Date: 2021-02-01T14:42:20+00:00

Actual source code: mfnkrylov.c
slepc-3.14.2 2021-02-01
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2020, 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==PETSC_DEFAULT) mfn->ncv = PetscMin(30,N);
     38:   if (mfn->max_it==PETSC_DEFAULT) mfn->max_it = 100;
     39:   MFNAllocateSolution(mfn,1);
     40:   return(0);
     41: }

     43: PetscErrorCode MFNSolve_Krylov(MFN mfn,Vec b,Vec x)
     44: {
     46:   PetscInt       n=0,m,ld,ldh,j;
     47:   PetscBLASInt   m_,inc=1;
     48:   Mat            G=NULL,H=NULL;
     49:   Vec            F=NULL;
     50:   PetscScalar    *array,*farray,*garray,*harray;
     51:   PetscReal      beta,betaold=0.0,nrm=1.0;
     52:   PetscBool      breakdown,set,flg,symm=PETSC_FALSE;

     55:   m  = mfn->ncv;
     56:   ld = m+1;
     57:   PetscCalloc1(ld*ld,&array);

     59:   /* set initial vector to b/||b|| */
     60:   BVInsertVec(mfn->V,0,b);
     61:   BVScaleColumn(mfn->V,0,1.0/mfn->bnorm);
     62:   VecSet(x,0.0);

     64:   /* Restart loop */
     65:   while (mfn->reason == MFN_CONVERGED_ITERATING) {
     66:     mfn->its++;

     68:     /* compute Arnoldi factorization */
     69:     BVMatArnoldi(mfn->V,mfn->transpose_solve?mfn->AT:mfn->A,array,ld,0,&m,&beta,&breakdown);

     71:     /* save previous Hessenberg matrix in G; allocate new storage for H and f(H) */
     72:     if (mfn->its>1) { G = H; H = NULL; }
     73:     ldh = n+m;
     74:     MFN_CreateVec(ldh,&F);
     75:     MFN_CreateDenseMat(ldh,&H);

     77:     /* glue together the previous H and the new H obtained with Arnoldi */
     78:     MatDenseGetArray(H,&harray);
     79:     for (j=0;j<m;j++) {
     80:       PetscArraycpy(harray+n+(j+n)*ldh,array+j*ld,m);
     81:     }
     82:     if (mfn->its>1) {
     83:       MatDenseGetArray(G,&garray);
     84:       for (j=0;j<n;j++) {
     85:         PetscArraycpy(harray+j*ldh,garray+j*n,n);
     86:       }
     87:       MatDenseRestoreArray(G,&garray);
     88:       MatDestroy(&G);
     89:       harray[n+(n-1)*ldh] = betaold;
     90:     }
     91:     MatDenseRestoreArray(H,&harray);

     93:     if (mfn->its==1) {
     94:       /* set symmetry flag of H from A */
     95:       MatIsHermitianKnown(mfn->A,&set,&flg);
     96:       symm = set? flg: PETSC_FALSE;
     97:       if (symm) {
     98:         MatSetOption(H,MAT_HERMITIAN,PETSC_TRUE);
     99:       }
    100:     }

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

    105:     /* x += ||b||*V*f(H)*e_1 */
    106:     VecGetArray(F,&farray);
    107:     PetscBLASIntCast(m,&m_);
    108:     nrm = BLASnrm2_(&m_,farray+n,&inc);   /* relative norm of the update ||u||/||b|| */
    109:     MFNMonitor(mfn,mfn->its,nrm);
    110:     for (j=0;j<m;j++) farray[j+n] *= mfn->bnorm;
    111:     BVSetActiveColumns(mfn->V,0,m);
    112:     BVMultVec(mfn->V,1.0,1.0,x,farray+n);
    113:     VecRestoreArray(F,&farray);

    115:     /* check convergence */
    116:     if (mfn->its >= mfn->max_it) mfn->reason = MFN_DIVERGED_ITS;
    117:     if (mfn->its>1) {
    118:       if (m<mfn->ncv || breakdown || beta==0.0 || nrm<mfn->tol) mfn->reason = MFN_CONVERGED_TOL;
    119:     }

    121:     /* restart with vector v_{m+1} */
    122:     if (mfn->reason == MFN_CONVERGED_ITERATING) {
    123:       BVCopyColumn(mfn->V,m,0);
    124:       n += m;
    125:       betaold = beta;
    126:     }
    127:   }

    129:   MatDestroy(&H);
    130:   MatDestroy(&G);
    131:   VecDestroy(&F);
    132:   PetscFree(array);
    133:   return(0);
    134: }

    136: SLEPC_EXTERN PetscErrorCode MFNCreate_Krylov(MFN mfn)
    137: {
    139:   mfn->ops->solve          = MFNSolve_Krylov;
    140:   mfn->ops->setup          = MFNSetUp_Krylov;
    141:   return(0);
    142: }