Actual source code: bvkrylov.c

slepc-3.15.2 2021-09-20
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2021, 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 - the upper Hessenberg matrix
 25: .  ldh - leading dimension of H
 26: .  k - number of locked columns
 27: -  m - dimension of the Arnoldi basis

 29:    Output Parameters:
 30: +  m - the modified dimension
 31: .  beta - (optional) norm of last vector before normalization
 32: -  breakdown - (optional) flag indicating that breakdown occurred

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

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

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

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

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

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

 54:    Level: advanced

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

 73:   BVCheckSizes(V,1);

 77:   if (k<0 || k>V->m) SETERRQ2(PetscObjectComm((PetscObject)V),PETSC_ERR_ARG_OUTOFRANGE,"Argument k has wrong value %D, should be between 0 and %D",k,V->m);
 78:   if (*m<1 || *m>V->m) SETERRQ2(PetscObjectComm((PetscObject)V),PETSC_ERR_ARG_OUTOFRANGE,"Argument m has wrong value %D, should be between 1 and %D",*m,V->m);
 79:   if (*m<=k) SETERRQ(PetscObjectComm((PetscObject)V),PETSC_ERR_ARG_OUTOFRANGE,"Argument m should be at least equal to k+1");

 81:   BVSetActiveColumns(V,0,*m);
 82:   for (j=k;j<*m;j++) {
 83:     BVMatMultColumn(V,A,j);
 84:     if (PetscUnlikely(j==V->N-1)) {   /* safeguard in case the full basis is requested */
 85:       BV_OrthogonalizeColumn_Safe(V,j+1,NULL,beta,&lindep);
 86:     } else {
 87:       BVOrthonormalizeColumn(V,j+1,PETSC_FALSE,beta,&lindep);
 88:     }
 89:     if (lindep) {
 90:       *m = j+1;
 91:       break;
 92:     }
 93:   }
 94:   if (breakdown) *breakdown = lindep;
 95:   if (lindep) { PetscInfo1(V,"Arnoldi finished early at m=%D\n",*m); }
 96:   /* extract Hessenberg matrix from the BV object */
 97:   BVGetBufferVec(V,&buf);
 98:   VecGetArray(buf,&a);
 99:   for (j=k;j<*m;j++) {
100:     PetscArraycpy(H+j*ldh,a+V->nc+(j+1)*(V->nc+V->m),j+2);
101:   }
102:   VecRestoreArray(buf,&a);

104:   PetscObjectStateIncrease((PetscObject)V);
105:   return(0);
106: }

108: /*@C
109:    BVMatLanczos - Computes a Lanczos factorization associated with a matrix.

111:    Collective on V

113:    Input Parameters:
114: +  V - basis vectors context
115: .  A - the matrix
116: .  alpha - diagonal entries of tridiagonal matrix
117: .  beta - subdiagonal entries of tridiagonal matrix
118: .  k - number of locked columns
119: -  m - dimension of the Lanczos basis

121:    Output Parameters:
122: +  m - the modified dimension
123: -  breakdown - (optional) flag indicating that breakdown occurred

125:    Notes:
126:    Computes an m-step Lanczos factorization for matrix A, with full
127:    reorthogonalization. At each Lanczos step, the corresponding Lanczos
128:    vector is orthogonalized with respect to all previous Lanczos vectors.
129:    This is equivalent to computing an m-step Arnoldi factorization and
130:    exploting symmetry of the operator.

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

135:                     A * V - V * T = beta_m*v_m * e_m^T

137:    where the columns of V are the Lanczos vectors (which are B-orthonormal),
138:    T is a real symmetric tridiagonal matrix, and e_m is the m-th vector of
139:    the canonical basis. The tridiagonal is stored as two arrays: alpha
140:    contains the diagonal elements, beta the off-diagonal. On exit, the last
141:    element of beta contains the B-norm of V[m] before normalization.
142:    The basis V must have at least m+1 columns, while the arrays alpha and
143:    beta must have space for at least m elements.

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

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

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

154:    Level: advanced

156: .seealso: BVMatArnoldi(), BVSetActiveColumns(), BVOrthonormalizeColumn()
157: @*/
158: PetscErrorCode BVMatLanczos(BV V,Mat A,PetscReal *alpha,PetscReal *beta,PetscInt k,PetscInt *m,PetscBool *breakdown)
159: {
161:   PetscScalar    *a;
162:   PetscInt       j;
163:   PetscBool      lindep=PETSC_FALSE;
164:   Vec            buf;

175:   BVCheckSizes(V,1);

179:   if (k<0 || k>V->m) SETERRQ2(PetscObjectComm((PetscObject)V),PETSC_ERR_ARG_OUTOFRANGE,"Argument k has wrong value %D, should be between 0 and %D",k,V->m);
180:   if (*m<1 || *m>V->m) SETERRQ2(PetscObjectComm((PetscObject)V),PETSC_ERR_ARG_OUTOFRANGE,"Argument m has wrong value %D, should be between 1 and %D",*m,V->m);
181:   if (*m<=k) SETERRQ(PetscObjectComm((PetscObject)V),PETSC_ERR_ARG_OUTOFRANGE,"Argument m should be at least equal to k+1");

183:   BVSetActiveColumns(V,0,*m);
184:   for (j=k;j<*m;j++) {
185:     BVMatMultColumn(V,A,j);
186:     if (PetscUnlikely(j==V->N-1)) {   /* safeguard in case the full basis is requested */
187:       BV_OrthogonalizeColumn_Safe(V,j+1,NULL,beta+j,&lindep);
188:     } else {
189:       BVOrthonormalizeColumn(V,j+1,PETSC_FALSE,beta+j,&lindep);
190:     }
191:     if (lindep) {
192:       *m = j+1;
193:       break;
194:     }
195:   }
196:   if (breakdown) *breakdown = lindep;

198:   /* extract Hessenberg matrix from the BV buffer */
199:   BVGetBufferVec(V,&buf);
200:   VecGetArray(buf,&a);
201:   for (j=k;j<*m;j++) alpha[j] = PetscRealPart(a[V->nc+j+(j+1)*(V->nc+V->m)]);
202:   VecRestoreArray(buf,&a);

204:   PetscObjectStateIncrease((PetscObject)V);
205:   return(0);
206: }