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

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Revision 588 - (show annotations)
Fri Mar 10 04:45:04 2006 UTC (13 years, 7 months ago) by gross
File MIME type: text/plain
File size: 14044 byte(s)
1D and 3D tests for eigenvalues_and_eigenvector added.
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 if (p<=0.) {
84 *ev2=trA;
85 *ev1=trA;
86 *ev0=trA;
87
88 } else {
89 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);
90 const register double sq_p=sqrt(p/3.);
91 register double z=-q/(2*pow(sq_p,3));
92 if (z<-1.) {
93 z=-1.;
94 } else if (z>1.) {
95 z=1.;
96 }
97 const register double alpha_3=acos(z)/3.;
98 *ev2=trA+2.*sq_p*cos(alpha_3);
99 *ev1=trA-2.*sq_p*cos(alpha_3+M_PI/3.);
100 *ev0=trA-2.*sq_p*cos(alpha_3-M_PI/3.);
101 }
102 }
103 /**
104 \brief
105 solves a 1x1 eigenvalue A*V=ev*V problem for symmetric A
106
107 \param A00 Input - A_00
108 \param ev0 Output - eigenvalue
109 \param V00 Output - eigenvector
110 \param tol Input - tolerance to identify to eigenvalues
111 */
112 inline
113 void eigenvalues_and_eigenvectors1(const double A00,double* ev0,double* V00,const double tol)
114 {
115 eigenvalues1(A00,ev0);
116 *V00=1.;
117 return;
118 }
119 /**
120 \brief
121 returns a non-zero vector in the kernel of [[A00,A01],[A01,A11]] assuming that the kernel dimension is at least 1.
122
123 \param A00 Input - matrix component
124 \param A10 Input - matrix component
125 \param A01 Input - matrix component
126 \param A11 Input - matrix component
127 \param V0 Output - vector component
128 \param V1 Output - vector component
129 */
130 inline
131 void vectorInKernel2(const double A00,const double A10,const double A01,const double A11,
132 double* V0, double*V1)
133 {
134 register double absA00=fabs(A00);
135 register double absA10=fabs(A10);
136 register double absA01=fabs(A01);
137 register double absA11=fabs(A11);
138 register double m=absA11>absA10 ? absA11 : absA10;
139 if (absA00>m || absA01>m) {
140 *V0=-A01;
141 *V1=A00;
142 } else {
143 if (m<=0) {
144 *V0=1.;
145 *V1=0.;
146 } else {
147 *V0=A11;
148 *V1=-A10;
149 }
150 }
151 }
152 /**
153 \brief
154 returns a non-zero vector in the kernel of [[A00,A01,A02],[A10,A11,A12],[A20,A21,A22]]
155 assuming that the kernel dimension is at least 1 and A00 is non zero.
156
157 \param A00 Input - matrix component
158 \param A10 Input - matrix component
159 \param A20 Input - matrix component
160 \param A01 Input - matrix component
161 \param A11 Input - matrix component
162 \param A21 Input - matrix component
163 \param A02 Input - matrix component
164 \param A12 Input - matrix component
165 \param A22 Input - matrix component
166 \param V0 Output - vector component
167 \param V1 Output - vector component
168 \param V2 Output - vector component
169 */
170 inline
171 void vectorInKernel3__nonZeroA00(const double A00,const double A10,const double A20,
172 const double A01,const double A11,const double A21,
173 const double A02,const double A12,const double A22,
174 double* V0,double* V1,double* V2)
175 {
176 double TEMP0,TEMP1;
177 register const double I00=1./A00;
178 register const double IA10=I00*A10;
179 register const double IA20=I00*A20;
180 vectorInKernel2(A11-IA10*A01,A12-IA10*A02,
181 A21-IA20*A01,A22-IA20*A02,&TEMP0,&TEMP1);
182 *V0=-(A10*TEMP0+A20*TEMP1);
183 *V1=A00*TEMP0;
184 *V2=A00*TEMP1;
185 }
186
187 /**
188 \brief
189 solves a 2x2 eigenvalue A*V=ev*V problem for symmetric A. Eigenvectors are
190 ordered by increasing value and eigen vectors are normalizeVector3d such that
191 length is zero and first non-zero component is positive.
192
193 \param A00 Input - A_00
194 \param A01 Input - A_01
195 \param A11 Input - A_11
196 \param ev0 Output - smallest eigenvalue
197 \param ev1 Output - eigenvalue
198 \param V00 Output - eigenvector componenent coresponding to ev0
199 \param V10 Output - eigenvector componenent coresponding to ev0
200 \param V01 Output - eigenvector componenent coresponding to ev1
201 \param V11 Output - eigenvector componenent coresponding to ev1
202 \param tol Input - tolerance to identify to eigenvalues
203 */
204 inline
205 void eigenvalues_and_eigenvectors2(const double A00,const double A01,const double A11,
206 double* ev0, double* ev1,
207 double* V00, double* V10, double* V01, double* V11,
208 const double tol)
209 {
210 double TEMP0,TEMP1;
211 eigenvalues2(A00,A01,A11,ev0,ev1);
212 const register double absev0=fabs(*ev0);
213 const register double absev1=fabs(*ev1);
214 register double max_ev=absev0>absev1 ? absev0 : absev1;
215 if (fabs((*ev0)-(*ev1))<tol*max_ev) {
216 *V00=1.;
217 *V10=0.;
218 *V01=0.;
219 *V11=1.;
220 } else {
221 vectorInKernel2(A00-(*ev0),A01,A01,A11-(*ev0),&TEMP0,&TEMP1);
222 const register double scale=1./sqrt(TEMP0*TEMP0+TEMP1*TEMP1);
223 if (TEMP0<0.) {
224 *V00=-TEMP0*scale;
225 *V10=-TEMP1*scale;
226 if (TEMP1<0.) {
227 *V01= *V10;
228 *V11=-(*V00);
229 } else {
230 *V01=-(*V10);
231 *V11= (*V10);
232 }
233 } else if (TEMP0>0.) {
234 *V00=TEMP0*scale;
235 *V10=TEMP1*scale;
236 if (TEMP1<0.) {
237 *V01=-(*V10);
238 *V11= (*V00);
239 } else {
240 *V01= (*V10);
241 *V11=-(*V00);
242 }
243 } else {
244 *V00=0.;
245 *V10=1;
246 *V11=0.;
247 *V01=1.;
248 }
249 }
250 }
251 /**
252 \brief
253 nomalizes a 3-d vector such that length is one and first non-zero component is positive.
254
255 \param V0 - vector componenent
256 \param V1 - vector componenent
257 \param V2 - vector componenent
258 */
259 inline
260 void normalizeVector3(double* V0,double* V1,double* V2)
261 {
262 register double s;
263 if (*V0>0) {
264 s=1./sqrt((*V0)*(*V0)+(*V1)*(*V1)+(*V2)*(*V2));
265 *V0*=s;
266 *V1*=s;
267 *V2*=s;
268 } else if (*V0<0) {
269 s=-1./sqrt((*V0)*(*V0)+(*V1)*(*V1)+(*V2)*(*V2));
270 *V0*=s;
271 *V1*=s;
272 *V2*=s;
273 } else {
274 if (*V1>0) {
275 s=1./sqrt((*V1)*(*V1)+(*V2)*(*V2));
276 *V1*=s;
277 *V2*=s;
278 } else if (*V1<0) {
279 s=-1./sqrt((*V1)*(*V1)+(*V2)*(*V2));
280 *V1*=s;
281 *V2*=s;
282 } else {
283 *V2=1.;
284 }
285 }
286 }
287 /**
288 \brief
289 solves a 2x2 eigenvalue A*V=ev*V problem for symmetric A. Eigenvectors are
290 ordered by increasing value and eigen vectors are normalizeVector3d such that
291 length is zero and first non-zero component is positive.
292
293 \param A00 Input - A_00
294 \param A01 Input - A_01
295 \param A11 Input - A_11
296 \param ev0 Output - smallest eigenvalue
297 \param ev1 Output - eigenvalue
298 \param V00 Output - eigenvector componenent coresponding to ev0
299 \param V10 Output - eigenvector componenent coresponding to ev0
300 \param V01 Output - eigenvector componenent coresponding to ev1
301 \param V11 Output - eigenvector componenent coresponding to ev1
302 \param tol Input - tolerance to identify to eigenvalues
303 */
304 inline
305 void eigenvalues_and_eigenvectors3(const double A00, const double A01, const double A02,
306 const double A11, const double A12, const double A22,
307 double* ev0, double* ev1, double* ev2,
308 double* V00, double* V10, double* V20,
309 double* V01, double* V11, double* V21,
310 double* V02, double* V12, double* V22,
311 const double tol)
312 {
313 register const double absA01=fabs(A01);
314 register const double absA02=fabs(A02);
315 register const double m=absA01>absA02 ? absA01 : absA02;
316 if (m<=0) {
317 double TEMP_V00,TEMP_V10,TEMP_V01,TEMP_V11,TEMP_EV0,TEMP_EV1;
318 eigenvalues_and_eigenvectors2(A11,A12,A22,
319 &TEMP_EV0,&TEMP_EV1,
320 &TEMP_V00,&TEMP_V10,&TEMP_V01,&TEMP_V11,tol);
321 if (A00<=TEMP_EV0) {
322 *V00=1.;
323 *V10=0.;
324 *V20=0.;
325 *V01=0.;
326 *V11=TEMP_V00;
327 *V21=TEMP_V10;
328 *V02=0.;
329 *V12=TEMP_V01;
330 *V22=TEMP_V11;
331 *ev0=A00;
332 *ev1=TEMP_EV0;
333 *ev2=TEMP_EV1;
334 } else if (A00>TEMP_EV1) {
335 *V02=1.;
336 *V12=0.;
337 *V22=0.;
338 *V00=0.;
339 *V10=TEMP_V00;
340 *V20=TEMP_V10;
341 *V01=0.;
342 *V11=TEMP_V01;
343 *V21=TEMP_V11;
344 *ev0=TEMP_EV0;
345 *ev1=TEMP_EV1;
346 *ev2=A00;
347 } else {
348 *V01=1.;
349 *V11=0.;
350 *V21=0.;
351 *V00=0.;
352 *V10=TEMP_V00;
353 *V20=TEMP_V10;
354 *V02=0.;
355 *V12=TEMP_V01;
356 *V22=TEMP_V11;
357 *ev0=TEMP_EV0;
358 *ev1=A00;
359 *ev2=TEMP_EV1;
360 }
361 } else {
362 eigenvalues3(A00,A01,A02,A11,A12,A22,ev0,ev1,ev2);
363 const register double absev0=fabs(*ev0);
364 const register double absev1=fabs(*ev1);
365 const register double absev2=fabs(*ev2);
366 register double max_ev=absev0>absev1 ? absev0 : absev1;
367 max_ev=max_ev>absev2 ? max_ev : absev2;
368 const register double d_01=fabs((*ev0)-(*ev1));
369 const register double d_12=fabs((*ev1)-(*ev2));
370 const register double max_d=d_01>d_12 ? d_01 : d_12;
371 if (max_d<=tol*max_ev) {
372 *V00=1.;
373 *V10=0;
374 *V20=0;
375 *V01=0;
376 *V11=1.;
377 *V21=0;
378 *V02=0;
379 *V12=0;
380 *V22=1.;
381 } else {
382 const register double S00=A00-(*ev0);
383 const register double absS00=fabs(S00);
384 if (fabs(S00)>m) {
385 vectorInKernel3__nonZeroA00(S00,A01,A02,A01,A11-(*ev0),A12,A02,A12,A22-(*ev0),V00,V10,V20);
386 } else if (absA02<m) {
387 vectorInKernel3__nonZeroA00(A01,A11-(*ev0),A12,S00,A01,A02,A02,A12,A22-(*ev0),V00,V10,V20);
388 } else {
389 vectorInKernel3__nonZeroA00(A02,A12,A22-(*ev0),S00,A01,A02,A01,A11-(*ev0),A12,V00,V10,V20);
390 }
391 normalizeVector3(V00,V10,V20);;
392 const register double T00=A00-(*ev2);
393 const register double absT00=fabs(T00);
394 if (fabs(T00)>m) {
395 vectorInKernel3__nonZeroA00(T00,A01,A02,A01,A11-(*ev2),A12,A02,A12,A22-(*ev2),V02,V12,V22);
396 } else if (absA02<m) {
397 vectorInKernel3__nonZeroA00(A01,A11-(*ev2),A12,T00,A01,A02,A02,A12,A22-(*ev2),V02,V12,V22);
398 } else {
399 vectorInKernel3__nonZeroA00(A02,A12,A22-(*ev2),T00,A01,A02,A01,A11-(*ev2),A12,V02,V12,V22);
400 }
401 const register double dot=(*V02)*(*V00)+(*V12)*(*V10)+(*V22)*(*V20);
402 *V02-=dot*(*V00);
403 *V12-=dot*(*V10);
404 *V22-=dot*(*V20);
405 normalizeVector3(V02,V12,V22);
406 *V01=(*V10)*(*V22)-(*V12)*(*V20);
407 *V11=(*V20)*(*V02)-(*V00)*(*V22);
408 *V21=(*V00)*(*V12)-(*V02)*(*V10);
409 normalizeVector3(V01,V11,V21);
410 }
411 }
412 }
413 } // end of namespace
414 #endif

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