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> 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: /* extract Hessenberg matrix from the BV object */
96: BVGetBufferVec(V,&buf);
97: VecGetArray(buf,&a);
98: for (j=k;j<*m;j++) {
99: PetscArraycpy(H+j*ldh,a+V->nc+(j+1)*(V->nc+V->m),j+2);
100: }
101: VecRestoreArray(buf,&a);
103: PetscObjectStateIncrease((PetscObject)V);
104: return(0);
105: }
107: /*@C
108: BVMatLanczos - Computes a Lanczos factorization associated with a matrix.
110: Collective on V
112: Input Parameters:
113: + V - basis vectors context
114: . A - the matrix
115: . alpha - diagonal entries of tridiagonal matrix
116: . beta - subdiagonal entries of tridiagonal matrix
117: . k - number of locked columns
118: - m - dimension of the Lanczos basis
120: Output Parameters:
121: + m - the modified dimension
122: - breakdown - (optional) flag indicating that breakdown occurred
124: Notes:
125: Computes an m-step Lanczos factorization for matrix A, with full
126: reorthogonalization. At each Lanczos step, the corresponding Lanczos
127: vector is orthogonalized with respect to all previous Lanczos vectors.
128: This is equivalent to computing an m-step Arnoldi factorization and
129: exploting symmetry of the operator.
131: The first k columns are assumed to be locked and therefore they are
132: not modified. On exit, the following relation is satisfied:
134: A * V - V * T = beta_m*v_m * e_m^T
136: where the columns of V are the Lanczos vectors (which are B-orthonormal),
137: T is a real symmetric tridiagonal matrix, and e_m is the m-th vector of
138: the canonical basis. The tridiagonal is stored as two arrays: alpha
139: contains the diagonal elements, beta the off-diagonal. On exit, the last
140: element of beta contains the B-norm of V[m] before normalization.
141: The basis V must have at least m+1 columns, while the arrays alpha and
142: beta must have space for at least m elements.
144: The breakdown flag indicates that orthogonalization failed, see
145: BVOrthonormalizeColumn(). In that case, on exit m contains the index of
146: the column that failed.
148: The values of k and m are not restricted to the active columns of V.
150: To create a Lanczos factorization from scratch, set k=0 and make sure the
151: first column contains the normalized initial vector.
153: Level: advanced
155: .seealso: BVMatArnoldi(), BVSetActiveColumns(), BVOrthonormalizeColumn()
156: @*/
157: PetscErrorCode BVMatLanczos(BV V,Mat A,PetscReal *alpha,PetscReal *beta,PetscInt k,PetscInt *m,PetscBool *breakdown)158: {
160: PetscScalar *a;
161: PetscInt j;
162: PetscBool lindep=PETSC_FALSE;
163: Vec buf;
174: BVCheckSizes(V,1);
178: 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);
179: 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);
180: if (*m<=k) SETERRQ(PetscObjectComm((PetscObject)V),PETSC_ERR_ARG_OUTOFRANGE,"Argument m should be at least equal to k+1");
182: BVSetActiveColumns(V,0,*m);
183: for (j=k;j<*m;j++) {
184: BVMatMultColumn(V,A,j);
185: if (PetscUnlikely(j==V->N-1)) { /* safeguard in case the full basis is requested */
186: BV_OrthogonalizeColumn_Safe(V,j+1,NULL,beta+j,&lindep);
187: } else {
188: BVOrthonormalizeColumn(V,j+1,PETSC_FALSE,beta+j,&lindep);
189: }
190: if (lindep) {
191: *m = j+1;
192: break;
193: }
194: }
195: if (breakdown) *breakdown = lindep;
197: /* extract Hessenberg matrix from the BV buffer */
198: BVGetBufferVec(V,&buf);
199: VecGetArray(buf,&a);
200: for (j=k;j<*m;j++) alpha[j] = PetscRealPart(a[V->nc+j+(j+1)*(V->nc+V->m)]);
201: VecRestoreArray(buf,&a);
203: PetscObjectStateIncrease((PetscObject)V);
204: return(0);
205: }