Actual source code: svdsolve.c
slepc-main 2024-11-09
1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-, Universitat Politecnica de Valencia, Spain
6: This file is part of SLEPc.
7: SLEPc is distributed under a 2-clause BSD license (see LICENSE).
8: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
9: */
10: /*
11: SVD routines related to the solution process
12: */
14: #include <slepc/private/svdimpl.h>
15: #include <slepc/private/bvimpl.h>
17: /*
18: SVDComputeVectors_Left - Compute left singular vectors as U=A*V.
19: Only done if the leftbasis flag is false. Assumes V is available.
20: */
21: PetscErrorCode SVDComputeVectors_Left(SVD svd)
22: {
23: Vec tl,omega2,u,v,w;
24: PetscInt i,oldsize;
25: VecType vtype;
26: const PetscScalar* varray;
28: PetscFunctionBegin;
29: if (!svd->leftbasis) {
30: /* generate left singular vectors on U */
31: if (!svd->U) PetscCall(SVDGetBV(svd,NULL,&svd->U));
32: PetscCall(BVGetSizes(svd->U,NULL,NULL,&oldsize));
33: if (!oldsize) {
34: if (!((PetscObject)svd->U)->type_name) PetscCall(BVSetType(svd->U,((PetscObject)svd->V)->type_name));
35: PetscCall(MatCreateVecsEmpty(svd->A,NULL,&tl));
36: PetscCall(BVSetSizesFromVec(svd->U,tl,svd->ncv));
37: PetscCall(VecDestroy(&tl));
38: }
39: PetscCall(BVSetActiveColumns(svd->V,0,svd->nconv));
40: PetscCall(BVSetActiveColumns(svd->U,0,svd->nconv));
41: if (!svd->ishyperbolic) PetscCall(BVMatMult(svd->V,svd->A,svd->U));
42: else if (svd->swapped) { /* compute right singular vectors as V=A'*Omega*U */
43: PetscCall(MatCreateVecs(svd->A,&w,NULL));
44: for (i=0;i<svd->nconv;i++) {
45: PetscCall(BVGetColumn(svd->V,i,&v));
46: PetscCall(BVGetColumn(svd->U,i,&u));
47: PetscCall(VecPointwiseMult(w,v,svd->omega));
48: PetscCall(MatMult(svd->A,w,u));
49: PetscCall(BVRestoreColumn(svd->V,i,&v));
50: PetscCall(BVRestoreColumn(svd->U,i,&u));
51: }
52: PetscCall(VecDestroy(&w));
53: } else { /* compute left singular vectors as usual U=A*V, and set-up Omega-orthogonalization of U */
54: PetscCall(BV_SetMatrixDiagonal(svd->U,svd->omega,svd->A));
55: PetscCall(BVMatMult(svd->V,svd->A,svd->U));
56: }
57: PetscCall(BVOrthogonalize(svd->U,NULL));
58: if (svd->ishyperbolic && !svd->swapped) { /* store signature after Omega-orthogonalization */
59: PetscCall(MatGetVecType(svd->A,&vtype));
60: PetscCall(VecCreate(PETSC_COMM_SELF,&omega2));
61: PetscCall(VecSetSizes(omega2,svd->nconv,svd->nconv));
62: PetscCall(VecSetType(omega2,vtype));
63: PetscCall(BVGetSignature(svd->U,omega2));
64: PetscCall(VecGetArrayRead(omega2,&varray));
65: for (i=0;i<svd->nconv;i++) {
66: svd->sign[i] = PetscRealPart(varray[i]);
67: if (PetscRealPart(varray[i])<0.0) PetscCall(BVScaleColumn(svd->U,i,-1.0));
68: }
69: PetscCall(VecRestoreArrayRead(omega2,&varray));
70: PetscCall(VecDestroy(&omega2));
71: }
72: }
73: PetscFunctionReturn(PETSC_SUCCESS);
74: }
76: PetscErrorCode SVDComputeVectors(SVD svd)
77: {
78: PetscFunctionBegin;
79: SVDCheckSolved(svd,1);
80: if (svd->state==SVD_STATE_SOLVED) PetscTryTypeMethod(svd,computevectors);
81: svd->state = SVD_STATE_VECTORS;
82: PetscFunctionReturn(PETSC_SUCCESS);
83: }
85: /*@
86: SVDSolve - Solves the singular value problem.
88: Collective
90: Input Parameter:
91: . svd - singular value solver context obtained from SVDCreate()
93: Options Database Keys:
94: + -svd_view - print information about the solver used
95: . -svd_view_mat0 - view the first matrix (A)
96: . -svd_view_mat1 - view the second matrix (B)
97: . -svd_view_signature - view the signature matrix (omega)
98: . -svd_view_vectors - view the computed singular vectors
99: . -svd_view_values - view the computed singular values
100: . -svd_converged_reason - print reason for convergence, and number of iterations
101: . -svd_error_absolute - print absolute errors of each singular triplet
102: . -svd_error_relative - print relative errors of each singular triplet
103: - -svd_error_norm - print errors relative to the matrix norms of each singular triplet
105: Notes:
106: All the command-line options listed above admit an optional argument specifying
107: the viewer type and options. For instance, use '-svd_view_mat0 binary:amatrix.bin'
108: to save the A matrix to a binary file, '-svd_view_values draw' to draw the computed
109: singular values graphically, or '-svd_error_relative :myerr.m:ascii_matlab' to save
110: the errors in a file that can be executed in Matlab.
112: Level: beginner
114: .seealso: SVDCreate(), SVDSetUp(), SVDDestroy()
115: @*/
116: PetscErrorCode SVDSolve(SVD svd)
117: {
118: PetscInt i,*workperm;
120: PetscFunctionBegin;
122: if (svd->state>=SVD_STATE_SOLVED) PetscFunctionReturn(PETSC_SUCCESS);
123: PetscCall(PetscLogEventBegin(SVD_Solve,svd,0,0,0));
125: /* call setup */
126: PetscCall(SVDSetUp(svd));
127: svd->its = 0;
128: svd->nconv = 0;
129: for (i=0;i<svd->ncv;i++) {
130: svd->sigma[i] = 0.0;
131: svd->errest[i] = 0.0;
132: svd->perm[i] = i;
133: }
134: PetscCall(SVDViewFromOptions(svd,NULL,"-svd_view_pre"));
136: switch (svd->problem_type) {
137: case SVD_STANDARD:
138: PetscUseTypeMethod(svd,solve);
139: break;
140: case SVD_GENERALIZED:
141: PetscUseTypeMethod(svd,solveg);
142: break;
143: case SVD_HYPERBOLIC:
144: PetscUseTypeMethod(svd,solveh);
145: break;
146: }
147: svd->state = SVD_STATE_SOLVED;
149: /* sort singular triplets */
150: if (svd->which == SVD_SMALLEST) PetscCall(PetscSortRealWithPermutation(svd->nconv,svd->sigma,svd->perm));
151: else {
152: PetscCall(PetscMalloc1(svd->nconv,&workperm));
153: for (i=0;i<svd->nconv;i++) workperm[i] = i;
154: PetscCall(PetscSortRealWithPermutation(svd->nconv,svd->sigma,workperm));
155: for (i=0;i<svd->nconv;i++) svd->perm[i] = workperm[svd->nconv-i-1];
156: PetscCall(PetscFree(workperm));
157: }
158: PetscCall(PetscLogEventEnd(SVD_Solve,svd,0,0,0));
160: /* various viewers */
161: PetscCall(SVDViewFromOptions(svd,NULL,"-svd_view"));
162: PetscCall(SVDConvergedReasonViewFromOptions(svd));
163: PetscCall(SVDErrorViewFromOptions(svd));
164: PetscCall(SVDValuesViewFromOptions(svd));
165: PetscCall(SVDVectorsViewFromOptions(svd));
166: PetscCall(MatViewFromOptions(svd->OP,(PetscObject)svd,"-svd_view_mat0"));
167: if (svd->isgeneralized) PetscCall(MatViewFromOptions(svd->OPb,(PetscObject)svd,"-svd_view_mat1"));
168: if (svd->ishyperbolic) PetscCall(VecViewFromOptions(svd->omega,(PetscObject)svd,"-svd_view_signature"));
170: /* Remove the initial subspaces */
171: svd->nini = 0;
172: svd->ninil = 0;
173: PetscFunctionReturn(PETSC_SUCCESS);
174: }
176: /*@
177: SVDGetIterationNumber - Gets the current iteration number. If the
178: call to SVDSolve() is complete, then it returns the number of iterations
179: carried out by the solution method.
181: Not Collective
183: Input Parameter:
184: . svd - the singular value solver context
186: Output Parameter:
187: . its - number of iterations
189: Note:
190: During the i-th iteration this call returns i-1. If SVDSolve() is
191: complete, then parameter "its" contains either the iteration number at
192: which convergence was successfully reached, or failure was detected.
193: Call SVDGetConvergedReason() to determine if the solver converged or
194: failed and why.
196: Level: intermediate
198: .seealso: SVDGetConvergedReason(), SVDSetTolerances()
199: @*/
200: PetscErrorCode SVDGetIterationNumber(SVD svd,PetscInt *its)
201: {
202: PetscFunctionBegin;
204: PetscAssertPointer(its,2);
205: *its = svd->its;
206: PetscFunctionReturn(PETSC_SUCCESS);
207: }
209: /*@
210: SVDGetConvergedReason - Gets the reason why the SVDSolve() iteration was
211: stopped.
213: Not Collective
215: Input Parameter:
216: . svd - the singular value solver context
218: Output Parameter:
219: . reason - negative value indicates diverged, positive value converged
220: (see SVDConvergedReason)
222: Options Database Key:
223: . -svd_converged_reason - print the reason to a viewer
225: Notes:
226: Possible values for reason are
227: + SVD_CONVERGED_TOL - converged up to tolerance
228: . SVD_CONVERGED_USER - converged due to a user-defined condition
229: . SVD_CONVERGED_MAXIT - reached the maximum number of iterations with SVD_CONV_MAXIT criterion
230: . SVD_DIVERGED_ITS - required more than max_it iterations to reach convergence
231: . SVD_DIVERGED_BREAKDOWN - generic breakdown in method
232: - SVD_DIVERGED_SYMMETRY_LOST - underlying indefinite eigensolver was not able to keep symmetry
234: Can only be called after the call to SVDSolve() is complete.
236: Level: intermediate
238: .seealso: SVDSetTolerances(), SVDSolve(), SVDConvergedReason
239: @*/
240: PetscErrorCode SVDGetConvergedReason(SVD svd,SVDConvergedReason *reason)
241: {
242: PetscFunctionBegin;
244: PetscAssertPointer(reason,2);
245: SVDCheckSolved(svd,1);
246: *reason = svd->reason;
247: PetscFunctionReturn(PETSC_SUCCESS);
248: }
250: /*@
251: SVDGetConverged - Gets the number of converged singular values.
253: Not Collective
255: Input Parameter:
256: . svd - the singular value solver context
258: Output Parameter:
259: . nconv - number of converged singular values
261: Note:
262: This function should be called after SVDSolve() has finished.
264: Level: beginner
266: .seealso: SVDSetDimensions(), SVDSolve(), SVDGetSingularTriplet()
267: @*/
268: PetscErrorCode SVDGetConverged(SVD svd,PetscInt *nconv)
269: {
270: PetscFunctionBegin;
272: PetscAssertPointer(nconv,2);
273: SVDCheckSolved(svd,1);
274: *nconv = svd->nconv;
275: PetscFunctionReturn(PETSC_SUCCESS);
276: }
278: /*@
279: SVDGetSingularTriplet - Gets the i-th triplet of the singular value decomposition
280: as computed by SVDSolve(). The solution consists in the singular value and its left
281: and right singular vectors.
283: Collective
285: Input Parameters:
286: + svd - singular value solver context
287: - i - index of the solution
289: Output Parameters:
290: + sigma - singular value
291: . u - left singular vector
292: - v - right singular vector
294: Note:
295: Both u or v can be NULL if singular vectors are not required.
296: Otherwise, the caller must provide valid Vec objects, i.e.,
297: they must be created by the calling program with e.g. MatCreateVecs().
299: The index i should be a value between 0 and nconv-1 (see SVDGetConverged()).
300: Singular triplets are indexed according to the ordering criterion established
301: with SVDSetWhichSingularTriplets().
303: In the case of GSVD, the solution consists in three vectors u,v,x that are
304: returned as follows. Vector x is returned in the right singular vector
305: (argument v) and has length equal to the number of columns of A and B.
306: The other two vectors are returned stacked on top of each other [u;v] in
307: the left singular vector argument, with length equal to m+n (number of rows
308: of A plus number of rows of B).
310: Level: beginner
312: .seealso: SVDSolve(), SVDGetConverged(), SVDSetWhichSingularTriplets()
313: @*/
314: PetscErrorCode SVDGetSingularTriplet(SVD svd,PetscInt i,PetscReal *sigma,Vec u,Vec v)
315: {
316: PetscInt M,N;
317: Vec w;
319: PetscFunctionBegin;
322: SVDCheckSolved(svd,1);
325: PetscCheck(i>=0,PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_OUTOFRANGE,"The index cannot be negative");
326: PetscCheck(i<svd->nconv,PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_OUTOFRANGE,"The index can be nconv-1 at most, see SVDGetConverged()");
327: if (sigma) *sigma = svd->sigma[svd->perm[i]];
328: if (u || v) {
329: if (!svd->isgeneralized) {
330: PetscCall(MatGetSize(svd->OP,&M,&N));
331: if (M<N) { w = u; u = v; v = w; }
332: }
333: PetscCall(SVDComputeVectors(svd));
334: if (u) PetscCall(BVCopyVec(svd->U,svd->perm[i],u));
335: if (v) PetscCall(BVCopyVec(svd->V,svd->perm[i],v));
336: }
337: PetscFunctionReturn(PETSC_SUCCESS);
338: }
340: /*
341: SVDComputeResidualNorms_Standard - Computes the norms of the left and
342: right residuals associated with the i-th computed singular triplet.
344: Input Parameters:
345: sigma - singular value
346: u,v - singular vectors
347: x,y - two work vectors with the same dimensions as u,v
348: */
349: static PetscErrorCode SVDComputeResidualNorms_Standard(SVD svd,PetscReal sigma,Vec u,Vec v,Vec x,Vec y,PetscReal *norm1,PetscReal *norm2)
350: {
351: PetscInt M,N;
353: PetscFunctionBegin;
354: /* norm1 = ||A*v-sigma*u||_2 */
355: if (norm1) {
356: PetscCall(MatMult(svd->OP,v,x));
357: PetscCall(VecAXPY(x,-sigma,u));
358: PetscCall(VecNorm(x,NORM_2,norm1));
359: }
360: /* norm2 = ||A^T*u-sigma*v||_2 */
361: if (norm2) {
362: PetscCall(MatGetSize(svd->OP,&M,&N));
363: if (M<N) PetscCall(MatMult(svd->A,u,y));
364: else PetscCall(MatMult(svd->AT,u,y));
365: PetscCall(VecAXPY(y,-sigma,v));
366: PetscCall(VecNorm(y,NORM_2,norm2));
367: }
368: PetscFunctionReturn(PETSC_SUCCESS);
369: }
371: /*
372: SVDComputeResidualNorms_Generalized - In GSVD, compute the residual norms
373: norm1 = ||s^2*A'*u-c*B'*B*x||_2 and norm2 = ||c^2*B'*v-s*A'*A*x||_2.
375: Input Parameters:
376: sigma - singular value
377: uv - left singular vectors [u;v]
378: x - right singular vector
379: y,z - two work vectors with the same dimension as x
380: */
381: static PetscErrorCode SVDComputeResidualNorms_Generalized(SVD svd,PetscReal sigma,Vec uv,Vec x,Vec y,Vec z,PetscReal *norm1,PetscReal *norm2)
382: {
383: Vec u,v,ax,bx,nest,aux[2];
384: PetscReal c,s;
386: PetscFunctionBegin;
387: PetscCall(MatCreateVecs(svd->OP,NULL,&u));
388: PetscCall(MatCreateVecs(svd->OPb,NULL,&v));
389: aux[0] = u;
390: aux[1] = v;
391: PetscCall(VecCreateNest(PetscObjectComm((PetscObject)svd),2,NULL,aux,&nest));
392: PetscCall(VecCopy(uv,nest));
394: s = 1.0/PetscSqrtReal(1.0+sigma*sigma);
395: c = sigma*s;
397: /* norm1 = ||s^2*A'*u-c*B'*B*x||_2 */
398: if (norm1) {
399: PetscCall(VecDuplicate(v,&bx));
400: PetscCall(MatMultHermitianTranspose(svd->OP,u,z));
401: PetscCall(MatMult(svd->OPb,x,bx));
402: PetscCall(MatMultHermitianTranspose(svd->OPb,bx,y));
403: PetscCall(VecAXPBY(y,s*s,-c,z));
404: PetscCall(VecNorm(y,NORM_2,norm1));
405: PetscCall(VecDestroy(&bx));
406: }
407: /* norm2 = ||c^2*B'*v-s*A'*A*x||_2 */
408: if (norm2) {
409: PetscCall(VecDuplicate(u,&ax));
410: PetscCall(MatMultHermitianTranspose(svd->OPb,v,z));
411: PetscCall(MatMult(svd->OP,x,ax));
412: PetscCall(MatMultHermitianTranspose(svd->OP,ax,y));
413: PetscCall(VecAXPBY(y,c*c,-s,z));
414: PetscCall(VecNorm(y,NORM_2,norm2));
415: PetscCall(VecDestroy(&ax));
416: }
418: PetscCall(VecDestroy(&v));
419: PetscCall(VecDestroy(&u));
420: PetscCall(VecDestroy(&nest));
421: PetscFunctionReturn(PETSC_SUCCESS);
422: }
424: /*
425: SVDComputeResidualNorms_Hyperbolic - Computes the norms of the left and
426: right residuals associated with the i-th computed singular triplet.
428: Input Parameters:
429: sigma - singular value
430: sign - corresponding element of the signature Omega2
431: u,v - singular vectors
432: x,y,z - three work vectors with the same dimensions as u,v,u
433: */
434: static PetscErrorCode SVDComputeResidualNorms_Hyperbolic(SVD svd,PetscReal sigma,PetscReal sign,Vec u,Vec v,Vec x,Vec y,Vec z,PetscReal *norm1,PetscReal *norm2)
435: {
436: PetscInt M,N;
438: PetscFunctionBegin;
439: /* norm1 = ||A*v-sigma*u||_2 */
440: if (norm1) {
441: PetscCall(MatMult(svd->OP,v,x));
442: PetscCall(VecAXPY(x,-sigma,u));
443: PetscCall(VecNorm(x,NORM_2,norm1));
444: }
445: /* norm2 = ||A^T*Omega*u-sigma*sign*v||_2 */
446: if (norm2) {
447: PetscCall(MatGetSize(svd->OP,&M,&N));
448: PetscCall(VecPointwiseMult(z,u,svd->omega));
449: if (M<N) PetscCall(MatMult(svd->A,z,y));
450: else PetscCall(MatMult(svd->AT,z,y));
451: PetscCall(VecAXPY(y,-sigma*sign,v));
452: PetscCall(VecNorm(y,NORM_2,norm2));
453: }
454: PetscFunctionReturn(PETSC_SUCCESS);
455: }
457: /*@
458: SVDComputeError - Computes the error (based on the residual norm) associated
459: with the i-th singular triplet.
461: Collective
463: Input Parameters:
464: + svd - the singular value solver context
465: . i - the solution index
466: - type - the type of error to compute
468: Output Parameter:
469: . error - the error
471: Notes:
472: The error can be computed in various ways, all of them based on the residual
473: norm obtained as sqrt(n1^2+n2^2) with n1 = ||A*v-sigma*u||_2 and
474: n2 = ||A^T*u-sigma*v||_2, where sigma is the singular value, u is the left
475: singular vector and v is the right singular vector.
477: In the case of the GSVD, the two components of the residual norm are
478: n1 = ||s^2*A'*u-c*B'*B*x||_2 and n2 = ||c^2*B'*v-s*A'*A*x||_2, where [u;v]
479: are the left singular vectors and x is the right singular vector, with
480: sigma=c/s.
482: Level: beginner
484: .seealso: SVDErrorType, SVDSolve()
485: @*/
486: PetscErrorCode SVDComputeError(SVD svd,PetscInt i,SVDErrorType type,PetscReal *error)
487: {
488: PetscReal sigma,norm1,norm2,c,s;
489: Vec u=NULL,v=NULL,x=NULL,y=NULL,z=NULL;
490: PetscReal vecnorm=1.0;
492: PetscFunctionBegin;
496: PetscAssertPointer(error,4);
497: SVDCheckSolved(svd,1);
499: /* allocate work vectors */
500: switch (svd->problem_type) {
501: case SVD_STANDARD:
502: PetscCall(SVDSetWorkVecs(svd,2,2));
503: u = svd->workl[0];
504: v = svd->workr[0];
505: x = svd->workl[1];
506: y = svd->workr[1];
507: break;
508: case SVD_GENERALIZED:
509: PetscCall(SVDSetWorkVecs(svd,1,3));
510: u = svd->workl[0];
511: v = svd->workr[0];
512: x = svd->workr[1];
513: y = svd->workr[2];
514: break;
515: case SVD_HYPERBOLIC:
516: PetscCall(SVDSetWorkVecs(svd,3,2));
517: u = svd->workl[0];
518: v = svd->workr[0];
519: x = svd->workl[1];
520: y = svd->workr[1];
521: z = svd->workl[2];
522: break;
523: }
525: /* compute residual norm */
526: PetscCall(SVDGetSingularTriplet(svd,i,&sigma,u,v));
527: switch (svd->problem_type) {
528: case SVD_STANDARD:
529: PetscCall(SVDComputeResidualNorms_Standard(svd,sigma,u,v,x,y,&norm1,&norm2));
530: break;
531: case SVD_GENERALIZED:
532: PetscCall(SVDComputeResidualNorms_Generalized(svd,sigma,u,v,x,y,&norm1,&norm2));
533: break;
534: case SVD_HYPERBOLIC:
535: PetscCall(SVDComputeResidualNorms_Hyperbolic(svd,sigma,svd->sign[svd->perm[i]],u,v,x,y,z,&norm1,&norm2));
536: break;
537: }
538: *error = SlepcAbs(norm1,norm2);
540: /* compute 2-norm of eigenvector of the cyclic form */
541: if (type!=SVD_ERROR_ABSOLUTE) {
542: switch (svd->problem_type) {
543: case SVD_STANDARD:
544: vecnorm = PETSC_SQRT2;
545: break;
546: case SVD_GENERALIZED:
547: PetscCall(VecNorm(v,NORM_2,&vecnorm));
548: vecnorm = PetscSqrtReal(1.0+vecnorm*vecnorm);
549: break;
550: case SVD_HYPERBOLIC:
551: PetscCall(VecNorm(u,NORM_2,&vecnorm));
552: vecnorm = PetscSqrtReal(1.0+vecnorm*vecnorm);
553: break;
554: }
555: }
557: /* compute error */
558: switch (type) {
559: case SVD_ERROR_ABSOLUTE:
560: break;
561: case SVD_ERROR_RELATIVE:
562: if (svd->isgeneralized) {
563: s = 1.0/PetscSqrtReal(1.0+sigma*sigma);
564: c = sigma*s;
565: norm1 /= c*vecnorm;
566: norm2 /= s*vecnorm;
567: *error = PetscMax(norm1,norm2);
568: } else *error /= sigma*vecnorm;
569: break;
570: case SVD_ERROR_NORM:
571: if (!svd->nrma) PetscCall(MatNorm(svd->OP,NORM_INFINITY,&svd->nrma));
572: if (svd->isgeneralized && !svd->nrmb) PetscCall(MatNorm(svd->OPb,NORM_INFINITY,&svd->nrmb));
573: *error /= PetscMax(svd->nrma,svd->nrmb)*vecnorm;
574: break;
575: default:
576: SETERRQ(PetscObjectComm((PetscObject)svd),PETSC_ERR_ARG_OUTOFRANGE,"Invalid error type");
577: }
578: PetscFunctionReturn(PETSC_SUCCESS);
579: }