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/*                      

   SLEPc eigensolver: "krylovschur"

   Method: Krylov-Schur

   Algorithm:

       Single-vector Krylov-Schur method for both symmetric and non-symmetric

       problems.

   References:

       [1] "Krylov-Schur Methods in SLEPc", SLEPc Technical Report STR-7,

           available at http://www.grycap.upv.es/slepc.

       [2] G.W. Stewart, "A Krylov-Schur Algorithm for Large Eigenproblems",

           SIAM J. Matrix Analysis and App., 23(3), pp. 601-614, 2001.

   Last update: Feb 2009

   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

   SLEPc - Scalable Library for Eigenvalue Problem Computations

   Copyright (c) 2002-2010, Universidad Politecnica de Valencia, Spain

   This file is part of SLEPc.

   SLEPc is free software: you can redistribute it and/or modify it under  the

   terms of version 3 of the GNU Lesser General Public License as published by

   the Free Software Foundation.

   SLEPc  is  distributed in the hope that it will be useful, but WITHOUT  ANY

   WARRANTY;  without even the implied warranty of MERCHANTABILITY or  FITNESS

   FOR  A  PARTICULAR PURPOSE. See the GNU Lesser General Public  License  for

   more details.

   You  should have received a copy of the GNU Lesser General  Public  License

   along with SLEPc. If not, see <http://www.gnu.org/licenses/>.

   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

*/

#include "private/epsimpl.h"                /*I "slepceps.h" I*/

#include "slepcblaslapack.h"

#undef __FUNCT__  

#define __FUNCT__ "ArrowTridFlip"

/*

   ArrowTridFlip - Solves the arrowhead-tridiagonal eigenproblem by flipping

   the matrix and tridiagonalizing the bottom part.

   On input:

     l is the size of diagonal part

     d contains diagonal elements (length n)

     e contains offdiagonal elements (length n-1)

   On output:

     d contains the eigenvalues in ascending order

     Q is the eigenvector matrix (order n)

   Workspace:

     S is workspace to store a copy of the full matrix (nxn reals)

*/

PetscErrorCode ArrowTridFlip(PetscInt n_,PetscInt l,PetscReal *d,PetscReal *e,PetscReal *Q,PetscReal *S)

{

#if defined(SLEPC_MISSING_LAPACK_SYTRD) || defined(SLEPC_MISSING_LAPACK_ORGTR) || defined(SLEPC_MISSING_LAPACK_STEQR)

  PetscFunctionBegin;

  SETERRQ(PETSC_ERR_SUP,"SYTRD/ORGTR/STEQR - Lapack routine is unavailable.");

#else

  PetscErrorCode ierr;

  PetscInt       i,j;

  PetscBLASInt   n,n1,n2,lwork,info;

  PetscFunctionBegin;

  ierr = PetscLogEventBegin(EPS_Dense,0,0,0,0);CHKERRQ(ierr);

  n = PetscBLASIntCast(n_);

  /* quick return */

  if (n == 1) {

    Q[0] = 1.0;

    PetscFunctionReturn(0);    

  }

  n1 = PetscBLASIntCast(l+1);    /* size of leading block, including residuals */

  n2 = PetscBLASIntCast(n-l-1);  /* size of trailing block */

  ierr = PetscMemzero(S,n*n*sizeof(PetscReal));CHKERRQ(ierr);

  /* Flip matrix S, copying the values saved in Q */

  for (i=0;i<n;i++)

    S[(n-1-i)+(n-1-i)*n] = d[i];

  for (i=0;i<l;i++)

    S[(n-1-i)+(n-1-l)*n] = e[i];

  for (i=l;i<n-1;i++)

    S[(n-1-i)+(n-1-i-1)*n] = e[i];

  /* Reduce (2,2)-block of flipped S to tridiagonal form */

  lwork = PetscBLASIntCast(n_*n_-n_);

  LAPACKsytrd_("L",&n1,S+n2*(n+1),&n,d,e,Q,Q+n,&lwork,&info);

  if (info) SETERRQ1(PETSC_ERR_LIB,"Error in Lapack xSYTRD %d",info);

  /* Flip back diag and subdiag, put them in d and e */

  for (i=0;i<n-1;i++) {

    d[n-i-1] = S[i+i*n];

    e[n-i-2] = S[i+1+i*n];

  }

  d[0] = S[n-1+(n-1)*n];

  /* Compute the orthogonal matrix used for tridiagonalization */

  LAPACKorgtr_("L",&n1,S+n2*(n+1),&n,Q,Q+n,&lwork,&info);

  if (info) SETERRQ1(PETSC_ERR_LIB,"Error in Lapack xORGTR %d",info);

  /* Create full-size Q, flipped back to original order */

  for (i=0;i<n;i++)

    for (j=0;j<n;j++)

      Q[i+j*n] = 0.0;

  for (i=n1;i<n;i++)

    Q[i+i*n] = 1.0;

  for (i=0;i<n1;i++)

    for (j=0;j<n1;j++)

      Q[i+j*n] = S[n-i-1+(n-j-1)*n];

  /* Solve the tridiagonal eigenproblem */

  LAPACKsteqr_("V",&n,d,e,Q,&n,S,&info);

  if (info) SETERRQ1(PETSC_ERR_LIB,"Error in Lapack xSTEQR %d",info);

  ierr = PetscLogEventEnd(EPS_Dense,0,0,0,0);CHKERRQ(ierr);

  PetscFunctionReturn(0);

#endif

}

#undef __FUNCT__  

#define __FUNCT__ "EPSProjectedKSSym"

/*

   EPSProjectedKSSym - Solves the projected eigenproblem in the Krylov-Schur

   method (symmetric case).

   On input:

     n is the matrix dimension

     l is the number of vectors kept in previous restart

     a contains diagonal elements (length n)

     b contains offdiagonal elements (length n-1)

   On output:

     eig is the sorted list of eigenvalues

     Q is the eigenvector matrix (order n, leading dimension n)

   Workspace:

     work is workspace to store a real square matrix of order n

     perm is workspace to store 2n integers

*/

PetscErrorCode EPSProjectedKSSym(EPS eps,PetscInt n,PetscInt l,PetscReal *a,PetscReal *b,PetscScalar *eig,PetscScalar *Q,PetscReal *work,PetscInt *perm)

{

  PetscErrorCode ierr;

  PetscInt       i,j,k,p;

  PetscReal      rtmp,*Qreal = (PetscReal*)Q;

  PetscFunctionBegin;

  /* Compute eigendecomposition of projected matrix */

  ierr = ArrowTridFlip(n,l,a,b,Qreal,work);CHKERRQ(ierr);

  /* Sort eigendecomposition according to eps->which */

  ierr = EPSSortEigenvaluesReal(eps,n,a,perm);CHKERRQ(ierr);

  for (i=0;i<n;i++)

    eig[i] = a[perm[i]];

  for (i=0;i<n;i++) {

    p = perm[i];

    if (p != i) {

      j = i + 1;

      while (perm[j] != i) j++;

      perm[j] = p; perm[i] = i;

      /* swap eigenvectors i and j */

      for (k=0;k<n;k++) {

        rtmp = Qreal[k+p*n]; Qreal[k+p*n] = Qreal[k+i*n]; Qreal[k+i*n] = rtmp;

      }

    }

  }

#if defined(PETSC_USE_COMPLEX)

  for (j=n-1;j>=0;j--)

    for (i=n-1;i>=0;i--)

      Q[i+j*n] = Qreal[i+j*n];

#endif

  PetscFunctionReturn(0);

}

#undef __FUNCT__  

#define __FUNCT__ "EPSSolve_KRYLOVSCHUR_SYMM"

PetscErrorCode EPSSolve_KRYLOVSCHUR_SYMM(EPS eps)

{

  PetscErrorCode ierr;

  PetscInt       i,k,l,lds,lt,nv,m;

  Vec            u=eps->work[0];

  PetscScalar    *Q;

  PetscReal      *a,*b,*work,beta;

  PetscInt       *iwork;

  PetscTruth     breakdown;

  PetscFunctionBegin;

  lds = PetscMin(eps->mpd,eps->ncv);

  ierr = PetscMalloc(lds*lds*sizeof(PetscReal),&work);CHKERRQ(ierr);

  ierr = PetscMalloc(lds*lds*sizeof(PetscScalar),&Q);CHKERRQ(ierr);

  ierr = PetscMalloc(2*lds*sizeof(PetscInt),&iwork);CHKERRQ(ierr);

  lt = PetscMin(eps->nev+eps->mpd,eps->ncv);

  ierr = PetscMalloc(lt*sizeof(PetscReal),&a);CHKERRQ(ierr);  

  ierr = PetscMalloc(lt*sizeof(PetscReal),&b);CHKERRQ(ierr);  

  /* Get the starting Lanczos vector */

  ierr = EPSGetStartVector(eps,0,eps->V[0],PETSC_NULL);CHKERRQ(ierr);

  l = 0;

  /* Restart loop */

  while (eps->reason == EPS_CONVERGED_ITERATING) {

    eps->its++;

    /* Compute an nv-step Lanczos factorization */

    m = PetscMin(eps->nconv+eps->mpd,eps->ncv);

    ierr = EPSFullLanczos(eps,a+l,b+l,eps->V,eps->nconv+l,&m,u,&breakdown);CHKERRQ(ierr);

    nv = m - eps->nconv;

    beta = b[nv-1];

    /* Solve projected problem and compute residual norm estimates */

    ierr = EPSProjectedKSSym(eps,nv,l,a,b,eps->eigr+eps->nconv,Q,work,iwork);CHKERRQ(ierr);

    /* Check convergence */

    ierr = EPSKrylovConvergence(eps,PETSC_TRUE,eps->nconv,nv,PETSC_NULL,nv,Q,eps->V+eps->nconv,nv,beta,1.0,&k,PETSC_NULL);CHKERRQ(ierr);

    if (eps->its >= eps->max_it) eps->reason = EPS_DIVERGED_ITS;

    if (k >= eps->nev) eps->reason = EPS_CONVERGED_TOL;

    /* Update l */

    if (eps->reason != EPS_CONVERGED_ITERATING || breakdown) l = 0;

    else l = (eps->nconv+nv-k)/2;

    if (eps->reason == EPS_CONVERGED_ITERATING) {

      if (breakdown) {

        /* Start a new Lanczos factorization */

        PetscInfo2(eps,"Breakdown in Krylov-Schur method (it=%i norm=%g)\n",eps->its,beta);

        ierr = EPSGetStartVector(eps,k,eps->V[k],&breakdown);CHKERRQ(ierr);

        if (breakdown) {

          eps->reason = EPS_DIVERGED_BREAKDOWN;

          PetscInfo(eps,"Unable to generate more start vectors\n");

        }

      } else {

        /* Prepare the Rayleigh quotient for restart */

        for (i=0;i<l;i++) {

          a[i] = PetscRealPart(eps->eigr[i+k]);

          b[i] = PetscRealPart(Q[nv-1+(i+k-eps->nconv)*nv]*beta);

        }

      }

    }

    /* Update the corresponding vectors V(:,idx) = V*Q(:,idx) */

    ierr = SlepcUpdateVectors(nv,eps->V+eps->nconv,0,k+l-eps->nconv,Q,nv,PETSC_FALSE);CHKERRQ(ierr);

    /* Normalize u and append it to V */

    if (eps->reason == EPS_CONVERGED_ITERATING && !breakdown) {

      ierr = VecAXPBY(eps->V[k+l],1.0/beta,0.0,u);CHKERRQ(ierr);

    }

    EPSMonitor(eps,eps->its,k,eps->eigr,eps->eigi,eps->errest,nv+eps->nconv);

    eps->nconv = k;

  }

  ierr = PetscFree(Q);CHKERRQ(ierr);

  ierr = PetscFree(a);CHKERRQ(ierr);

  ierr = PetscFree(b);CHKERRQ(ierr);

  ierr = PetscFree(work);CHKERRQ(ierr);

  ierr = PetscFree(iwork);CHKERRQ(ierr);

  PetscFunctionReturn(0);

}