Actual source code: bvkrylov.c

slepc-3.17.0 2022-03-31
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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:   PetscFunctionReturn(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: .  alpha - diagonal entries of tridiagonal matrix
123: .  beta - subdiagonal entries of tridiagonal matrix
124: -  k - number of locked columns

126:    Input/Output Parameter:
127: .  m - dimension of the Lanczos basis, may be modified

129:    Output Parameter:
130: .  breakdown - (optional) flag indicating that breakdown occurred

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

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

142: $                    A * V - V * T = beta_m*v_m * e_m^T

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

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

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

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

161:    Level: advanced

163: .seealso: BVMatArnoldi(), BVSetActiveColumns(), BVOrthonormalizeColumn()
164: @*/
165: PetscErrorCode BVMatLanczos(BV V,Mat A,PetscReal *alpha,PetscReal *beta,PetscInt k,PetscInt *m,PetscBool *breakdown)
166: {
167:   PetscScalar    *a;
168:   PetscInt       j;
169:   PetscBool      lindep=PETSC_FALSE;
170:   Vec            buf;

180:   BVCheckSizes(V,1);


188:   for (j=k;j<*m;j++) {
189:     BVMatMultColumn(V,A,j);
190:     if (PetscUnlikely(j==V->N-1)) BV_OrthogonalizeColumn_Safe(V,j+1,NULL,beta+j,&lindep); /* safeguard in case the full basis is requested */
191:     else BVOrthonormalizeColumn(V,j+1,PETSC_FALSE,beta+j,&lindep);
192:     if (PetscUnlikely(lindep)) {
193:       *m = j+1;
194:       break;
195:     }
196:   }
197:   if (breakdown) *breakdown = lindep;
198:   if (lindep) PetscInfo(V,"Lanczos finished early at m=%" PetscInt_FMT "\n",*m);

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

206:   PetscObjectStateIncrease((PetscObject)V);
207:   PetscFunctionReturn(0);
208: }