Date: 2022-11-02T11:46:21+00:00

Actual source code: bvkrylov.c
slepc-3.18.1 2022-11-02
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      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:    BV routines related to Krylov decompositions
     12: */

     14: #include <slepc/private/bvimpl.h>

     16: /*@
     17:    BVMatArnoldi - Computes an Arnoldi factorization associated with a matrix.

     19:    Collective on V

     21:    Input Parameters:
     22: +  V - basis vectors context
     23: .  A - the matrix
     24: .  H - (optional) the upper Hessenberg matrix
     25: .  k - number of locked columns
     26: -  m - dimension of the Arnoldi basis, may be modified

     28:    Output Parameters:
     29: +  beta - (optional) norm of last vector before normalization
     30: -  breakdown - (optional) flag indicating that breakdown occurred

     32:    Notes:
     33:    Computes an m-step Arnoldi factorization for matrix A. The first k columns
     34:    are assumed to be locked and therefore they are not modified. On exit, the
     35:    following relation is satisfied

     37: $                    A * V - V * H = beta*v_m * e_m^T

     39:    where the columns of V are the Arnoldi vectors (which are orthonormal), H is
     40:    an upper Hessenberg matrix, e_m is the m-th vector of the canonical basis.
     41:    On exit, beta contains the norm of V[m] before normalization.

     43:    The breakdown flag indicates that orthogonalization failed, see
     44:    BVOrthonormalizeColumn(). In that case, on exit m contains the index of
     45:    the column that failed.

     47:    The values of k and m are not restricted to the active columns of V.

     49:    To create an Arnoldi factorization from scratch, set k=0 and make sure the
     50:    first column contains the normalized initial vector.

     52:    Level: advanced

     54: .seealso: BVMatLanczos(), BVSetActiveColumns(), BVOrthonormalizeColumn()
     55: @*/
     56: PetscErrorCode BVMatArnoldi(BV V,Mat A,Mat H,PetscInt k,PetscInt *m,PetscReal *beta,PetscBool *breakdown)
     57: {
     58:   PetscScalar       *h;
     59:   const PetscScalar *a;
     60:   PetscInt          j,ldh,rows,cols;
     61:   PetscBool         lindep=PETSC_FALSE;
     62:   Vec               buf;

     70:   BVCheckSizes(V,1);

     77:   if (H) {
     81:     MatGetSize(H,&rows,&cols);
     82:     MatDenseGetLDA(H,&ldh);
     85:   }

     87:   for (j=k;j<*m;j++) {
     88:     BVMatMultColumn(V,A,j);
     89:     if (PetscUnlikely(j==V->N-1)) BV_OrthogonalizeColumn_Safe(V,j+1,NULL,beta,&lindep); /* safeguard in case the full basis is requested */
     90:     else BVOrthonormalizeColumn(V,j+1,PETSC_FALSE,beta,&lindep);
     91:     if (PetscUnlikely(lindep)) {
     92:       *m = j+1;
     93:       break;
     94:     }
     95:   }
     96:   if (breakdown) *breakdown = lindep;
     97:   if (lindep) PetscInfo(V,"Arnoldi finished early at m=%" PetscInt_FMT "\n",*m);

     99:   if (H) {
    100:     MatDenseGetArray(H,&h);
    101:     BVGetBufferVec(V,&buf);
    102:     VecGetArrayRead(buf,&a);
    103:     for (j=k;j<*m-1;j++) PetscArraycpy(h+j*ldh,a+V->nc+(j+1)*(V->nc+V->m),j+2);
    104:     PetscArraycpy(h+(*m-1)*ldh,a+V->nc+(*m)*(V->nc+V->m),*m);
    105:     if (ldh>*m) h[(*m)+(*m-1)*ldh] = a[V->nc+(*m)+(*m)*(V->nc+V->m)];
    106:     VecRestoreArrayRead(buf,&a);
    107:     MatDenseRestoreArray(H,&h);
    108:   }

    110:   PetscObjectStateIncrease((PetscObject)V);
    111:   return 0;
    112: }

    114: /*@C
    115:    BVMatLanczos - Computes a Lanczos factorization associated with a matrix.

    117:    Collective on V

    119:    Input Parameters:
    120: +  V - basis vectors context
    121: .  A - the matrix
    122: .  T - (optional) the tridiagonal matrix
    123: .  k - number of locked columns
    124: -  m - dimension of the Lanczos basis, may be modified

    126:    Output Parameters:
    127: +  beta - (optional) norm of last vector before normalization
    128: -  breakdown - (optional) flag indicating that breakdown occurred

    130:    Notes:
    131:    Computes an m-step Lanczos factorization for matrix A, with full
    132:    reorthogonalization. At each Lanczos step, the corresponding Lanczos
    133:    vector is orthogonalized with respect to all previous Lanczos vectors.
    134:    This is equivalent to computing an m-step Arnoldi factorization and
    135:    exploting symmetry of the operator.

    137:    The first k columns are assumed to be locked and therefore they are
    138:    not modified. On exit, the following relation is satisfied

    140: $                    A * V - V * T = beta*v_m * e_m^T

    142:    where the columns of V are the Lanczos vectors (which are B-orthonormal),
    143:    T is a real symmetric tridiagonal matrix, and e_m is the m-th vector of
    144:    the canonical basis. On exit, beta contains the B-norm of V[m] before
    145:    normalization. The T matrix is stored in a special way, its first column
    146:    contains the diagonal elements, and its second column the off-diagonal
    147:    ones. In complex scalars, the elements are stored as PetscReal and thus
    148:    occupy only the first column of the Mat object. This is the same storage
    149:    scheme used in matrix DS_MAT_T obtained with DSGetMat().

    151:    The breakdown flag indicates that orthogonalization failed, see
    152:    BVOrthonormalizeColumn(). In that case, on exit m contains the index of
    153:    the column that failed.

    155:    The values of k and m are not restricted to the active columns of V.

    157:    To create a Lanczos factorization from scratch, set k=0 and make sure the
    158:    first column contains the normalized initial vector.

    160:    Level: advanced

    162: .seealso: BVMatArnoldi(), BVSetActiveColumns(), BVOrthonormalizeColumn(), DSGetMat()
    163: @*/
    164: PetscErrorCode BVMatLanczos(BV V,Mat A,Mat T,PetscInt k,PetscInt *m,PetscReal *beta,PetscBool *breakdown)
    165: {
    166:   PetscScalar       *t;
    167:   const PetscScalar *a;
    168:   PetscReal         *alpha,*betat;
    169:   PetscInt          j,ldt,rows,cols,mincols=PetscDefined(USE_COMPLEX)?1:2;
    170:   PetscBool         lindep=PETSC_FALSE;
    171:   Vec               buf;

    179:   BVCheckSizes(V,1);

    186:   if (T) {
    190:     MatGetSize(T,&rows,&cols);
    191:     MatDenseGetLDA(T,&ldt);
    194:   }

    196:   for (j=k;j<*m;j++) {
    197:     BVMatMultColumn(V,A,j);
    198:     if (PetscUnlikely(j==V->N-1)) BV_OrthogonalizeColumn_Safe(V,j+1,NULL,beta,&lindep); /* safeguard in case the full basis is requested */
    199:     else BVOrthonormalizeColumn(V,j+1,PETSC_FALSE,beta,&lindep);
    200:     if (PetscUnlikely(lindep)) {
    201:       *m = j+1;
    202:       break;
    203:     }
    204:   }
    205:   if (breakdown) *breakdown = lindep;
    206:   if (lindep) PetscInfo(V,"Lanczos finished early at m=%" PetscInt_FMT "\n",*m);

    208:   if (T) {
    209:     MatDenseGetArray(T,&t);
    210:     alpha = (PetscReal*)t;
    211:     betat = alpha+ldt;
    212:     BVGetBufferVec(V,&buf);
    213:     VecGetArrayRead(buf,&a);
    214:     for (j=k;j<*m;j++) {
    215:       alpha[j] = PetscRealPart(a[V->nc+j+(j+1)*(V->nc+V->m)]);
    216:       betat[j] = PetscRealPart(a[V->nc+j+1+(j+1)*(V->nc+V->m)]);
    217:     }
    218:     VecRestoreArrayRead(buf,&a);
    219:     MatDenseRestoreArray(T,&t);
    220:   }

    222:   PetscObjectStateIncrease((PetscObject)V);
    223:   return 0;
    224: }