Subversion Repositories slepc-dev

/*                      

     This file contains routines for handling small-size dense problems.

     All routines are simply wrappers to LAPACK routines. Matrices passed in

     as arguments are assumed to be square matrices stored in column-major

     format with a leading dimension equal to the number of rows.

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

      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 "src/eps/epsimpl.h" /*I "slepceps.h" I*/

#include "slepcblaslapack.h"

#undef __FUNCT__  

#define __FUNCT__ "EPSDenseNHEP"

/*@

   EPSDenseNHEP - Solves a dense standard non-Hermitian Eigenvalue Problem.

   Not Collective

   Input Parameters:

+  n  - dimension of the eigenproblem

-  A  - pointer to the array containing the matrix values

   Output Parameters:

+  w  - pointer to the array to store the computed eigenvalues

.  wi - imaginary part of the eigenvalues (only when using real numbers)

.  V  - pointer to the array to store right eigenvectors

-  W  - pointer to the array to store left eigenvectors

   Notes:

   If either V or W are PETSC_NULL then the corresponding eigenvectors are

   not computed.

   Matrix A is overwritten.

   This routine uses LAPACK routines xGEEVX.

   Level: developer

.seealso: EPSDenseGNHEP(), EPSDenseHEP(), EPSDenseGHEP()

@*/

PetscErrorCode EPSDenseNHEP(int n,PetscScalar *A,PetscScalar *w,PetscScalar *wi,PetscScalar *V,PetscScalar *W)

{

#if defined(SLEPC_MISSING_LAPACK_GEEVX)

  PetscFunctionBegin;

  SETERRQ(PETSC_ERR_SUP,"GEEVX - Lapack routine is unavailable.");

#else

  PetscErrorCode ierr;

  PetscReal      abnrm,*scale,dummy;

  PetscScalar    *work;

  int            ilo,ihi,lwork = 4*n,info;

  const char     *jobvr,*jobvl;

#if defined(PETSC_USE_COMPLEX)

  PetscReal      *rwork;

#else

  int            idummy;

#endif 

  PetscFunctionBegin;

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

  if (V) jobvr = "V";

  else jobvr = "N";

  if (W) jobvl = "V";

  else jobvl = "N";

  ierr  = PetscMalloc(lwork*sizeof(PetscScalar),&work);CHKERRQ(ierr);

  ierr  = PetscMalloc(n*sizeof(PetscReal),&scale);CHKERRQ(ierr);

#if defined(PETSC_USE_COMPLEX)

  ierr  = PetscMalloc(2*n*sizeof(PetscReal),&rwork);CHKERRQ(ierr);

  LAPACKgeevx_("B",jobvl,jobvr,"N",&n,A,&n,w,W,&n,V,&n,&ilo,&ihi,scale,&abnrm,&dummy,&dummy,work,&lwork,rwork,&info,1,1,1,1);

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

  ierr = PetscFree(rwork);CHKERRQ(ierr);

#else

  LAPACKgeevx_("B",jobvl,jobvr,"N",&n,A,&n,w,wi,W,&n,V,&n,&ilo,&ihi,scale,&abnrm,&dummy,&dummy,work,&lwork,&idummy,&info,1,1,1,1);

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

#endif 

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

  ierr = PetscFree(scale);CHKERRQ(ierr);

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

  PetscFunctionReturn(0);

#endif 

}

#undef __FUNCT__  

#define __FUNCT__ "EPSDenseGNHEP"

/*@

   EPSDenseGNHEP - Solves a dense Generalized non-Hermitian Eigenvalue Problem.

   Not Collective

   Input Parameters:

+  n  - dimension of the eigenproblem

.  A  - pointer to the array containing the matrix values for A

-  B  - pointer to the array containing the matrix values for B

   Output Parameters:

+  w  - pointer to the array to store the computed eigenvalues

.  wi - imaginary part of the eigenvalues (only when using real numbers)

.  V  - pointer to the array to store right eigenvectors

-  W  - pointer to the array to store left eigenvectors

   Notes:

   If either V or W are PETSC_NULL then the corresponding eigenvectors are

   not computed.

   Matrices A and B are overwritten.

   This routine uses LAPACK routines xGGEVX.

   Level: developer

.seealso: EPSDenseNHEP(), EPSDenseHEP(), EPSDenseGHEP()

@*/

PetscErrorCode EPSDenseGNHEP(int n,PetscScalar *A,PetscScalar *B,PetscScalar *w,PetscScalar *wi,PetscScalar *V,PetscScalar *W)

{

#if defined(SLEPC_MISSING_LAPACK_GGEVX)

  PetscFunctionBegin;

  SETERRQ(PETSC_ERR_SUP,"GGEVX - Lapack routine is unavailable.");

#else

  PetscErrorCode ierr;

  PetscReal      *rscale,*lscale,abnrm,bbnrm,dummy;

  PetscScalar    *alpha,*beta,*work;

  int            i,ilo,ihi,idummy,info;

  const char     *jobvr,*jobvl;

#if defined(PETSC_USE_COMPLEX)

  PetscReal      *rwork;

  int            lwork = 2*n;

#else

  PetscReal      *alphai;

  int            lwork = 6*n;

#endif 

  PetscFunctionBegin;

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

  if (V) jobvr = "V";

  else jobvr = "N";

  if (W) jobvl = "V";

  else jobvl = "N";

  ierr  = PetscMalloc(n*sizeof(PetscScalar),&alpha);CHKERRQ(ierr);

  ierr  = PetscMalloc(n*sizeof(PetscScalar),&beta);CHKERRQ(ierr);

  ierr  = PetscMalloc(n*sizeof(PetscReal),&rscale);CHKERRQ(ierr);

  ierr  = PetscMalloc(n*sizeof(PetscReal),&lscale);CHKERRQ(ierr);

  ierr  = PetscMalloc(lwork*sizeof(PetscScalar),&work);CHKERRQ(ierr);

#if defined(PETSC_USE_COMPLEX)

  ierr  = PetscMalloc(6*n*sizeof(PetscReal),&rwork);CHKERRQ(ierr);

  LAPACKggevx_("B",jobvl,jobvr,"N",&n,A,&n,B,&n,alpha,beta,W,&n,V,&n,&ilo,&ihi,lscale,rscale,&abnrm,&bbnrm,&dummy,&dummy,work,&lwork,rwork,&idummy,&idummy,&info,1,1,1,1);

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

  for (i=0;i<n;i++) {

    w[i] = alpha[i]/beta[i];

  }

  ierr = PetscFree(rwork);CHKERRQ(ierr);

#else

  ierr  = PetscMalloc(n*sizeof(PetscReal),&alphai);CHKERRQ(ierr);

  LAPACKggevx_("B",jobvl,jobvr,"N",&n,A,&n,B,&n,alpha,alphai,beta,W,&n,V,&n,&ilo,&ihi,lscale,rscale,&abnrm,&bbnrm,&dummy,&dummy,work,&lwork,&idummy,&idummy,&info,1,1,1,1);

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

  for (i=0;i<n;i++) {

    w[i] = alpha[i]/beta[i];

    wi[i] = alphai[i]/beta[i];

  }

  ierr = PetscFree(alphai);CHKERRQ(ierr);

#endif 

  ierr = PetscFree(alpha);CHKERRQ(ierr);

  ierr = PetscFree(beta);CHKERRQ(ierr);

  ierr = PetscFree(rscale);CHKERRQ(ierr);

  ierr = PetscFree(lscale);CHKERRQ(ierr);

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

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

  PetscFunctionReturn(0);

#endif

}

#undef __FUNCT__  

#define __FUNCT__ "EPSDenseHEP"

/*@

   EPSDenseHEP - Solves a dense standard Hermitian Eigenvalue Problem.

   Not Collective

   Input Parameters:

+  n   - dimension of the eigenproblem

.  A   - pointer to the array containing the matrix values

-  lda - leading dimension of A

   Output Parameters:

+  w  - pointer to the array to store the computed eigenvalues

-  V  - pointer to the array to store the eigenvectors

   Notes:

   If V is PETSC_NULL then the eigenvectors are not computed.

   Matrix A is overwritten.

   This routine uses LAPACK routines DSYEVR or ZHEEVR.

   Level: developer

.seealso: EPSDenseNHEP(), EPSDenseGNHEP(), EPSDenseGHEP()

@*/

PetscErrorCode EPSDenseHEP(int n,PetscScalar *A,int lda,PetscReal *w,PetscScalar *V)

{

#if defined(SLEPC_MISSING_LAPACK_SYEVR) || defined(SLEPC_MISSING_LAPACK_HEEVR)

  PetscFunctionBegin;

  SETERRQ(PETSC_ERR_SUP,"DSYEVR/ZHEEVR - Lapack routine is unavailable.");

#else

  PetscErrorCode ierr;

  PetscReal      abstol = 0.0,vl,vu;

  PetscScalar    *work;

  int            il,iu,m,*isuppz,*iwork,liwork = 10*n,info;

  const char     *jobz;

#if defined(PETSC_USE_COMPLEX)

  PetscReal      *rwork;

  int            lwork = 18*n,lrwork = 24*n;

#else

  int            lwork = 26*n;

#endif 

  PetscFunctionBegin;

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

  if (V) jobz = "V";

  else jobz = "N";

  ierr  = PetscMalloc(2*n*sizeof(int),&isuppz);CHKERRQ(ierr);

  ierr  = PetscMalloc(lwork*sizeof(PetscScalar),&work);CHKERRQ(ierr);

  ierr  = PetscMalloc(liwork*sizeof(int),&iwork);CHKERRQ(ierr);

#if defined(PETSC_USE_COMPLEX)

  ierr  = PetscMalloc(lrwork*sizeof(PetscReal),&rwork);CHKERRQ(ierr);

  LAPACKsyevr_(jobz,"A","L",&n,A,&lda,&vl,&vu,&il,&iu,&abstol,&m,w,V,&n,isuppz,work,&lwork,rwork,&lrwork,iwork,&liwork,&info,1,1,1);

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

  ierr = PetscFree(rwork);CHKERRQ(ierr);

#else

  LAPACKsyevr_(jobz,"A","L",&n,A,&lda,&vl,&vu,&il,&iu,&abstol,&m,w,V,&n,isuppz,work,&lwork,iwork,&liwork,&info,1,1,1);

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

#endif 

  ierr = PetscFree(isuppz);CHKERRQ(ierr);

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

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

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

  PetscFunctionReturn(0);

#endif

}

#undef __FUNCT__  

#define __FUNCT__ "EPSDenseGHEP"

/*@

   EPSDenseGHEP - Solves a dense Generalized Hermitian Eigenvalue Problem.

   Not Collective

   Input Parameters:

+  n  - dimension of the eigenproblem

.  A  - pointer to the array containing the matrix values for A

-  B  - pointer to the array containing the matrix values for B

   Output Parameters:

+  w  - pointer to the array to store the computed eigenvalues

-  V  - pointer to the array to store the eigenvectors

   Notes:

   If V is PETSC_NULL then the eigenvectors are not computed.

   Matrices A and B are overwritten.

   This routine uses LAPACK routines DSYGVD or ZHEGVD.

   Level: developer

.seealso: EPSDenseNHEP(), EPSDenseGNHEP(), EPSDenseHEP()

@*/

PetscErrorCode EPSDenseGHEP(int n,PetscScalar *A,PetscScalar *B,PetscReal *w,PetscScalar *V)

{

#if defined(SLEPC_MISSING_LAPACK_SYGVD) || defined(SLEPC_MISSING_LAPACK_HEGVD)

  PetscFunctionBegin;

  SETERRQ(PETSC_ERR_SUP,"DSYGVD/ZHEGVD - Lapack routine is unavailable.");

#else

  PetscErrorCode ierr;

  PetscScalar    *work;

  int            itype = 1,*iwork,info,

                 liwork = V ? 5*n+3 : 1;

  const char     *jobz;

#if defined(PETSC_USE_COMPLEX)

  PetscReal      *rwork;

  int            lwork  = V ? n*n+2*n     : n+1,

                 lrwork = V ? 2*n*n+5*n+1 : n;

#else

  int            lwork  = V ? 2*n*n+6*n+1 : 2*n+1;

#endif 

  PetscFunctionBegin;

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

  if (V) jobz = "V";

  else jobz = "N";  

  ierr  = PetscMalloc(lwork*sizeof(PetscScalar),&work);CHKERRQ(ierr);

  ierr  = PetscMalloc(liwork*sizeof(int),&iwork);CHKERRQ(ierr);

#if defined(PETSC_USE_COMPLEX)

  ierr  = PetscMalloc(lrwork*sizeof(PetscReal),&rwork);CHKERRQ(ierr);

  LAPACKsygvd_(&itype,jobz,"U",&n,A,&n,B,&n,w,work,&lwork,rwork,&lrwork,iwork,&liwork,&info,1,1);

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

  ierr = PetscFree(rwork);CHKERRQ(ierr);

#else

  LAPACKsygvd_(&itype,jobz,"U",&n,A,&n,B,&n,w,work,&lwork,iwork,&liwork,&info,1,1);

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

#endif 

  if (V) {

    ierr = PetscMemcpy(V,A,n*n*sizeof(PetscScalar));CHKERRQ(ierr);

  }

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

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

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

  PetscFunctionReturn(0);

#endif 

}

#undef __FUNCT__  

#define __FUNCT__ "EPSDenseHessenberg"

/*@

   EPSDenseHessenberg - Computes the Hessenberg form of a dense matrix.

   Not Collective

   Input Parameters:

+  n     - dimension of the matrix

.  k     - first active column

-  lda   - leading dimension of A

   Input/Output Parameters:

+  A  - on entry, the full matrix; on exit, the upper Hessenberg matrix (H)

-  Q  - on exit, orthogonal matrix of vectors A = Q*H*Q'

   Notes:

   Only active columns (from k to n) are computed.

   Both A and Q are overwritten.

   This routine uses LAPACK routines xGEHRD and xORGHR/xUNGHR.

   Level: developer

.seealso: EPSDenseSchur(), EPSSortDenseSchur(), EPSDenseTridiagonal()

@*/

PetscErrorCode EPSDenseHessenberg(int n,int k,PetscScalar *A,int lda,PetscScalar *Q)

{

#if defined(SLEPC_MISSING_LAPACK_GEHRD) || defined(SLEPC_MISSING_LAPACK_ORGHR) || defined(SLEPC_MISSING_LAPACK_UNGHR)

  PetscFunctionBegin;

  SETERRQ(PETSC_ERR_SUP,"GEHRD,ORGHR/UNGHR - Lapack routines are unavailable.");

#else

  PetscScalar    *tau,*work;

  PetscErrorCode ierr;

  int            i,j,ilo,lwork,info;

  PetscFunctionBegin;

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

  ierr = PetscMalloc(n*sizeof(PetscScalar),&tau);CHKERRQ(ierr);

  lwork = n;

  ierr = PetscMalloc(lwork*sizeof(PetscScalar),&work);CHKERRQ(ierr);

  ilo = k+1;

  LAPACKgehrd_(&n,&ilo,&n,A,&lda,tau,work,&lwork,&info);

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

  for (j=0;j<n-1;j++) {

    for (i=j+2;i<n;i++) {

      Q[i+j*n] = A[i+j*lda];

      A[i+j*lda] = 0.0;

    }      

  }

  LAPACKorghr_(&n,&ilo,&n,Q,&n,tau,work,&lwork,&info);

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

  ierr = PetscFree(tau);CHKERRQ(ierr);

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

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

  PetscFunctionReturn(0);

#endif

}

#undef __FUNCT__  

#define __FUNCT__ "EPSDenseSchur"

/*@

   EPSDenseSchur - Computes the upper (quasi-)triangular form of a dense

   upper Hessenberg matrix.

   Not Collective

   Input Parameters:

+  n   - dimension of the matrix

.  k   - first active column

-  ldh - leading dimension of H

   Input/Output Parameters:

+  H  - on entry, the upper Hessenber matrix; on exit, the upper

        (quasi-)triangular matrix (T)

-  Z  - on entry, initial transformation matrix; on exit, orthogonal

        matrix of Schur vectors

   Output Parameters:

+  wr - pointer to the array to store the computed eigenvalues

-  wi - imaginary part of the eigenvalues (only when using real numbers)

   Notes:

   This function computes the (real) Schur decomposition of an upper

   Hessenberg matrix H: H*Z = Z*T,  where T is an upper (quasi-)triangular

   matrix (returned in H), and Z is the orthogonal matrix of Schur vectors.

   Eigenvalues are extracted from the diagonal blocks of T and returned in

   wr,wi. Transformations are accumulated in Z so that on entry it can

   contain the transformation matrix associated to the Hessenberg reduction.

   Only active columns (from k to n) are computed.

   Both H and Z are overwritten.

   This routine uses LAPACK routines xHSEQR.

   Level: developer

.seealso: EPSDenseHessenberg(), EPSSortDenseSchur(), EPSDenseTridiagonal()

@*/

PetscErrorCode EPSDenseSchur(int n,int k,PetscScalar *H,int ldh,PetscScalar *Z,PetscScalar *wr,PetscScalar *wi)

{

#if defined(SLEPC_MISSING_LAPACK_HSEQR)

  PetscFunctionBegin;

  SETERRQ(PETSC_ERR_SUP,"HSEQR - Lapack routine is unavailable.");

#else

  PetscErrorCode ierr;

  int ilo,lwork,info;

  PetscScalar *work;

#if !defined(PETSC_USE_COMPLEX)

  int j;

#endif

  PetscFunctionBegin;

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

  lwork = n;

  ierr = PetscMalloc(lwork*sizeof(PetscScalar),&work);CHKERRQ(ierr);

  ilo = k+1;

#if !defined(PETSC_USE_COMPLEX)

  LAPACKhseqr_("S","V",&n,&ilo,&n,H,&ldh,wr,wi,Z,&n,work,&lwork,&info,1,1);

  for (j=0;j<k;j++) {

    if (j==n-1 || H[j*ldh+j+1] == 0.0) {

      /* real eigenvalue */

      wr[j] = H[j*ldh+j];

      wi[j] = 0.0;

    } else {

      /* complex eigenvalue */

      wr[j] = H[j*ldh+j];

      wr[j+1] = H[j*ldh+j];

      wi[j] = sqrt(PetscAbsReal(H[j*ldh+j+1])) *

              sqrt(PetscAbsReal(H[(j+1)*ldh+j]));

      wi[j+1] = -wi[j];

      j++;

    }

  }

#else

  LAPACKhseqr_("S","V",&n,&ilo,&n,H,&ldh,wr,Z,&n,work,&lwork,&info,1,1);

#endif

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

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

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

  PetscFunctionReturn(0);

#endif

}

#undef __FUNCT__  

#define __FUNCT__ "EPSSortDenseSchur"

/*@

   EPSSortDenseSchur - Reorders the Schur decomposition computed by

   EPSDenseSchur().

   Not Collective

   Input Parameters:

+  n     - dimension of the matrix

.  k     - first active column

.  ldt   - leading dimension of T

-  which - eigenvalue sort order

   Input/Output Parameters:

+  T  - the upper (quasi-)triangular matrix

-  Z  - the orthogonal matrix of Schur vectors

   Output Parameters:

+  wr - pointer to the array to store the computed eigenvalues

-  wi - imaginary part of the eigenvalues (only when using real numbers)

   Notes:

   This function reorders the eigenvalues in wr,wi located in positions k

   to n according to the sort order specified in which. The Schur

   decomposition Z*T*Z^T, is also reordered by means of rotations so that

   eigenvalues in the diagonal blocks of T follow the same order.

   Both T and Z are overwritten.

   This routine uses LAPACK routines xTREXC.

   Level: developer

.seealso: EPSDenseHessenberg(), EPSDenseSchur(), EPSDenseTridiagonal()

@*/

PetscErrorCode EPSSortDenseSchur(int n,int k,PetscScalar *T,int ldt,PetscScalar *Z,PetscScalar *wr,PetscScalar *wi,EPSWhich which)

{

#if defined(SLEPC_MISSING_LAPACK_TREXC)

  PetscFunctionBegin;

  SETERRQ(PETSC_ERR_SUP,"TREXC - Lapack routine is unavailable.");

#else

  int i,j,ifst,ilst,info,pos;

#if !defined(PETSC_USE_COMPLEX)

  PetscScalar *work;

#endif

  PetscReal   value,v;

  PetscErrorCode ierr;

  PetscFunctionBegin;

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

#if !defined(PETSC_USE_COMPLEX)

  ierr = PetscMalloc(n*sizeof(PetscScalar),&work);CHKERRQ(ierr);

#endif

  for (i=k;i<n-1;i++) {

    switch(which) {

      case EPS_LARGEST_MAGNITUDE:

      case EPS_SMALLEST_MAGNITUDE:

        value = SlepcAbsEigenvalue(wr[i],wi[i]);

        break;

      case EPS_LARGEST_REAL:

      case EPS_SMALLEST_REAL:

        value = PetscRealPart(wr[i]);

        break;

      case EPS_LARGEST_IMAGINARY:

      case EPS_SMALLEST_IMAGINARY:

#if !defined(PETSC_USE_COMPLEX)

        value = PetscAbsReal(wi[i]);

#else

        value = PetscImaginaryPart(wr[i]);

#endif

        break;

      default: SETERRQ(1,"Wrong value of which");

    }

    pos = 0;

    for (j=i+1;j<n;j++) {

      switch(which) {

        case EPS_LARGEST_MAGNITUDE:

        case EPS_SMALLEST_MAGNITUDE:

          v = SlepcAbsEigenvalue(wr[j],wi[j]);

          break;

        case EPS_LARGEST_REAL:

        case EPS_SMALLEST_REAL:

          v = PetscRealPart(wr[j]);

          break;

        case EPS_LARGEST_IMAGINARY:

        case EPS_SMALLEST_IMAGINARY:

#if !defined(PETSC_USE_COMPLEX)

          v = PetscAbsReal(wi[j]);

#else

          v = PetscImaginaryPart(wr[j]);

#endif

          break;

        default: SETERRQ(1,"Wrong value of which");

      }

      switch(which) {

        case EPS_LARGEST_MAGNITUDE:

        case EPS_LARGEST_REAL:

        case EPS_LARGEST_IMAGINARY:

          if (v > value) {

            value = v;

            pos = j;

          }

          break;

        case EPS_SMALLEST_MAGNITUDE:

        case EPS_SMALLEST_REAL:

        case EPS_SMALLEST_IMAGINARY:

          if (v < value) {

            value = v;

            pos = j;

          }

          break;

        default: SETERRQ(1,"Wrong value of which");

      }

#if !defined(PETSC_USE_COMPLEX)

      if (wi[j] != 0) j++;

#endif

    }

    if (pos) {

      ifst = pos + 1;

      ilst = i + 1;

#if !defined(PETSC_USE_COMPLEX)

      LAPACKtrexc_("V",&n,T,&ldt,Z,&n,&ifst,&ilst,work,&info,1);

#else

      LAPACKtrexc_("V",&n,T,&ldt,Z,&n,&ifst,&ilst,&info,1);

#endif

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

      for (j=k;j<n;j++) {

#if !defined(PETSC_USE_COMPLEX)

        if (j==n-1 || T[j*ldt+j+1] == 0.0) {

          /* real eigenvalue */

          wr[j] = T[j*ldt+j];

          wi[j] = 0.0;

        } else {

          /* complex eigenvalue */

          wr[j] = T[j*ldt+j];

          wr[j+1] = T[j*ldt+j];

          wi[j] = sqrt(PetscAbsReal(T[j*ldt+j+1])) *

                  sqrt(PetscAbsReal(T[(j+1)*ldt+j]));

          wi[j+1] = -wi[j];

          j++;

        }

#else

        wr[j] = T[j*(ldt+1)];

#endif

      }

    }

#if !defined(PETSC_USE_COMPLEX)

    if (wi[i] != 0) i++;

#endif

  }

#if !defined(PETSC_USE_COMPLEX)

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

#endif

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

  PetscFunctionReturn(0);

#endif 

}

#undef __FUNCT__  

#define __FUNCT__ "EPSDenseTridiagonal"

/*@

   EPSDenseTridiagonal - Solves a real tridiagonal Hermitian Eigenvalue Problem.

   Not Collective

   Input Parameters:

+  n   - dimension of the eigenproblem

.  A   - pointer to the array containing the matrix values

-  lda - leading dimension of A

   Output Parameters:

+  w  - pointer to the array to store the computed eigenvalues

-  V  - pointer to the array to store the eigenvectors

   Notes:

   If V is PETSC_NULL then the eigenvectors are not computed.

   This routine use LAPACK routines DSTEVR.

   Level: developer

.seealso: EPSDenseNHEP(), EPSDenseHEP(), EPSDenseGNHEP(), EPSDenseGHEP()

@*/

PetscErrorCode EPSDenseTridiagonal(int n,PetscScalar *A,int lda,PetscReal *w,PetscScalar *V)

{

#if defined(SLEPC_MISSING_LAPACK_DSTEVR)

  PetscFunctionBegin;

  SETERRQ(PETSC_ERR_SUP,"DSTEVR - Lapack routine is unavailable.");

#else

  PetscErrorCode ierr;

  PetscReal      abstol = 0.0,vl,vu,*D,*E,*work;

  int            i,il,iu,m,*isuppz,lwork = 20*n,*iwork,liwork = 10*n,info;

  const char     *jobz;

#if defined(PETSC_USE_COMPLEX)

  int            j;

  PetscReal      *VV;

#endif