Actual source code: bvkrylov.c

slepc-3.13.1 2020-04-12
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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:    BV routines related to Krylov decompositions
 12: */

 14: #include <slepc/private/bvimpl.h>          /*I   "slepcbv.h"   I*/

 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:     BVOrthonormalizeColumn(V,j+1,PETSC_FALSE,beta,&lindep);
 85:     if (lindep) {
 86:       *m = j+1;
 87:       break;
 88:     }
 89:   }
 90:   if (breakdown) *breakdown = lindep;
 91:   /* extract Hessenberg matrix from the BV object */
 92:   BVGetBufferVec(V,&buf);
 93:   VecGetArray(buf,&a);
 94:   for (j=k;j<*m;j++) {
 95:     PetscArraycpy(H+j*ldh,a+V->nc+(j+1)*(V->nc+V->m),j+2);
 96:   }
 97:   VecRestoreArray(buf,&a);

 99:   PetscObjectStateIncrease((PetscObject)V);
100:   return(0);
101: }

103: /*@C
104:    BVMatLanczos - Computes a Lanczos factorization associated with a matrix.

106:    Collective on V

108:    Input Parameters:
109: +  V - basis vectors context
110: .  A - the matrix
111: .  alpha - diagonal entries of tridiagonal matrix
112: .  beta - subdiagonal entries of tridiagonal matrix
113: .  k - number of locked columns
114: -  m - dimension of the Lanczos basis

116:    Output Parameters:
117: +  m - the modified dimension
118: -  breakdown - (optional) flag indicating that breakdown occurred

120:    Notes:
121:    Computes an m-step Lanczos factorization for matrix A, with full
122:    reorthogonalization. At each Lanczos step, the corresponding Lanczos
123:    vector is orthogonalized with respect to all previous Lanczos vectors.
124:    This is equivalent to computing an m-step Arnoldi factorization and
125:    exploting symmetry of the operator.

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

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

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

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

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

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

149:    Level: advanced

151: .seealso: BVMatArnoldi(), BVSetActiveColumns(), BVOrthonormalizeColumn()
152: @*/
153: PetscErrorCode BVMatLanczos(BV V,Mat A,PetscReal *alpha,PetscReal *beta,PetscInt k,PetscInt *m,PetscBool *breakdown)
154: {
156:   PetscScalar    *a;
157:   PetscInt       j;
158:   PetscBool      lindep=PETSC_FALSE;
159:   Vec            buf;

170:   BVCheckSizes(V,1);

174:   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);
175:   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);
176:   if (*m<=k) SETERRQ(PetscObjectComm((PetscObject)V),PETSC_ERR_ARG_OUTOFRANGE,"Argument m should be at least equal to k+1");

178:   BVSetActiveColumns(V,0,*m);
179:   for (j=k;j<*m;j++) {
180:     BVMatMultColumn(V,A,j);
181:     BVOrthonormalizeColumn(V,j+1,PETSC_FALSE,beta+j,&lindep);
182:     if (lindep) {
183:       *m = j+1;
184:       break;
185:     }
186:   }
187:   if (breakdown) *breakdown = lindep;

189:   /* extract Hessenberg matrix from the BV buffer */
190:   BVGetBufferVec(V,&buf);
191:   VecGetArray(buf,&a);
192:   for (j=k;j<*m;j++) alpha[j] = PetscRealPart(a[V->nc+j+(j+1)*(V->nc+V->m)]);
193:   VecRestoreArray(buf,&a);

195:   PetscObjectStateIncrease((PetscObject)V);
196:   return(0);
197: }