Actual source code: mfnkrylov.c

slepc-3.11.2 2019-07-30
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2019, 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:    SLEPc matrix function solver: "krylov"

 13:    Method: Arnoldi with Eiermann-Ernst restart

 15:    Algorithm:

 17:        Build Arnoldi approximations using f(H) for the Hessenberg matrix H,
 18:        restart by discarding the Krylov basis but keeping H.

 20:    References:

 22:        [1] M. Eiermann and O. Ernst, "A restarted Krylov subspace method
 23:            for the evaluation of matrix functions", SIAM J. Numer. Anal.
 24:            44(6):2481-2504, 2006.
 25: */

 27: #include <slepc/private/mfnimpl.h>
 28: #include <slepcblaslapack.h>

 30: PetscErrorCode MFNSetUp_Krylov(MFN mfn)
 31: {
 33:   PetscInt       N;

 36:   MatGetSize(mfn->A,&N,NULL);
 37:   if (!mfn->ncv) mfn->ncv = PetscMin(30,N);
 38:   if (!mfn->max_it) mfn->max_it = 100;
 39:   MFNAllocateSolution(mfn,1);
 40:   return(0);
 41: }

 43: PetscErrorCode MFNBasicArnoldi(MFN mfn,PetscScalar *H,PetscInt ldh,PetscInt k,PetscInt *M,PetscReal *beta,PetscBool *breakdown)
 44: {
 46:   PetscScalar    *a;
 47:   PetscInt       j,nc,n,m = *M;
 48:   Vec            vj,vj1,buf;

 51:   BVSetActiveColumns(mfn->V,0,m);
 52:   for (j=k;j<m;j++) {
 53:     BVGetColumn(mfn->V,j,&vj);
 54:     BVGetColumn(mfn->V,j+1,&vj1);
 55:     MatMult(mfn->A,vj,vj1);
 56:     BVRestoreColumn(mfn->V,j,&vj);
 57:     BVRestoreColumn(mfn->V,j+1,&vj1);
 58:     BVOrthonormalizeColumn(mfn->V,j+1,PETSC_FALSE,beta,breakdown);
 59:     if (*breakdown) {
 60:       *M = j+1;
 61:       break;
 62:     }
 63:   }
 64:   /* extract Hessenberg matrix from the BV object */
 65:   BVGetNumConstraints(mfn->V,&nc);
 66:   BVGetSizes(mfn->V,NULL,NULL,&n);
 67:   BVGetBufferVec(mfn->V,&buf);
 68:   VecGetArray(buf,&a);
 69:   for (j=k;j<*M;j++) {
 70:     PetscMemcpy(H+j*ldh,a+nc+(j+1)*(nc+n),(j+2)*sizeof(PetscScalar));
 71:   }
 72:   VecRestoreArray(buf,&a);
 73:   return(0);
 74: }

 76: PetscErrorCode MFNSolve_Krylov(MFN mfn,Vec b,Vec x)
 77: {
 79:   PetscInt       n=0,m,ld,ldh,j;
 80:   PetscBLASInt   m_,inc=1;
 81:   Mat            G=NULL,H=NULL;
 82:   Vec            F=NULL;
 83:   PetscScalar    *array,*farray,*garray,*harray;
 84:   PetscReal      beta,betaold=0.0,nrm=1.0;
 85:   PetscBool      breakdown,set,flg,symm=PETSC_FALSE;

 88:   m  = mfn->ncv;
 89:   ld = m+1;
 90:   PetscCalloc1(ld*ld,&array);

 92:   /* set initial vector to b/||b|| */
 93:   BVInsertVec(mfn->V,0,b);
 94:   BVScaleColumn(mfn->V,0,1.0/mfn->bnorm);
 95:   VecSet(x,0.0);

 97:   /* Restart loop */
 98:   while (mfn->reason == MFN_CONVERGED_ITERATING) {
 99:     mfn->its++;

101:     /* compute Arnoldi factorization */
102:     MFNBasicArnoldi(mfn,array,ld,0,&m,&beta,&breakdown);

104:     /* save previous Hessenberg matrix in G; allocate new storage for H and f(H) */
105:     if (mfn->its>1) { G = H; H = NULL; }
106:     ldh = n+m;
107:     MFN_CreateVec(ldh,&F);
108:     MFN_CreateDenseMat(ldh,&H);

110:     /* glue together the previous H and the new H obtained with Arnoldi */
111:     MatDenseGetArray(H,&harray);
112:     for (j=0;j<m;j++) {
113:       PetscMemcpy(harray+n+(j+n)*ldh,array+j*ld,m*sizeof(PetscScalar));
114:     }
115:     if (mfn->its>1) {
116:       MatDenseGetArray(G,&garray);
117:       for (j=0;j<n;j++) {
118:         PetscMemcpy(harray+j*ldh,garray+j*n,n*sizeof(PetscScalar));
119:       }
120:       MatDenseRestoreArray(G,&garray);
121:       MatDestroy(&G);
122:       harray[n+(n-1)*ldh] = betaold;
123:     }
124:     MatDenseRestoreArray(H,&harray);

126:     if (mfn->its==1) {
127:       /* set symmetry flag of H from A */
128:       MatIsHermitianKnown(mfn->A,&set,&flg);
129:       symm = set? flg: PETSC_FALSE;
130:       if (symm) {
131:         MatSetOption(H,MAT_HERMITIAN,PETSC_TRUE);
132:       }
133:     }

135:     /* evaluate f(H) */
136:     FNEvaluateFunctionMatVec(mfn->fn,H,F);

138:     /* x += ||b||*V*f(H)*e_1 */
139:     VecGetArray(F,&farray);
140:     PetscBLASIntCast(m,&m_);
141:     nrm = BLASnrm2_(&m_,farray+n,&inc);   /* relative norm of the update ||u||/||b|| */
142:     MFNMonitor(mfn,mfn->its,nrm);
143:     for (j=0;j<m;j++) farray[j+n] *= mfn->bnorm;
144:     BVSetActiveColumns(mfn->V,0,m);
145:     BVMultVec(mfn->V,1.0,1.0,x,farray+n);
146:     VecRestoreArray(F,&farray);

148:     /* check convergence */
149:     if (mfn->its >= mfn->max_it) mfn->reason = MFN_DIVERGED_ITS;
150:     if (mfn->its>1) {
151:       if (m<mfn->ncv || breakdown || beta==0.0 || nrm<mfn->tol) mfn->reason = MFN_CONVERGED_TOL;
152:     }

154:     /* restart with vector v_{m+1} */
155:     if (mfn->reason == MFN_CONVERGED_ITERATING) {
156:       BVCopyColumn(mfn->V,m,0);
157:       n += m;
158:       betaold = beta;
159:     }
160:   }

162:   MatDestroy(&H);
163:   MatDestroy(&G);
164:   VecDestroy(&F);
165:   PetscFree(array);
166:   return(0);
167: }

169: SLEPC_EXTERN PetscErrorCode MFNCreate_Krylov(MFN mfn)
170: {
172:   mfn->ops->solve          = MFNSolve_Krylov;
173:   mfn->ops->setup          = MFNSetUp_Krylov;
174:   return(0);
175: }