0<0) {
261 s=-1./sqrt((*V0)*(*V0)+(*V1)*(*V1)+(*V2)*(*V2));
262 *V0*=s;
263 *V1*=s;
264 *V2*=s;
265 } else {
266 if (*V1>0) {
267 s=1./sqrt((*V1)*(*V1)+(*V2)*(*V2));
268 *V1*=s;
269 *V2*=s;
270 } else if (*V1<0) {
271 s=-1./sqrt((*V1)*(*V1)+(*V2)*(*V2));
272 *V1*=s;
273 *V2*=s;
274 } else {
275 *V2=1.;
276 }
277 }
278 }
279 /**
280 \brief
281 solves a 2x2 eigenvalue A*V=ev*V problem for symmetric A. Eigenvectors are
282 ordered by increasing value and eigen vectors are normalizeVector3d such that
283 length is zero and first non-zero component is positive.
284
285 \param A00 Input - A_00
286 \param A01 Input - A_01
287 \param A11 Input - A_11
288 \param ev0 Output - smallest eigenvalue
289 \param ev1 Output - eigenvalue
290 \param V00 Output - eigenvector componenent coresponding to ev0
291 \param V10 Output - eigenvector componenent coresponding to ev0
292 \param V01 Output - eigenvector componenent coresponding to ev1
293 \param V11 Output - eigenvector componenent coresponding to ev1
294 \param tol Input - tolerance to identify to eigenvalues
295 */
296 inline
297 void eigenvalues_and_eigenvectors3(const double A00, const double A01, const double A02,
298 const double A11, const double A12, const double A22,
299 double* ev0, double* ev1, double* ev2,
300 double* V00, double* V10, double* V20,
301 double* V01, double* V11, double* V21,
302 double* V02, double* V12, double* V22,
303 const double tol)
304 {
305 register const double absA01=fabs(A01);
306 register const double absA02=fabs(A02);
307 register const double m=absA01>absA02 ? absA01 : absA02;
308 if (m<=0) {
309 double TEMP_V00,TEMP_V10,TEMP_V01,TEMP_V11,TEMP_EV0,TEMP_EV1;
310 eigenvalues_and_eigenvectors2(A11,A12,A22,
311 &TEMP_EV0,&TEMP_EV1,
312 &TEMP_V00,&TEMP_V10,&TEMP_V01,&TEMP_V11,tol);
313 if (A00<=TEMP_EV0) {
314 *V00=1.;
315 *V10=0.;
316 *V20=0.;
317 *V01=0.;
318 *V11=TEMP_V00;
319 *V21=TEMP_V10;
320 *V02=0.;
321 *V12=TEMP_V01;
322 *V22=TEMP_V11;
323 *ev0=A00;
324 *ev1=TEMP_EV0;
325 *ev2=TEMP_EV1;
326 } else if (A00>TEMP_EV1) {
327 *V00=TEMP_V00;
328 *V10=TEMP_V10;
329 *V20=0.;
330 *V01=TEMP_V01;
331 *V11=TEMP_V11;
332 *V21=0.;
333 *V02=0.;
334 *V12=0.;
335 *V22=1.;
336 *ev0=TEMP_EV0;
337 *ev1=TEMP_EV1;
338 *ev0=A00;
339 } else {
340 *V00=TEMP_V00;
341 *V10=0;
342 *V20=TEMP_V10;
343 *V01=0.;
344 *V11=1.;
345 *V21=0.;
346 *V02=TEMP_V01;
347 *V12=0.;
348 *V22=TEMP_V11;
349 *ev0=TEMP_EV0;
350 *ev1=A00;
351 *ev2=TEMP_EV1;
352 }
353 } else {
354 eigenvalues3(A00,A01,A02,A11,A12,A22,ev0,ev1,ev2);
355 const register double absev0=fabs(*ev0);
356 const register double absev1=fabs(*ev1);
357 const register double absev2=fabs(*ev2);
358 register double max_ev=absev0>absev1 ? absev0 : absev1;
359 max_ev=max_ev>absev2 ? max_ev : absev2;
360 const register double d_01=fabs((*ev0)-(*ev1));
361 const register double d_12=fabs((*ev1)-(*ev2));
362 const register double max_d=d_01>d_12 ? d_01 : d_12;
363 if (max_d<=tol*max_ev) {
364 *V00=1.;
365 *V10=0;
366 *V20=0;
367 *V01=0;
368 *V11=1.;
369 *V21=0;
370 *V02=0;
371 *V12=0;
372 *V22=1.;
373 } else {
374 const register double S00=A00-(*ev0);
375 const register double absS00=fabs(S00);
376 if (fabs(S00)>m) {
377 vectorInKernel3__nonZeroA00(S00,A01,A02,A01,A11-(*ev0),A12,A02,A12,A22-(*ev0),V00,V10,V20);
378 } else if (absA02<m) {
379 vectorInKernel3__nonZeroA00(A01,A11-(*ev0),A12,S00,A01,A02,A02,A12,A22-(*ev0),V00,V10,V20);
380 } else {
381 vectorInKernel3__nonZeroA00(A02,A12,A22-(*ev0),S00,A01,A02,A01,A11-(*ev0),A12,V00,V10,V20);
382 }
383 normalizeVector3(V00,V10,V20);;
384 const register double T00=A00-(*ev2);
385 const register double absT00=fabs(T00);
386 if (fabs(T00)>m) {
387 vectorInKernel3__nonZeroA00(T00,A01,A02,A01,A11-(*ev2),A12,A02,A12,A22-(*ev2),V02,V12,V22);
388 } else if (absA02<m) {
389 vectorInKernel3__nonZeroA00(A01,A11-(*ev2),A12,T00,A01,A02,A02,A12,A22-(*ev2),V02,V12,V22);
390 } else {
391 vectorInKernel3__nonZeroA00(A02,A12,A22-(*ev2),T00,A01,A02,A01,A11-(*ev2),A12,V02,V12,V22);
392 }
393 const register double dot=(*V02)*(*V00)+(*V12)*(*V10)+(*V22)*(*V20);
394 *V02-=dot*(*V00);
395 *V12-=dot*(*V10);
396 *V22-=dot*(*V20);
397 normalizeVector3(V02,V12,V22);
398 *V01=(*V10)*(*V22)-(*V12)*(*V20);
399 *V11=(*V20)*(*V02)-(*V00)*(*V22);
400 *V21=(*V00)*(*V12)-(*V02)*(*V10);
401 normalizeVector3(V01,V11,V21);
402 }
403 }
404 }
405 } // end of namespace
406 #endif

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