Actual source code: krylov.c

  1: /*
  2:    Common subroutines for all Krylov-type solvers.

  4:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  5:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  6:    Copyright (c) 2002-2013, Universitat Politecnica de Valencia, Spain

  8:    This file is part of SLEPc.

 10:    SLEPc is free software: you can redistribute it and/or modify it under  the
 11:    terms of version 3 of the GNU Lesser General Public License as published by
 12:    the Free Software Foundation.

 14:    SLEPc  is  distributed in the hope that it will be useful, but WITHOUT  ANY
 15:    WARRANTY;  without even the implied warranty of MERCHANTABILITY or  FITNESS
 16:    FOR  A  PARTICULAR PURPOSE. See the GNU Lesser General Public  License  for
 17:    more details.

 19:    You  should have received a copy of the GNU Lesser General  Public  License
 20:    along with SLEPc. If not, see <http://www.gnu.org/licenses/>.
 21:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 22: */

 24: #include <slepc-private/epsimpl.h>
 25: #include <slepc-private/slepcimpl.h>
 26: #include <slepcblaslapack.h>

 30: /*
 31:    EPSBasicArnoldi - Computes an m-step Arnoldi factorization. The first k
 32:    columns are assumed to be locked and therefore they are not modified. On
 33:    exit, the following relation is satisfied:

 35:                     OP * V - V * H = f * e_m^T

 37:    where the columns of V are the Arnoldi vectors (which are B-orthonormal),
 38:    H is an upper Hessenberg matrix, f is the residual vector and e_m is
 39:    the m-th vector of the canonical basis. The vector f is B-orthogonal to
 40:    the columns of V. On exit, beta contains the B-norm of f and the next
 41:    Arnoldi vector can be computed as v_{m+1} = f / beta.
 42: */
 43: PetscErrorCode EPSBasicArnoldi(EPS eps,PetscBool trans,PetscScalar *H,PetscInt ldh,Vec *V,PetscInt k,PetscInt *M,Vec f,PetscReal *beta,PetscBool *breakdown)
 44: {
 46:   PetscInt       j,m = *M;
 47:   PetscReal      norm;

 50:   for (j=k;j<m-1;j++) {
 51:     if (trans) {
 52:       STApplyTranspose(eps->st,V[j],V[j+1]);
 53:     } else {
 54:       STApply(eps->st,V[j],V[j+1]);
 55:     }
 56:     IPOrthogonalize(eps->ip,eps->nds,eps->defl,j+1,NULL,V,V[j+1],H+ldh*j,&norm,breakdown);
 57:     H[j+1+ldh*j] = norm;
 58:     if (*breakdown) {
 59:       *M = j+1;
 60:       *beta = norm;
 61:       return(0);
 62:     } else {
 63:       VecScale(V[j+1],1/norm);
 64:     }
 65:   }
 66:   if (trans) {
 67:     STApplyTranspose(eps->st,V[m-1],f);
 68:   } else {
 69:     STApply(eps->st,V[m-1],f);
 70:   }
 71:   IPOrthogonalize(eps->ip,eps->nds,eps->defl,m,NULL,V,f,H+ldh*(m-1),beta,NULL);
 72:   return(0);
 73: }

 77: /*
 78:    EPSKrylovConvergence - Implements the loop that checks for convergence
 79:    in Krylov methods.

 81:    Input Parameters:
 82:      eps   - the eigensolver; some error estimates are updated in eps->errest
 83:      getall - whether all residuals must be computed
 84:      kini  - initial value of k (the loop variable)
 85:      nits  - number of iterations of the loop
 86:      V     - set of basis vectors (used only if trueresidual is activated)
 87:      nv    - number of vectors to process (dimension of Q, columns of V)
 88:      beta  - norm of f (the residual vector of the Arnoldi/Lanczos factorization)
 89:      corrf - correction factor for residual estimates (only in harmonic KS)

 91:    Output Parameters:
 92:      kout  - the first index where the convergence test failed
 93: */
 94: PetscErrorCode EPSKrylovConvergence(EPS eps,PetscBool getall,PetscInt kini,PetscInt nits,Vec *V,PetscInt nv,PetscReal beta,PetscReal corrf,PetscInt *kout)
 95: {
 97:   PetscInt       k,newk,marker,ld;
 98:   PetscScalar    re,im,*Zr,*Zi,*X;
 99:   PetscReal      resnorm;
100:   PetscBool      isshift,refined;
101:   Vec            x,y;

104:   if (eps->trueres) {
105:     VecDuplicate(eps->t,&x);
106:     VecDuplicate(eps->t,&y);
107:   }
108:   DSGetLeadingDimension(eps->ds,&ld);
109:   DSGetRefined(eps->ds,&refined);
110:   PetscObjectTypeCompare((PetscObject)eps->st,STSHIFT,&isshift);
111:   marker = -1;
112:   if (eps->trackall) getall = PETSC_TRUE;
113:   for (k=kini;k<kini+nits;k++) {
114:     /* eigenvalue */
115:     re = eps->eigr[k];
116:     im = eps->eigi[k];
117:     if (eps->trueres || isshift) {
118:       STBackTransform(eps->st,1,&re,&im);
119:     }
120:     newk = k;
121:     DSVectors(eps->ds,DS_MAT_X,&newk,&resnorm);
122:     if (eps->trueres) {
123:       DSGetArray(eps->ds,DS_MAT_X,&X);
124:       Zr = X+k*ld;
125:       if (newk==k+1) Zi = X+newk*ld;
126:       else Zi = NULL;
127:       EPSComputeRitzVector(eps,Zr,Zi,V,nv,x,y);
128:       DSRestoreArray(eps->ds,DS_MAT_X,&X);
129:       EPSComputeResidualNorm_Private(eps,re,im,x,y,&resnorm);
130:     }
131:     else if (!refined) resnorm *= beta*corrf;
132:     /* error estimate */
133:     (*eps->converged)(eps,re,im,resnorm,&eps->errest[k],eps->convergedctx);
134:     if (marker==-1 && eps->errest[k] >= eps->tol) marker = k;
135:     if (newk==k+1) {
136:       eps->errest[k+1] = eps->errest[k];
137:       k++;
138:     }
139:     if (marker!=-1 && !getall) break;
140:   }
141:   if (marker!=-1) k = marker;
142:   *kout = k;
143:   if (eps->trueres) {
144:     VecDestroy(&x);
145:     VecDestroy(&y);
146:   }
147:   return(0);
148: }

152: /*
153:    EPSFullLanczos - Computes an m-step Lanczos factorization with full
154:    reorthogonalization.  At each Lanczos step, the corresponding Lanczos
155:    vector is orthogonalized with respect to all previous Lanczos vectors.
156:    This is equivalent to computing an m-step Arnoldi factorization and
157:    exploting symmetry of the operator.

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

162:                     OP * V - V * T = f * e_m^T

164:    where the columns of V are the Lanczos vectors (which are B-orthonormal),
165:    T is a real symmetric tridiagonal matrix, f is the residual vector and e_m
166:    is the m-th vector of the canonical basis. The tridiagonal is stored as
167:    two arrays: alpha contains the diagonal elements, beta the off-diagonal.
168:    The vector f is B-orthogonal to the columns of V. On exit, the last element
169:    of beta contains the B-norm of f and the next Lanczos vector can be
170:    computed as v_{m+1} = f / beta(end).

172: */
173: PetscErrorCode EPSFullLanczos(EPS eps,PetscReal *alpha,PetscReal *beta,Vec *V,PetscInt k,PetscInt *M,Vec f,PetscBool *breakdown)
174: {
176:   PetscInt       j,m = *M;
177:   PetscReal      norm;
178:   PetscScalar    *hwork,lhwork[100];

181:   if (m > 100) {
182:     PetscMalloc((eps->nds+m)*sizeof(PetscScalar),&hwork);
183:   } else hwork = lhwork;

185:   for (j=k;j<m-1;j++) {
186:     STApply(eps->st,V[j],V[j+1]);
187:     IPOrthogonalize(eps->ip,eps->nds,eps->defl,j+1,NULL,V,V[j+1],hwork,&norm,breakdown);
188:     alpha[j] = PetscRealPart(hwork[j]);
189:     beta[j] = norm;
190:     if (*breakdown) {
191:       *M = j+1;
192:       if (m > 100) {
193:         PetscFree(hwork);
194:       }
195:       return(0);
196:     } else {
197:       VecScale(V[j+1],1.0/norm);
198:     }
199:   }
200:   STApply(eps->st,V[m-1],f);
201:   IPOrthogonalize(eps->ip,eps->nds,eps->defl,m,NULL,V,f,hwork,&norm,NULL);
202:   alpha[m-1] = PetscRealPart(hwork[m-1]);
203:   beta[m-1] = norm;

205:   if (m > 100) {
206:     PetscFree(hwork);
207:   }
208:   return(0);
209: }