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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: Oct 2006

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

      SLEPc - Scalable Library for Eigenvalue Problem Computations

      Copyright (c) 2002-2007, Universidad Politecnica de Valencia, Spain

      This file is part of SLEPc. See the README file for conditions of use

      and additional information.

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

*/

#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

  PetscInt       i,j;

  PetscBLASInt   n1,n2,lwork,info;

  PetscFunctionBegin;

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

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

 /* Clean matrix S */

  for (i=0;i<n;i++)

    for (j=0;j<n;j++)

      S[i+j*n] = 0.0;

  /* 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;i++)

    S[(n-1-i)+(n-1-i-1)*n] = e[i];

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

  lwork = 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);

  PetscFunctionReturn(0);

#endif

}

#undef __FUNCT__  

#define __FUNCT__ "EPSProjectedKSSym"

/*

   EPSProjectedKSSym - Solves the projected eigenproblem in the Krylov-Schur

   method (symmetric case).

   On input:

     l is the number of vectors kept in previous restart (0 means first restart)

     S is the projected matrix (order n, leading dimension is lds)

   On output:

     S is diagonal with diagonal elements (eigenvalues) sorted appropriately

     eig is the sorted list of eigenvalues

     Q is the eigenvector matrix (order n)

   Workspace:

     work is workspace to store 2n reals and 2n integers

*/

PetscErrorCode EPSProjectedKSSym(EPS eps,PetscInt n,PetscInt l,PetscScalar *S,PetscInt lds,PetscScalar *eig,PetscScalar *Q,PetscReal *work)

{

  PetscErrorCode ierr;

  PetscInt       i,j;

  PetscReal      *ritz = work;

  PetscReal      *e = work+n;

  PetscInt       *perm = ((PetscInt*)(work+n))+n;

  PetscReal      *Sreal = (PetscReal*)S, *Qreal = (PetscReal*)Q;

  PetscFunctionBegin;

  /* Compute eigendecomposition of S, S <- Q S Q' */

  if (l==0) {

    ierr = EPSDenseTridiagonal(n,S,lds,ritz,Q);CHKERRQ(ierr);

  } else {

    for (i=0;i<n;i++)

      ritz[i] = S[i+i*lds];

    for (i=0;i<l;i++)

      e[i] = S[l+i*lds];

    for (i=l;i<n;i++)

      e[i] = S[i+1+i*lds];

    ierr = ArrowTridFlip(n,l,ritz,e,Qreal,Sreal);CHKERRQ(ierr);

#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

  }

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

  ierr = EPSSortEigenvaluesReal(n,ritz,eps->which,n,perm,e);CHKERRQ(ierr);

  for (i=0;i<n;i++)

    eig[i] = ritz[perm[i]];

  for (j=0;j<n;j++)

    for (i=0;i<n;i++)

      S[i+j*lds] = Q[i+j*n];

  for (j=0;j<n;j++)

    for (i=0;i<n;i++)

      Q[i+j*n] = S[i+perm[j]*lds];

  /* Rebuild S from eig */

  for (i=0;i<n;i++) {

    S[i+i*lds] = eig[i];

    for (j=i+1;j<n;j++)

      S[j+i*lds] = 0.0;

  }

  PetscFunctionReturn(0);

}

#undef __FUNCT__  

#define __FUNCT__ "EPSSolve_KRYLOVSCHUR_SYMM"

PetscErrorCode EPSSolve_KRYLOVSCHUR_SYMM(EPS eps)

{

  PetscErrorCode ierr;

  PetscInt       i,k,l,n,lwork;

  Vec            u=eps->work[1];

  PetscScalar    *S=eps->T,*Q;

  PetscReal      beta,*work;

  PetscTruth     breakdown;

  PetscFunctionBegin;

  ierr = PetscMemzero(S,eps->ncv*eps->ncv*sizeof(PetscScalar));CHKERRQ(ierr);

  ierr = PetscMalloc(eps->ncv*eps->ncv*sizeof(PetscScalar),&Q);CHKERRQ(ierr);

  lwork = 2*eps->ncv*sizeof(PetscReal) + 2*eps->ncv*sizeof(PetscInt);

  ierr = PetscMalloc(lwork,&work);CHKERRQ(ierr);

  /* Get the starting Arnoldi 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 Arnoldi factorization */

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

    ierr = EPSBasicArnoldi(eps,PETSC_FALSE,S,eps->ncv,eps->V,eps->nconv+l,&eps->nv,u,&beta,&breakdown);CHKERRQ(ierr);

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

    n = eps->nv-eps->nconv;

    ierr = EPSProjectedKSSym(eps,n,l,S+eps->nconv*(eps->ncv+1),eps->ncv,eps->eigr+eps->nconv,Q,work);CHKERRQ(ierr);

    for (i=eps->nconv;i<eps->nv;i++)

      eps->errest[i] = beta*PetscAbsScalar(Q[(i-eps->nconv+1)*n-1]) / PetscAbsScalar(eps->eigr[i]);

    /* Check convergence */

    k = eps->nconv;

    while (k<eps->nv && eps->errest[k]<eps->tol) k++;    

    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->nv-k)/2;

    if (eps->reason == EPS_CONVERGED_ITERATING) {

      if (breakdown) {

        /* Start a new Arnoldi 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=k;i<k+l;i++) {

          S[i*eps->ncv+k+l] = Q[(i-eps->nconv+1)*n-1]*beta;

        }

      }

    }

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

    ierr = EPSUpdateVectors(n,eps->V+eps->nconv,0,k+l-eps->nconv,Q,eps->AV);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);

    }

    eps->nconv = k;

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

  }

  ierr = PetscFree(Q);CHKERRQ(ierr);

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

  PetscFunctionReturn(0);

}