AnasaziBlockKrylovSchur.hpp

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// @HEADER
// *****************************************************************************
//                 Anasazi: Block Eigensolvers Package
//
// Copyright 2004 NTESS and the Anasazi contributors.
// SPDX-License-Identifier: BSD-3-Clause
// *****************************************************************************
// @HEADER

#ifndef ANASAZI_BLOCK_KRYLOV_SCHUR_HPP
#define ANASAZI_BLOCK_KRYLOV_SCHUR_HPP

#include "AnasaziTypes.hpp"

#include "AnasaziEigensolver.hpp"
#include "AnasaziMultiVecTraits.hpp"
#include "AnasaziOperatorTraits.hpp"
#include "Teuchos_ScalarTraits.hpp"
#include "AnasaziHelperTraits.hpp"

#include "AnasaziOrthoManager.hpp"

#include "Teuchos_LAPACK.hpp"
#include "Teuchos_BLAS.hpp"
#include "Teuchos_SerialDenseMatrix.hpp"
#include "Teuchos_ParameterList.hpp"
#include "Teuchos_TimeMonitor.hpp"

namespace Anasazi {



  template <class ScalarType, class MulVec>
  struct BlockKrylovSchurState {
    int curDim;
    Teuchos::RCP<const MulVec> V;
    Teuchos::RCP<const Teuchos::SerialDenseMatrix<int,ScalarType> > H;
    Teuchos::RCP<const Teuchos::SerialDenseMatrix<int,ScalarType> > S;
    Teuchos::RCP<const Teuchos::SerialDenseMatrix<int,ScalarType> > Q;
    BlockKrylovSchurState() : curDim(0), V(Teuchos::null),
                              H(Teuchos::null), S(Teuchos::null),
                              Q(Teuchos::null) {}
  };




  class BlockKrylovSchurInitFailure : public AnasaziError {public:
    BlockKrylovSchurInitFailure(const std::string& what_arg) : AnasaziError(what_arg)
    {}};

  class BlockKrylovSchurOrthoFailure : public AnasaziError {public:
    BlockKrylovSchurOrthoFailure(const std::string& what_arg) : AnasaziError(what_arg)
    {}};
  


  template <class ScalarType, class MV, class OP>
  class BlockKrylovSchur : public Eigensolver<ScalarType,MV,OP> { 
  public:

    
    BlockKrylovSchur( const Teuchos::RCP<Eigenproblem<ScalarType,MV,OP> > &problem, 
                      const Teuchos::RCP<SortManager<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> > &sorter,
                      const Teuchos::RCP<OutputManager<ScalarType> > &printer,
                      const Teuchos::RCP<StatusTest<ScalarType,MV,OP> > &tester,
                      const Teuchos::RCP<OrthoManager<ScalarType,MV> > &ortho,
                      Teuchos::ParameterList &params 
                    );
    
    virtual ~BlockKrylovSchur() {};



    
    void iterate();

    void initialize(BlockKrylovSchurState<ScalarType,MV>& state);

    void initialize();

    bool isInitialized() const { return initialized_; }

    BlockKrylovSchurState<ScalarType,MV> getState() const {
      BlockKrylovSchurState<ScalarType,MV> state;
      state.curDim = curDim_;
      state.V = V_;
      state.H = H_;
      state.Q = Q_;
      state.S = schurH_;
      return state;
    }
    




    int getNumIters() const { return(iter_); }

    void resetNumIters() { iter_=0; }

    Teuchos::RCP<const MV> getRitzVectors() { return ritzVectors_; }

    std::vector<Value<ScalarType> > getRitzValues() { 
      std::vector<Value<ScalarType> > ret = ritzValues_;
      ret.resize(ritzIndex_.size());
      return ret;
    }

    std::vector<int> getRitzIndex() { return ritzIndex_; }

    std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> getResNorms() {
      std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> ret(0);
      return ret;
    }

    std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> getRes2Norms() {
      std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> ret(0);
      return ret;
    }

    std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> getRitzRes2Norms() {
      std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> ret = ritzResiduals_;
      ret.resize(ritzIndex_.size());
      return ret;
    }




    void setStatusTest(Teuchos::RCP<StatusTest<ScalarType,MV,OP> > test);

    Teuchos::RCP<StatusTest<ScalarType,MV,OP> > getStatusTest() const;

    const Eigenproblem<ScalarType,MV,OP>& getProblem() const { return(*problem_); };

    void setSize(int blockSize, int numBlocks);

    void setBlockSize(int blockSize);

    void setStepSize(int stepSize);

    void setNumRitzVectors(int numRitzVecs);

    int getStepSize() const { return(stepSize_); }

    int getBlockSize() const { return(blockSize_); }

    int getNumRitzVectors() const { return(numRitzVecs_); }

    int getCurSubspaceDim() const {
      if (!initialized_) return 0;
      return curDim_;
    }

    int getMaxSubspaceDim() const { return (problem_->isHermitian()?blockSize_*numBlocks_:blockSize_*numBlocks_+1); }


    void setAuxVecs(const Teuchos::Array<Teuchos::RCP<const MV> > &auxvecs);

    Teuchos::Array<Teuchos::RCP<const MV> > getAuxVecs() const {return auxVecs_;}



    
    void currentStatus(std::ostream &os);



    
    bool isRitzVecsCurrent() const { return ritzVecsCurrent_; }

    bool isRitzValsCurrent() const { return ritzValsCurrent_; }
    
    bool isSchurCurrent() const { return schurCurrent_; }
    


    
    void computeRitzVectors();

    void computeRitzValues();
    
    void computeSchurForm( const bool sort = true );


  private:
    //
    // Convenience typedefs
    //
    typedef MultiVecTraits<ScalarType,MV> MVT;
    typedef OperatorTraits<ScalarType,MV,OP> OPT;
    typedef Teuchos::ScalarTraits<ScalarType> SCT;
    typedef typename SCT::magnitudeType MagnitudeType;
    typedef typename std::vector<ScalarType>::iterator STiter;
    typedef typename std::vector<MagnitudeType>::iterator MTiter;
    const MagnitudeType MT_ONE;  
    const MagnitudeType MT_ZERO; 
    const MagnitudeType NANVAL;
    const ScalarType ST_ONE;
    const ScalarType ST_ZERO;
    //
    // Internal structs
    //
    struct CheckList {
      bool checkV;
      bool checkArn;
      bool checkAux;
      CheckList() : checkV(false), checkArn(false), checkAux(false) {};
    };
    //
    // Internal methods
    //
    std::string accuracyCheck(const CheckList &chk, const std::string &where) const;
    void sortSchurForm( Teuchos::SerialDenseMatrix<int,ScalarType>& H,
                        Teuchos::SerialDenseMatrix<int,ScalarType>& Q,
                        std::vector<int>& order );
    //
    // Classes inputed through constructor that define the eigenproblem to be solved.
    //
    const Teuchos::RCP<Eigenproblem<ScalarType,MV,OP> >     problem_;
    const Teuchos::RCP<SortManager<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> > sm_;
    const Teuchos::RCP<OutputManager<ScalarType> >          om_;
    Teuchos::RCP<StatusTest<ScalarType,MV,OP> >             tester_;
    const Teuchos::RCP<OrthoManager<ScalarType,MV> >        orthman_;
    //
    // Information obtained from the eigenproblem
    //
    Teuchos::RCP<const OP> Op_;
    //
    // Internal timers
    //
    Teuchos::RCP<Teuchos::Time> timerOp_, timerSortRitzVal_,
                                        timerCompSF_, timerSortSF_,
                                        timerCompRitzVec_, timerOrtho_;
    //
    // Counters
    //
    int count_ApplyOp_;

    //
    // Algorithmic parameters.
    //
    // blockSize_ is the solver block size; it controls the number of eigenvectors that 
    // we compute, the number of residual vectors that we compute, and therefore the number
    // of vectors added to the basis on each iteration.
    int blockSize_;
    // numBlocks_ is the size of the allocated space for the Krylov basis, in blocks.
    int numBlocks_; 
    // stepSize_ dictates how many iterations are performed before eigenvectors and eigenvalues
    // are computed again
    int stepSize_;
    
    // 
    // Current solver state
    //
    // initialized_ specifies that the basis vectors have been initialized and the iterate() routine
    // is capable of running; _initialize is controlled  by the initialize() member method
    // For the implications of the state of initialized_, please see documentation for initialize()
    bool initialized_;
    //
    // curDim_ reflects how much of the current basis is valid 
    // NOTE: for Hermitian, 0 <= curDim_ <= blockSize_*numBlocks_
    //   for non-Hermitian, 0 <= curDim_ <= blockSize_*numBlocks_ + 1
    // this also tells us how many of the values in _theta are valid Ritz values
    int curDim_;
    //
    // State Multivecs
    Teuchos::RCP<MV> ritzVectors_, V_;
    int numRitzVecs_;
    //
    // Projected matrices
    // H_ : Projected matrix from the Krylov-Schur factorization AV = VH + FB^T
    //
    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > H_;
    // 
    // Schur form of Projected matrices (these are only updated when the Ritz values/vectors are updated).
    // schurH_: Schur form reduction of H
    // Q_: Schur vectors of H
    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > schurH_;
    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > Q_;
    // 
    // Auxiliary vectors
    Teuchos::Array<Teuchos::RCP<const MV> > auxVecs_;
    int numAuxVecs_;
    //
    // Number of iterations that have been performed.
    int iter_;
    //
    // State flags
    bool ritzVecsCurrent_, ritzValsCurrent_, schurCurrent_;
    // 
    // Current eigenvalues, residual norms
    std::vector<Value<ScalarType> > ritzValues_;
    std::vector<MagnitudeType> ritzResiduals_;
    // 
    // Current index vector for Ritz values and vectors
    std::vector<int> ritzIndex_;  // computed by BKS
    std::vector<int> ritzOrder_;  // returned from sort manager
    //
    // Number of Ritz pairs to be printed upon output, if possible
    int numRitzPrint_;
  };


  // Constructor
  template <class ScalarType, class MV, class OP>
  BlockKrylovSchur<ScalarType,MV,OP>::BlockKrylovSchur(
        const Teuchos::RCP<Eigenproblem<ScalarType,MV,OP> > &problem, 
        const Teuchos::RCP<SortManager<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> > &sorter,
        const Teuchos::RCP<OutputManager<ScalarType> > &printer,
        const Teuchos::RCP<StatusTest<ScalarType,MV,OP> > &tester,
        const Teuchos::RCP<OrthoManager<ScalarType,MV> > &ortho,
        Teuchos::ParameterList &params
        ) :
    MT_ONE(Teuchos::ScalarTraits<MagnitudeType>::one()),
    MT_ZERO(Teuchos::ScalarTraits<MagnitudeType>::zero()),
    NANVAL(Teuchos::ScalarTraits<MagnitudeType>::nan()),
    ST_ONE(Teuchos::ScalarTraits<ScalarType>::one()),
    ST_ZERO(Teuchos::ScalarTraits<ScalarType>::zero()),
    // problem, tools
    problem_(problem), 
    sm_(sorter),
    om_(printer),
    tester_(tester),
    orthman_(ortho),
    // timers, counters
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
    timerOp_(Teuchos::TimeMonitor::getNewTimer("Anasazi: BlockKrylovSchur::Operation Op*x")),
    timerSortRitzVal_(Teuchos::TimeMonitor::getNewTimer("Anasazi: BlockKrylovSchur::Sorting Ritz values")),
    timerCompSF_(Teuchos::TimeMonitor::getNewTimer("Anasazi: BlockKrylovSchur::Computing Schur form")),
    timerSortSF_(Teuchos::TimeMonitor::getNewTimer("Anasazi: BlockKrylovSchur::Sorting Schur form")),
    timerCompRitzVec_(Teuchos::TimeMonitor::getNewTimer("Anasazi: BlockKrylovSchur::Computing Ritz vectors")),
    timerOrtho_(Teuchos::TimeMonitor::getNewTimer("Anasazi: BlockKrylovSchur::Orthogonalization")),
#endif
    count_ApplyOp_(0),
    // internal data
    blockSize_(0),
    numBlocks_(0),
    stepSize_(0),
    initialized_(false),
    curDim_(0),
    numRitzVecs_(0),
    auxVecs_( Teuchos::Array<Teuchos::RCP<const MV> >(0) ), 
    numAuxVecs_(0),
    iter_(0),
    ritzVecsCurrent_(false),
    ritzValsCurrent_(false),
    schurCurrent_(false),
    numRitzPrint_(0)
  {     
    TEUCHOS_TEST_FOR_EXCEPTION(problem_ == Teuchos::null,std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: user specified null problem pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(sm_ == Teuchos::null,std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: user passed null sort manager pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(om_ == Teuchos::null,std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: user passed null output manager pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(tester_ == Teuchos::null,std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: user passed null status test pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(orthman_ == Teuchos::null,std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: user passed null orthogonalization manager pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(problem_->isProblemSet() == false, std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: user specified problem is not set.");
    TEUCHOS_TEST_FOR_EXCEPTION(sorter == Teuchos::null,std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: user specified null sort manager pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(printer == Teuchos::null,std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: user specified null output manager pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(tester == Teuchos::null,std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: user specified null status test pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(ortho == Teuchos::null,std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: user specified null ortho manager pointer.");

    // Get problem operator
    Op_ = problem_->getOperator();

    // get the step size
    TEUCHOS_TEST_FOR_EXCEPTION(!params.isParameter("Step Size"), std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::constructor: mandatory parameter 'Step Size' is not specified.");
    int ss = params.get("Step Size",numBlocks_);
    setStepSize(ss);

    // set the block size and allocate data
    int bs = params.get("Block Size", 1);
    int nb = params.get("Num Blocks", 3*problem_->getNEV());
    setSize(bs,nb);

    // get the number of Ritz vectors to compute and allocate data.
    // --> if this parameter is not specified in the parameter list, then it's assumed that no Ritz vectors will be computed.
    int numRitzVecs = params.get("Number of Ritz Vectors", 0);
    setNumRitzVectors( numRitzVecs );

    // get the number of Ritz values to print out when currentStatus is called.
    numRitzPrint_ = params.get("Print Number of Ritz Values", bs);
  }


  // Set the block size
  // This simply calls setSize(), modifying the block size while retaining the number of blocks.
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::setBlockSize (int blockSize) 
  {
    setSize(blockSize,numBlocks_);
  }


  // Set the step size.
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::setStepSize (int stepSize)
  {
    TEUCHOS_TEST_FOR_EXCEPTION(stepSize <= 0, std::invalid_argument, "Anasazi::BlockKrylovSchur::setStepSize(): new step size must be positive and non-zero.");
    stepSize_ = stepSize;
  }

  // Set the number of Ritz vectors to compute.
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::setNumRitzVectors (int numRitzVecs)
  {
    // This routine only allocates space; it doesn't not perform any computation
    // any change in size will invalidate the state of the solver.

    TEUCHOS_TEST_FOR_EXCEPTION(numRitzVecs < 0, std::invalid_argument, "Anasazi::BlockKrylovSchur::setNumRitzVectors(): number of Ritz vectors to compute must be positive.");

    // Check to see if the number of requested Ritz vectors has changed.
    if (numRitzVecs != numRitzVecs_) {
      if (numRitzVecs) {
        ritzVectors_ = Teuchos::null;
        ritzVectors_ = MVT::Clone(*V_, numRitzVecs);
      } else {
        ritzVectors_ = Teuchos::null;
      }
      numRitzVecs_ = numRitzVecs;
      ritzVecsCurrent_ = false;
    }      
  }

  // Set the block size and make necessary adjustments.
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::setSize (int blockSize, int numBlocks) 
  {
    // This routine only allocates space; it doesn't not perform any computation
    // any change in size will invalidate the state of the solver.

    TEUCHOS_TEST_FOR_EXCEPTION(numBlocks <= 0 || blockSize <= 0, std::invalid_argument, "Anasazi::BlockKrylovSchur::setSize was passed a non-positive argument.");
    TEUCHOS_TEST_FOR_EXCEPTION(numBlocks < 3, std::invalid_argument, "Anasazi::BlockKrylovSchur::setSize(): numBlocks must be at least three.");
    if (blockSize == blockSize_ && numBlocks == numBlocks_) {
      // do nothing
      return;
    }

    blockSize_ = blockSize;
    numBlocks_ = numBlocks;

    Teuchos::RCP<const MV> tmp;
    // grab some Multivector to Clone
    // in practice, getInitVec() should always provide this, but it is possible to use a 
    // Eigenproblem with nothing in getInitVec() by manually initializing with initialize(); 
    // in case of that strange scenario, we will try to Clone from V_; first resort to getInitVec(), 
    // because we would like to clear the storage associated with V_ so we have room for the new V_
    if (problem_->getInitVec() != Teuchos::null) {
      tmp = problem_->getInitVec();
    }
    else {
      tmp = V_;
      TEUCHOS_TEST_FOR_EXCEPTION(tmp == Teuchos::null,std::invalid_argument,
          "Anasazi::BlockKrylovSchur::setSize(): eigenproblem did not specify initial vectors to clone from.");
    }


    // blockSize*numBlocks dependent
    //
    int newsd;
    if (problem_->isHermitian()) {
      newsd = blockSize_*numBlocks_;
    } else {
      newsd = blockSize_*numBlocks_+1;
    }
    // check that new size is valid
    TEUCHOS_TEST_FOR_EXCEPTION(static_cast<ptrdiff_t>(newsd) > MVT::GetGlobalLength(*tmp),std::invalid_argument,
        "Anasazi::BlockKrylovSchur::setSize(): maximum basis size is larger than problem dimension.");

    ritzValues_.resize(newsd);
    ritzResiduals_.resize(newsd,MT_ONE);
    ritzOrder_.resize(newsd);
    V_ = Teuchos::null;
    V_ = MVT::Clone(*tmp,newsd+blockSize_);
    H_ = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(newsd+blockSize_,newsd) );
    Q_ = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(newsd,newsd) );

    initialized_ = false;
    curDim_ = 0;
  }


  // Set the auxiliary vectors
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::setAuxVecs(const Teuchos::Array<Teuchos::RCP<const MV> > &auxvecs) {
    typedef typename Teuchos::Array<Teuchos::RCP<const MV> >::iterator tarcpmv;

    // set new auxiliary vectors
    auxVecs_ = auxvecs;
    
    if (om_->isVerbosity( Debug ) ) {
      // Check almost everything here
      CheckList chk;
      chk.checkAux = true;
      om_->print( Debug, accuracyCheck(chk, ": in setAuxVecs()") );
    }

    numAuxVecs_ = 0;
    for (tarcpmv i=auxVecs_.begin(); i != auxVecs_.end(); i++) {
      numAuxVecs_ += MVT::GetNumberVecs(**i);
    }
    
    // If the solver has been initialized, X and P are not necessarily orthogonal to new auxiliary vectors
    if (numAuxVecs_ > 0 && initialized_) {
      initialized_ = false;
    }
  }

  /* Initialize the state of the solver
   * 
   * POST-CONDITIONS:
   *
   * V_ is orthonormal, orthogonal to auxVecs_, for first curDim_ vectors
   *
   */

  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::initialize(BlockKrylovSchurState<ScalarType,MV>& newstate)
  {
    // NOTE: memory has been allocated by setBlockSize(). Use SetBlock below; do not Clone

    std::vector<int> bsind(blockSize_);
    for (int i=0; i<blockSize_; i++) bsind[i] = i;

    // in BlockKrylovSchur, V and H are required.  
    // if either doesn't exist, then we will start with the initial vector.
    //
    // inconsistent multivectors widths and lengths will not be tolerated, and
    // will be treated with exceptions.
    //
    std::string errstr("Anasazi::BlockKrylovSchur::initialize(): specified multivectors must have a consistent length and width.");

    // set up V,H: if the user doesn't specify both of these these, 
    // we will start over with the initial vector.
    if (newstate.V != Teuchos::null && newstate.H != Teuchos::null) {

      // initialize V_,H_, and curDim_

      TEUCHOS_TEST_FOR_EXCEPTION( MVT::GetGlobalLength(*newstate.V) != MVT::GetGlobalLength(*V_),
                          std::invalid_argument, errstr );
      if (newstate.V != V_) {
        TEUCHOS_TEST_FOR_EXCEPTION( MVT::GetNumberVecs(*newstate.V) < blockSize_,
            std::invalid_argument, errstr );
        TEUCHOS_TEST_FOR_EXCEPTION( MVT::GetNumberVecs(*newstate.V) > getMaxSubspaceDim(),
            std::invalid_argument, errstr );
      }
      TEUCHOS_TEST_FOR_EXCEPTION( newstate.curDim > getMaxSubspaceDim(),
                          std::invalid_argument, errstr );

      curDim_ = newstate.curDim;
      int lclDim = MVT::GetNumberVecs(*newstate.V);

      // check size of H
      TEUCHOS_TEST_FOR_EXCEPTION(newstate.H->numRows() < curDim_ || newstate.H->numCols() < curDim_, std::invalid_argument, errstr);
      
      if (curDim_ == 0 && lclDim > blockSize_) {
        om_->stream(Warnings) << "Anasazi::BlockKrylovSchur::initialize(): the solver was initialized with a kernel of " << lclDim << std::endl
                                  << "The block size however is only " << blockSize_ << std::endl
                                  << "The last " << lclDim - blockSize_ << " vectors of the kernel will be overwritten on the first call to iterate()." << std::endl;
      }


      // copy basis vectors from newstate into V
      if (newstate.V != V_) {
        std::vector<int> nevind(lclDim);
        for (int i=0; i<lclDim; i++) nevind[i] = i;
        MVT::SetBlock(*newstate.V,nevind,*V_);
      }

      // put data into H_, make sure old information is not still hanging around.
      if (newstate.H != H_) {
        H_->putScalar( ST_ZERO );
        Teuchos::SerialDenseMatrix<int,ScalarType> newH(Teuchos::View,*newstate.H,curDim_+blockSize_,curDim_);
        Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > lclH;
        lclH = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*H_,curDim_+blockSize_,curDim_) );
        lclH->assign(newH);

        // done with local pointers
        lclH = Teuchos::null;
      }

    }
    else {
      // user did not specify a basis V
      // get vectors from problem or generate something, projectAndNormalize, call initialize() recursively
      Teuchos::RCP<const MV> ivec = problem_->getInitVec();
      TEUCHOS_TEST_FOR_EXCEPTION(ivec == Teuchos::null,std::invalid_argument,
                         "Anasazi::BlockKrylovSchur::initialize(): eigenproblem did not specify initial vectors to clone from.");

      int lclDim = MVT::GetNumberVecs(*ivec);
      bool userand = false;
      if (lclDim < blockSize_) {
        // we need at least blockSize_ vectors
        // use a random multivec
        userand = true;
      }
              
      if (userand) {
        // make an index
        std::vector<int> dimind2(lclDim);
        for (int i=0; i<lclDim; i++) { dimind2[i] = i; }

        // alloc newV as a view of the first block of V
        Teuchos::RCP<MV> newV1 = MVT::CloneViewNonConst(*V_,dimind2);

        // copy the initial vectors into the first lclDim vectors of V
        MVT::SetBlock(*ivec,dimind2,*newV1);

        // resize / reinitialize the index vector        
        dimind2.resize(blockSize_-lclDim);
        for (int i=0; i<blockSize_-lclDim; i++) { dimind2[i] = lclDim + i; }

        // initialize the rest of the vectors with random vectors
        Teuchos::RCP<MV> newV2 = MVT::CloneViewNonConst(*V_,dimind2);
        MVT::MvRandom(*newV2);
      }
      else {
        // alloc newV as a view of the first block of V
        Teuchos::RCP<MV> newV1 = MVT::CloneViewNonConst(*V_,bsind);
       
        // get a view of the first block of initial vectors
        Teuchos::RCP<const MV> ivecV = MVT::CloneView(*ivec,bsind);
 
        // assign ivec to first part of newV
        MVT::SetBlock(*ivecV,bsind,*newV1);
      }

      // get pointer into first block of V
      Teuchos::RCP<MV> newV = MVT::CloneViewNonConst(*V_,bsind);

      // remove auxVecs from newV and normalize newV
      if (auxVecs_.size() > 0) {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor lcltimer( *timerOrtho_ );
#endif
        
        Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > dummy;
        int rank = orthman_->projectAndNormalize(*newV,auxVecs_);
        TEUCHOS_TEST_FOR_EXCEPTION( rank != blockSize_,BlockKrylovSchurInitFailure,
                            "Anasazi::BlockKrylovSchur::initialize(): couldn't generate initial basis of full rank." );
      }
      else {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor lcltimer( *timerOrtho_ );
#endif

        int rank = orthman_->normalize(*newV);
        TEUCHOS_TEST_FOR_EXCEPTION( rank != blockSize_,BlockKrylovSchurInitFailure,
                            "Anasazi::BlockKrylovSchur::initialize(): couldn't generate initial basis of full rank." );
      }

      // set curDim
      curDim_ = 0;

      // clear pointer
      newV = Teuchos::null;
    }

    // The Ritz vectors/values and Schur form are no longer current.
    ritzVecsCurrent_ = false;
    ritzValsCurrent_ = false;
    schurCurrent_ = false;

    // the solver is initialized
    initialized_ = true;

    if (om_->isVerbosity( Debug ) ) {
      // Check almost everything here
      CheckList chk;
      chk.checkV = true;
      chk.checkArn = true;
      chk.checkAux = true;
      om_->print( Debug, accuracyCheck(chk, ": after initialize()") );
    }

    // Print information on current status
    if (om_->isVerbosity(Debug)) {
      currentStatus( om_->stream(Debug) );
    }
    else if (om_->isVerbosity(IterationDetails)) {
      currentStatus( om_->stream(IterationDetails) );
    }
  }


  // initialize the solver with default state
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::initialize()
  {
    BlockKrylovSchurState<ScalarType,MV> empty;
    initialize(empty);
  }


  // Perform BlockKrylovSchur iterations until the StatusTest tells us to stop.
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::iterate()
  {
    //
    // Allocate/initialize data structures
    //
    if (initialized_ == false) {
      initialize();
    }

    // Compute the current search dimension. 
    // If the problem is non-Hermitian and the blocksize is one, let the solver use the extra vector.
    int searchDim = blockSize_*numBlocks_;
    if (problem_->isHermitian() == false) {
      searchDim++;
    } 

    // iterate until the status test tells us to stop.
    //
    // also break if our basis is full
    //
    while (tester_->checkStatus(this) != Passed && curDim_+blockSize_ <= searchDim) {

      iter_++;

      // F can be found at the curDim_ block, but the next block is at curDim_ + blockSize_.
      int lclDim = curDim_ + blockSize_; 

      // Get the current part of the basis.
      std::vector<int> curind(blockSize_);
      for (int i=0; i<blockSize_; i++) { curind[i] = lclDim + i; }
      Teuchos::RCP<MV> Vnext = MVT::CloneViewNonConst(*V_,curind);

      // Get a view of the previous vectors
      // this is used for orthogonalization and for computing V^H K H
      for (int i=0; i<blockSize_; i++) { curind[i] = curDim_ + i; }
      Teuchos::RCP<const MV> Vprev = MVT::CloneView(*V_,curind);
      // om_->stream(Debug) << "Vprev: " << std::endl;
      // MVT::MvPrint(*Vprev,om_->stream(Debug));

      // Compute the next vector in the Krylov basis:  Vnext = Op*Vprev
      {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor lcltimer( *timerOp_ );
#endif
        OPT::Apply(*Op_,*Vprev,*Vnext);
        count_ApplyOp_ += blockSize_;
      }
      // om_->stream(Debug) << "Vnext: " << std::endl;
      // MVT::MvPrint(*Vnext,om_->stream(Debug));
      Vprev = Teuchos::null;
      
      // Remove all previous Krylov-Schur basis vectors and auxVecs from Vnext
      {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor lcltimer( *timerOrtho_ );
#endif
        
        // Get a view of all the previous vectors
        std::vector<int> prevind(lclDim);
        for (int i=0; i<lclDim; i++) { prevind[i] = i; }
        Vprev = MVT::CloneView(*V_,prevind);
        Teuchos::Array<Teuchos::RCP<const MV> > AVprev(1, Vprev);
        
        // Get a view of the part of the Hessenberg matrix needed to hold the ortho coeffs.
        Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> >
          subH = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>
                               ( Teuchos::View,*H_,lclDim,blockSize_,0,curDim_ ) );
        Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > AsubH;
        AsubH.append( subH );
        
        // Add the auxiliary vectors to the current basis vectors if any exist
        if (auxVecs_.size() > 0) {
          for (Array_size_type i=0; i<auxVecs_.size(); i++) {
            AVprev.append( auxVecs_[i] );
            AsubH.append( Teuchos::null );
          }
        }
        
        // Get a view of the part of the Hessenberg matrix needed to hold the norm coeffs.
        // om_->stream(Debug) << "V before ortho: " << std::endl; 
        // MVT::MvPrint(*Vprev,om_->stream(Debug));
        Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> >
          subR = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>
                               ( Teuchos::View,*H_,blockSize_,blockSize_,lclDim,curDim_ ) );
        int rank = orthman_->projectAndNormalize(*Vnext,AVprev,AsubH,subR);
        // om_->stream(Debug) << "Vnext after ortho: " << std::endl;
        // MVT::MvPrint(*Vnext,om_->stream(Debug));
        // om_->stream(Debug) << "subH: " << std::endl << *subH << std::endl;
        // om_->stream(Debug) << "subR: " << std::endl << *subR << std::endl;
        // om_->stream(Debug) << "H:    " << std::endl << *H_ << std::endl;
        TEUCHOS_TEST_FOR_EXCEPTION(rank != blockSize_,BlockKrylovSchurOrthoFailure,
                           "Anasazi::BlockKrylovSchur::iterate(): couldn't generate basis of full rank.");
      }
      //
      // V has been extended, and H has been extended. 
      //
      // Update basis dim and release all pointers.
      Vnext = Teuchos::null;
      curDim_ += blockSize_;
      // The Ritz vectors/values and Schur form are no longer current.
      ritzVecsCurrent_ = false;
      ritzValsCurrent_ = false;
      schurCurrent_ = false;
      //
      // Update Ritz values and residuals if needed
      if (!(iter_%stepSize_)) {
        computeRitzValues();
      }
      
      // When required, monitor some orthogonalities
      if (om_->isVerbosity( Debug ) ) {
        // Check almost everything here
        CheckList chk;
        chk.checkV = true;
        chk.checkArn = true;
        om_->print( Debug, accuracyCheck(chk, ": after local update") );
      }
      else if (om_->isVerbosity( OrthoDetails ) ) {
        CheckList chk;
        chk.checkV = true;
        om_->print( OrthoDetails, accuracyCheck(chk, ": after local update") );
      }
      
      // Print information on current iteration
      if (om_->isVerbosity(Debug)) {
        currentStatus( om_->stream(Debug) );
      }
      else if (om_->isVerbosity(IterationDetails)) {
        currentStatus( om_->stream(IterationDetails) );
      }
      
    } // end while (statusTest == false)
    
  } // end of iterate()


  // Check accuracy, orthogonality, and other debugging stuff
  // 
  // bools specify which tests we want to run (instead of running more than we actually care about)
  //
  // checkV : V orthonormal
  //          orthogonal to auxvecs
  // checkAux: check that auxiliary vectors are actually orthonormal
  //
  // checkArn: check the Arnoldi factorization
  //
  // NOTE:  This method needs to check the current dimension of the subspace, since it is possible to
  //        call this method when curDim_ = 0 (after initialization).
  template <class ScalarType, class MV, class OP>
  std::string BlockKrylovSchur<ScalarType,MV,OP>::accuracyCheck( const CheckList &chk, const std::string &where ) const 
  {
    std::stringstream os;
    os.precision(2);
    os.setf(std::ios::scientific, std::ios::floatfield);
    MagnitudeType tmp;

    os << " Debugging checks: iteration " << iter_ << where << std::endl;

    // index vectors for V and F
    std::vector<int> lclind(curDim_);
    for (int i=0; i<curDim_; i++) lclind[i] = i;
    std::vector<int> bsind(blockSize_);
    for (int i=0; i<blockSize_; i++) { bsind[i] = curDim_ + i; }
    
    Teuchos::RCP<const MV> lclV,lclF;
    Teuchos::RCP<MV> lclAV;
    if (curDim_)
      lclV = MVT::CloneView(*V_,lclind);
    lclF = MVT::CloneView(*V_,bsind);

    if (chk.checkV) {
      if (curDim_) {
        tmp = orthman_->orthonormError(*lclV);
        os << " >> Error in V^H M V == I  : " << tmp << std::endl;
      }
      tmp = orthman_->orthonormError(*lclF);
      os << " >> Error in F^H M F == I  : " << tmp << std::endl;
      if (curDim_) {
        tmp = orthman_->orthogError(*lclV,*lclF);
        os << " >> Error in V^H M F == 0  : " << tmp << std::endl;
      }
      for (Array_size_type i=0; i<auxVecs_.size(); i++) {
        if (curDim_) {
          tmp = orthman_->orthogError(*lclV,*auxVecs_[i]);
          os << " >> Error in V^H M Aux[" << i << "] == 0 : " << tmp << std::endl;
        }
        tmp = orthman_->orthogError(*lclF,*auxVecs_[i]);
        os << " >> Error in F^H M Aux[" << i << "] == 0 : " << tmp << std::endl;
      }
    }
    
    if (chk.checkArn) {

      if (curDim_) {
        // Compute AV      
        lclAV = MVT::Clone(*V_,curDim_);
        {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
          Teuchos::TimeMonitor lcltimer( *timerOp_ );
#endif
          OPT::Apply(*Op_,*lclV,*lclAV);
        }
        
        // Compute AV - VH
        Teuchos::SerialDenseMatrix<int,ScalarType> subH(Teuchos::View,*H_,curDim_,curDim_);
        MVT::MvTimesMatAddMv( -ST_ONE, *lclV, subH, ST_ONE, *lclAV );
        
        // Compute FB_k^T - (AV-VH)
        Teuchos::SerialDenseMatrix<int,ScalarType> curB(Teuchos::View,*H_,
                                                        blockSize_,curDim_, curDim_ );
        MVT::MvTimesMatAddMv( -ST_ONE, *lclF, curB, ST_ONE, *lclAV );
        
        // Compute || FE_k^T - (AV-VH) ||
        std::vector<MagnitudeType> arnNorms( curDim_ );
        orthman_->norm( *lclAV, arnNorms );
        
        for (int i=0; i<curDim_; i++) {        
        os << " >> Error in Krylov-Schur factorization (R = AV-VS-FB^H), ||R[" << i << "]|| : " << arnNorms[i] << std::endl;
        }
      }
    }

    if (chk.checkAux) {
      for (Array_size_type i=0; i<auxVecs_.size(); i++) {
        tmp = orthman_->orthonormError(*auxVecs_[i]);
        os << " >> Error in Aux[" << i << "]^H M Aux[" << i << "] == I : " << tmp << std::endl;
        for (Array_size_type j=i+1; j<auxVecs_.size(); j++) {
          tmp = orthman_->orthogError(*auxVecs_[i],*auxVecs_[j]);
          os << " >> Error in Aux[" << i << "]^H M Aux[" << j << "] == 0 : " << tmp << std::endl;
        }
      }
    }

    os << std::endl;

    return os.str();
  }

  /* Get the current approximate eigenvalues, i.e. Ritz values.
   * 
   * POST-CONDITIONS:
   *
   * ritzValues_ contains Ritz w.r.t. V, H
   * Q_ contains the Schur vectors w.r.t. H
   * schurH_ contains the Schur matrix w.r.t. H
   * ritzOrder_ contains the current ordering from sort manager
   */

  template <class ScalarType, class MV, class OP>  
  void BlockKrylovSchur<ScalarType,MV,OP>::computeRitzValues()
  {
    // Can only call this if the solver is initialized
    if (initialized_) {

      // This just updates the Ritz values and residuals.
      // --> ritzValsCurrent_ will be set to 'true' by this method.
      if (!ritzValsCurrent_) {
        // Compute the current Ritz values, through computing the Schur form
        //   without updating the current projection matrix or sorting the Schur form.
        computeSchurForm( false );
      }
    }
  }

  /* Get the current approximate eigenvectors, i.e. Ritz vectors.
   * 
   * POST-CONDITIONS:
   *
   * ritzValues_ contains Ritz w.r.t. V, H
   * ritzVectors_ is first blockSize_ Ritz vectors w.r.t. V, H
   * Q_ contains the Schur vectors w.r.t. H
   * schurH_ contains the Schur matrix w.r.t. H
   * ritzOrder_ contains the current ordering from sort manager
   */

  template <class ScalarType, class MV, class OP>  
  void BlockKrylovSchur<ScalarType,MV,OP>::computeRitzVectors()
  {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor LocalTimer(*timerCompRitzVec_);
#endif

    TEUCHOS_TEST_FOR_EXCEPTION(numRitzVecs_==0, std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::computeRitzVectors(): no Ritz vectors were required from this solver.");

    TEUCHOS_TEST_FOR_EXCEPTION(curDim_ < numRitzVecs_, std::invalid_argument,
                       "Anasazi::BlockKrylovSchur::computeRitzVectors(): the current subspace is not large enough to compute the number of requested Ritz vectors.");


    // Check to see if the current subspace dimension is non-trivial and the solver is initialized
    if (curDim_ && initialized_) {

      // Check to see if the Ritz vectors are current.
      if (!ritzVecsCurrent_) {
        
        // Check to see if the Schur factorization of H (schurH_, Q) is current and sorted.
        if (!schurCurrent_) {
          // Compute the Schur factorization of the current H, which will not directly change H,
          // the factorization will be sorted and placed in (schurH_, Q)
          computeSchurForm( true );
        }
        
        // After the Schur form is computed, then the Ritz values are current.
        // Thus, I can check the Ritz index vector to see if I have enough space for the Ritz vectors requested.
        TEUCHOS_TEST_FOR_EXCEPTION(ritzIndex_[numRitzVecs_-1]==1, std::logic_error,
                           "Anasazi::BlockKrylovSchur::computeRitzVectors(): the number of required Ritz vectors splits a complex conjugate pair.");

        Teuchos::LAPACK<int,ScalarType> lapack;
        Teuchos::LAPACK<int,MagnitudeType> lapack_mag;

        // Compute the Ritz vectors.
        // --> For a Hermitian problem this is simply the current basis times the first numRitzVecs_ Schur vectors
        //     
        // --> For a non-Hermitian problem, this involves solving the projected eigenproblem, then
        //     placing the product of the current basis times the first numRitzVecs_ Schur vectors times the
        //     eigenvectors of interest into the Ritz vectors.

        // Get a view of the current Krylov-Schur basis vectors and Schur vectors
        std::vector<int> curind( curDim_ );
        for (int i=0; i<curDim_; i++) { curind[i] = i; }
        Teuchos::RCP<const MV> Vtemp = MVT::CloneView( *V_, curind );
        if (problem_->isHermitian()) {
          // Get a view into the current Schur vectors
          Teuchos::SerialDenseMatrix<int,ScalarType> subQ( Teuchos::View, *Q_, curDim_, numRitzVecs_ );

          // Compute the current Ritz vectors      
          MVT::MvTimesMatAddMv( ST_ONE, *Vtemp, subQ, ST_ZERO, *ritzVectors_ );
          
          // Double check that no complex Ritz values have snuck into the set of converged nev.
          bool complexRitzVals = false;
          for (int i=0; i<numRitzVecs_; i++) {
            if (ritzIndex_[i] != 0) {
              complexRitzVals = true;
              break;
            }
          }
          if (complexRitzVals)
            om_->stream(Warnings) << " Eigenproblem is Hermitian and complex eigenvalues have converged, corresponding eigenvectors will be incorrect!!!" 
                                  << std::endl;
        } else {

          // Get a view into the current Schur vectors.
          Teuchos::SerialDenseMatrix<int,ScalarType> subQ( Teuchos::View, *Q_, curDim_, curDim_ );
          
          // Get a set of work vectors to hold the current Ritz vectors.
          Teuchos::RCP<MV> tmpritzVectors_ = MVT::Clone( *V_, curDim_ );

          // Compute the current Krylov-Schur vectors.
          MVT::MvTimesMatAddMv( ST_ONE, *Vtemp, subQ, ST_ZERO, *tmpritzVectors_ );          

          //  Now compute the eigenvectors of the Schur form
          //  Reset the dense matrix and compute the eigenvalues of the Schur form.
          //
          // Allocate the work space. This space will be used below for calls to:
          // * TREVC (requires 3*N for real, 2*N for complex) 

          int lwork = 3*curDim_;
          std::vector<ScalarType> work( lwork );
          std::vector<MagnitudeType> rwork( curDim_ );
          char side = 'R';
          int mm, info = 0; 
          const int ldvl = 1;
          ScalarType vl[ ldvl ];
          Teuchos::SerialDenseMatrix<int,ScalarType> copyQ( Teuchos::Copy, *Q_, curDim_, curDim_ );
          lapack.TREVC( side, curDim_, schurH_->values(), schurH_->stride(), vl, ldvl,
                        copyQ.values(), copyQ.stride(), curDim_, &mm, &work[0], &rwork[0], &info );
          TEUCHOS_TEST_FOR_EXCEPTION(info != 0, std::logic_error,
                             "Anasazi::BlockKrylovSchur::computeRitzVectors(): TREVC(n==" << curDim_ << ") returned info " << info << " != 0.");

          // Get a view into the eigenvectors of the Schur form
          Teuchos::SerialDenseMatrix<int,ScalarType> subCopyQ( Teuchos::View, copyQ, curDim_, numRitzVecs_ );
          
          // Convert back to Ritz vectors of the operator.
          curind.resize( numRitzVecs_ );  // This is already initialized above
          Teuchos::RCP<MV> view_ritzVectors = MVT::CloneViewNonConst( *ritzVectors_, curind );
          MVT::MvTimesMatAddMv( ST_ONE, *tmpritzVectors_, subCopyQ, ST_ZERO, *view_ritzVectors );

          // Compute the norm of the new Ritz vectors
          std::vector<MagnitudeType> ritzNrm( numRitzVecs_ );
          MVT::MvNorm( *view_ritzVectors, ritzNrm );

          // Release memory used to compute Ritz vectors before scaling the current vectors.
          tmpritzVectors_ = Teuchos::null;
          view_ritzVectors = Teuchos::null;
          
          // Scale the Ritz vectors to have Euclidean norm.
          ScalarType ritzScale = ST_ONE;
          for (int i=0; i<numRitzVecs_; i++) {
            
            // If this is a conjugate pair then normalize by the real and imaginary parts.
            if (ritzIndex_[i] == 1 ) {
              ritzScale = ST_ONE/lapack_mag.LAPY2(ritzNrm[i],ritzNrm[i+1]);
              std::vector<int> newind(2);
              newind[0] = i; newind[1] = i+1;
              tmpritzVectors_ = MVT::CloneCopy( *ritzVectors_, newind );
              view_ritzVectors = MVT::CloneViewNonConst( *ritzVectors_, newind );
              MVT::MvAddMv( ritzScale, *tmpritzVectors_, ST_ZERO, *tmpritzVectors_, *view_ritzVectors );

              // Increment counter for imaginary part
              i++;
            } else {

              // This is a real Ritz value, normalize the vector
              std::vector<int> newind(1);
              newind[0] = i;
              tmpritzVectors_ = MVT::CloneCopy( *ritzVectors_, newind );
              view_ritzVectors = MVT::CloneViewNonConst( *ritzVectors_, newind );
              MVT::MvAddMv( ST_ONE/ritzNrm[i], *tmpritzVectors_, ST_ZERO, *tmpritzVectors_, *view_ritzVectors );
            }              
          }
          
        } // if (problem_->isHermitian()) 
        
        // The current Ritz vectors have been computed.
        ritzVecsCurrent_ = true;
        
      } // if (!ritzVecsCurrent_)      
    } // if (curDim_)    
  } // computeRitzVectors()

  
  // Set a new StatusTest for the solver.
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::setStatusTest(Teuchos::RCP<StatusTest<ScalarType,MV,OP> > test) {
    TEUCHOS_TEST_FOR_EXCEPTION(test == Teuchos::null,std::invalid_argument,
        "Anasazi::BlockKrylovSchur::setStatusTest() was passed a null StatusTest.");
    tester_ = test;
  }

  // Get the current StatusTest used by the solver.
  template <class ScalarType, class MV, class OP>
  Teuchos::RCP<StatusTest<ScalarType,MV,OP> > BlockKrylovSchur<ScalarType,MV,OP>::getStatusTest() const {
    return tester_;
  }
  

  /* Get the current approximate eigenvalues, i.e. Ritz values.
   * 
   * POST-CONDITIONS:
   *
   * ritzValues_ contains Ritz w.r.t. V, H
   * Q_ contains the Schur vectors w.r.t. H
   * schurH_ contains the Schur matrix w.r.t. H
   * ritzOrder_ contains the current ordering from sort manager
   * schurCurrent_ = true if sort = true; i.e. the Schur form is sorted according to the index
   *  vector returned by the sort manager.
   */
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::computeSchurForm( const bool sort )
  {
    // local timer
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor LocalTimer(*timerCompSF_);
#endif

    // Check to see if the dimension of the factorization is greater than zero.
    if (curDim_) {

      // Check to see if the Schur factorization is current.
      if (!schurCurrent_) {
        
        // Check to see if the Ritz values are current
        // --> If they are then the Schur factorization is current but not sorted.
        if (!ritzValsCurrent_) {
          Teuchos::LAPACK<int,ScalarType> lapack; 
          Teuchos::LAPACK<int,MagnitudeType> lapack_mag;
          Teuchos::BLAS<int,ScalarType> blas;
          Teuchos::BLAS<int,MagnitudeType> blas_mag;
          
          // Get a view into Q, the storage for H's Schur vectors.
          Teuchos::SerialDenseMatrix<int,ScalarType> subQ( Teuchos::View, *Q_, curDim_, curDim_ );
          
          // Get a copy of H to compute/sort the Schur form.
          schurH_ = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( Teuchos::Copy, *H_, curDim_, curDim_ ) );
          //
          //---------------------------------------------------
          // Compute the Schur factorization of subH
          // ---> Use driver GEES to first reduce to upper Hessenberg 
          //         form and then compute Schur form, outputting Ritz values
          //---------------------------------------------------
          //
          // Allocate the work space. This space will be used below for calls to:
          // * GEES  (requires 3*N for real, 2*N for complex)
          // * TREVC (requires 3*N for real, 2*N for complex) 
          // * TREXC (requires N for real, none for complex)
          // Furthermore, GEES requires a real array of length curDim_ (for complex datatypes)
          //
          int lwork = 3*curDim_;
          std::vector<ScalarType> work( lwork );
          std::vector<MagnitudeType> rwork( curDim_ );
          std::vector<MagnitudeType> tmp_rRitzValues( curDim_ );
          std::vector<MagnitudeType> tmp_iRitzValues( curDim_ );
          std::vector<int> bwork( curDim_ );
          int info = 0, sdim = 0; 
          char jobvs = 'V';
          lapack.GEES( jobvs,curDim_, schurH_->values(), schurH_->stride(), &sdim, &tmp_rRitzValues[0],
                       &tmp_iRitzValues[0], subQ.values(), subQ.stride(), &work[0], lwork, 
                       &rwork[0], &bwork[0], &info );
 
          TEUCHOS_TEST_FOR_EXCEPTION(info != 0, std::logic_error,
                             "Anasazi::BlockKrylovSchur::computeSchurForm(): GEES(n==" << curDim_ << ") returned info " << info << " != 0.");
         
          // Check if imaginary part is detected by the Schur factorization of subH for Hermitian eigenproblems
          // NOTE: Because of full orthogonalization, there will be small entries above the block tridiagonal in the block Hessenberg matrix.
          //       The spectrum of this matrix may include imaginary eigenvalues with small imaginary part, which will mess up the Schur
          //       form sorting below.
          bool hermImagDetected = false;
          if (problem_->isHermitian()) {
            for (int i=0; i<curDim_; i++)
            {
              if (tmp_iRitzValues[i] != MT_ZERO)
              {
                hermImagDetected = true;
                break;
              }
            }
            if (hermImagDetected) {
              // Warn the user that complex eigenvalues have been detected.
              om_->stream(Warnings) << " Eigenproblem is Hermitian and complex eigenvalues have been detected!!!" << std::endl; 
              // Compute || H - H' || to indicate how bad the symmetry is in the projected eigenproblem
              Teuchos::SerialDenseMatrix<int,ScalarType> localH( Teuchos::View, *H_, curDim_, curDim_ );
              Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > tLocalH;
              if (Teuchos::ScalarTraits<ScalarType>::isComplex)
                tLocalH = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( localH, Teuchos::CONJ_TRANS ) );
              else
                tLocalH = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( localH, Teuchos::TRANS ) );
              (*tLocalH) -= localH;
              MagnitudeType normF = tLocalH->normFrobenius();
              MagnitudeType norm1 = tLocalH->normOne();
              om_->stream(Warnings) << " Symmetry error in projected eigenproblem:  || S - S' ||_F = " << normF 
                                    << ", || S - S' ||_1 = " << norm1 << std::endl;
            }
          } 
          //
          //---------------------------------------------------
          // Use the Krylov-Schur factorization to compute the current Ritz residuals 
          // for ALL the eigenvalues estimates (Ritz values)
          //           || Ax - x\theta || = || U_m+1*B_m+1^H*Q*s || 
          //                              = || B_m+1^H*Q*s ||
          //
          // where U_m+1 is the current Krylov-Schur basis, Q are the Schur vectors, and x = U_m+1*Q*s
          // NOTE: This means that s = e_i if the problem is hermitian, else the eigenvectors
          //       of the Schur form need to be computed.
          //
          // First compute H_{m+1,m}*B_m^T, then determine what 's' is.
          //---------------------------------------------------
          //
          // Get current B_m+1
          Teuchos::SerialDenseMatrix<int,ScalarType> curB(Teuchos::View, *H_,
                                                          blockSize_, curDim_, curDim_ );
          //
          // Compute B_m+1^H*Q
          Teuchos::SerialDenseMatrix<int,ScalarType> subB( blockSize_, curDim_ );
          blas.GEMM( Teuchos::NO_TRANS, Teuchos::NO_TRANS, blockSize_, curDim_, curDim_, ST_ONE, 
                     curB.values(), curB.stride(), subQ.values(), subQ.stride(), 
                     ST_ZERO, subB.values(), subB.stride() );
          //
          // Determine what 's' is and compute Ritz residuals.
          //
          ScalarType* b_ptr = subB.values();
          if (problem_->isHermitian() && !hermImagDetected) {
            //
            // 's' is the i-th canonical basis vector.
            //
            for (int i=0; i<curDim_ ; i++) {
              ritzResiduals_[i] = blas.NRM2(blockSize_, b_ptr + i*blockSize_, 1);
            }   
          } else {
            //
            //  Compute S: the eigenvectors of the block upper triangular, Schur matrix.
            //
            char side = 'R';
            int mm;
            const int ldvl = 1;
            ScalarType vl[ ldvl ];
            Teuchos::SerialDenseMatrix<int,ScalarType> S( curDim_, curDim_ );
            lapack.TREVC( side, curDim_, schurH_->values(), schurH_->stride(), vl, ldvl,
                          S.values(), S.stride(), curDim_, &mm, &work[0], &rwork[0], &info );
            
            TEUCHOS_TEST_FOR_EXCEPTION(info != 0, std::logic_error,
                               "Anasazi::BlockKrylovSchur::computeSchurForm(): TREVC(n==" << curDim_ << ") returned info " << info << " != 0.");
            //
            // Scale the eigenvectors so that their Euclidean norms are all one.
            //
            HelperTraits<ScalarType>::scaleRitzVectors( tmp_iRitzValues, &S );
            //
            // Compute ritzRes = *B_m+1^H*Q*S where the i-th column of S is 's' for the i-th Ritz-value
            //
            Teuchos::SerialDenseMatrix<int,ScalarType> ritzRes( blockSize_, curDim_ );
            blas.GEMM( Teuchos::NO_TRANS, Teuchos::NO_TRANS, blockSize_, curDim_, curDim_, ST_ONE, 
                       subB.values(), subB.stride(), S.values(), S.stride(), 
                       ST_ZERO, ritzRes.values(), ritzRes.stride() );

            /* TO DO:  There's be an incorrect assumption made in the computation of the Ritz residuals.
                       This assumption is that the next vector in the Krylov subspace is Euclidean orthonormal.
                       It may not be normalized using Euclidean norm.
            Teuchos::RCP<MV> ritzResVecs = MVT::Clone( *V_, curDim_ );
            std::vector<int> curind(blockSize_);
            for (int i=0; i<blockSize_; i++) { curind[i] = curDim_ + i; }
            Teuchos::RCP<MV> Vtemp = MVT::CloneView(*V_,curind);     
            
            MVT::MvTimesMatAddMv( ST_ONE, *Vtemp, ritzRes, ST_ZERO, *ritzResVecs );
            std::vector<MagnitudeType> ritzResNrms(curDim_);
            MVT::MvNorm( *ritzResVecs, ritzResNrms );
            i = 0;
            while( i < curDim_ ) {
              if ( tmp_ritzValues[curDim_+i] != MT_ZERO ) {
                ritzResiduals_[i] = lapack_mag.LAPY2( ritzResNrms[i], ritzResNrms[i+1] );
                ritzResiduals_[i+1] = ritzResiduals_[i];
                i = i+2;
              } else {
                ritzResiduals_[i] = ritzResNrms[i];
                i++;
              }
            }
            */
            //
            // Compute the Ritz residuals for each Ritz value.
            // 
            HelperTraits<ScalarType>::computeRitzResiduals( tmp_iRitzValues, ritzRes, &ritzResiduals_ );
          }
          //
          // Sort the Ritz values.
          //
          {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
            Teuchos::TimeMonitor LocalTimer2(*timerSortRitzVal_);
#endif
            int i=0;
            if (problem_->isHermitian() && !hermImagDetected) {
              //
              // Sort using just the real part of the Ritz values.
              sm_->sort(tmp_rRitzValues, Teuchos::rcpFromRef(ritzOrder_), curDim_); // don't catch exception
              ritzIndex_.clear();
              while ( i < curDim_ ) {
                // The Ritz value is not complex.
                ritzValues_[i].set(tmp_rRitzValues[i], MT_ZERO);
                ritzIndex_.push_back(0);
                i++;
              }
            }
            else {
              //
              // Sort using both the real and imaginary parts of the Ritz values.
              sm_->sort(tmp_rRitzValues, tmp_iRitzValues, Teuchos::rcpFromRef(ritzOrder_) , curDim_);
              HelperTraits<ScalarType>::sortRitzValues( tmp_rRitzValues, tmp_iRitzValues, &ritzValues_, &ritzOrder_, &ritzIndex_ );
            }
            //
            // Sort the ritzResiduals_ based on the ordering from the Sort Manager.
            std::vector<MagnitudeType> ritz2( curDim_ );
            for (i=0; i<curDim_; i++) { ritz2[i] = ritzResiduals_[ ritzOrder_[i] ]; }
            blas_mag.COPY( curDim_, &ritz2[0], 1, &ritzResiduals_[0], 1 );
            
            // The Ritz values have now been updated.
            ritzValsCurrent_ = true;
          }

        } // if (!ritzValsCurrent_) ...
        // 
        //---------------------------------------------------
        // * The Ritz values and residuals have been updated at this point.
        // 
        // * The Schur factorization of the projected matrix has been computed,
        //   and is stored in (schurH_, Q_).
        //
        // Now the Schur factorization needs to be sorted.
        //---------------------------------------------------
        //
        // Sort the Schur form using the ordering from the Sort Manager.
        if (sort) {
          sortSchurForm( *schurH_, *Q_, ritzOrder_ );    
          //
          // Indicate the Schur form in (schurH_, Q_) is current and sorted
          schurCurrent_ = true;
        }
      } // if (!schurCurrent_) ...
  
    } // if (curDim_) ...
  
  } // computeSchurForm( ... )
  

  // Sort the Schur form of H stored in (H,Q) using the ordering vector.
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::sortSchurForm( Teuchos::SerialDenseMatrix<int,ScalarType>& H,
                                                          Teuchos::SerialDenseMatrix<int,ScalarType>& Q,
                                                          std::vector<int>& order ) 
  {
    // local timer
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor LocalTimer(*timerSortSF_);
#endif
    //
    //---------------------------------------------------
    // Reorder real Schur factorization, remember to add one to the indices for the
    // fortran call and determine offset.  The offset is necessary since the TREXC
    // method reorders in a nonsymmetric fashion, thus we use the reordering in
    // a stack-like fashion.  Also take into account conjugate pairs, which may mess
    // up the reordering, since the pair is moved if one of the pair is moved.
    //---------------------------------------------------
    //
    int i = 0, nevtemp = 0;
    char compq = 'V';
    std::vector<int> offset2( curDim_ );
    std::vector<int> order2( curDim_ );

    // LAPACK objects.
    Teuchos::LAPACK<int,ScalarType> lapack; 
    int lwork = 3*curDim_;
    std::vector<ScalarType> work( lwork );

    while (i < curDim_) {
      if ( ritzIndex_[i] != 0 ) { // This is the first value of a complex conjugate pair
        offset2[nevtemp] = 0;
        for (int j=i; j<curDim_; j++) {
          if (order[j] > order[i]) { offset2[nevtemp]++; }
        }
        order2[nevtemp] = order[i];
        i = i+2;
      } else {
        offset2[nevtemp] = 0;
        for (int j=i; j<curDim_; j++) {
          if (order[j] > order[i]) { offset2[nevtemp]++; }
        }
        order2[nevtemp] = order[i];
        i++;
      }
      nevtemp++;
    }
    ScalarType *ptr_h = H.values();
    ScalarType *ptr_q = Q.values();
    int ldh = H.stride(), ldq = Q.stride();
    int info = 0;
    for (i=nevtemp-1; i>=0; i--) {
      int ifst = order2[i]+1+offset2[i];
      int ilst = 1;
      lapack.TREXC( compq, curDim_, ptr_h, ldh, ptr_q, ldq, &ifst,
                    &ilst, &work[0], &info );
      TEUCHOS_TEST_FOR_EXCEPTION(info != 0, std::logic_error,
                         "Anasazi::BlockKrylovSchur::computeSchurForm(): TREXC(n==" << curDim_ << ") returned info " << info << " != 0.");
    }
  }

  // Print the current status of the solver
  template <class ScalarType, class MV, class OP>
  void BlockKrylovSchur<ScalarType,MV,OP>::currentStatus(std::ostream &os) 
  {
    using std::endl;

    os.setf(std::ios::scientific, std::ios::floatfield);
    os.precision(6);
    os <<"================================================================================" << endl;
    os << endl;
    os <<"                         BlockKrylovSchur Solver Status" << endl;
    os << endl;
    os <<"The solver is "<<(initialized_ ? "initialized." : "not initialized.") << endl;
    os <<"The number of iterations performed is " <<iter_<<endl;
    os <<"The block size is         " << blockSize_<<endl;
    os <<"The number of blocks is   " << numBlocks_<<endl;
    os <<"The current basis size is " << curDim_<<endl;
    os <<"The number of auxiliary vectors is " << numAuxVecs_ << endl;
    os <<"The number of operations Op*x   is "<<count_ApplyOp_<<endl;

    os.setf(std::ios_base::right, std::ios_base::adjustfield);

    os << endl;
    if (initialized_) {
      os <<"CURRENT RITZ VALUES             "<<endl;
      if (ritzIndex_.size() != 0) {
        int numPrint = (curDim_ < numRitzPrint_? curDim_: numRitzPrint_);
        if (problem_->isHermitian()) {
          os << std::setw(20) << "Ritz Value" 
             << std::setw(20) << "Ritz Residual"
             << endl;
          os <<"--------------------------------------------------------------------------------"<<endl;
          for (int i=0; i<numPrint; i++) {
            os << std::setw(20) << ritzValues_[i].realpart 
               << std::setw(20) << ritzResiduals_[i] 
               << endl;
          }
        } else {
          os << std::setw(24) << "Ritz Value" 
             << std::setw(30) << "Ritz Residual"
             << endl;
          os <<"--------------------------------------------------------------------------------"<<endl;
          for (int i=0; i<numPrint; i++) {
            // Print out the real eigenvalue.
            os << std::setw(15) << ritzValues_[i].realpart;
            if (ritzValues_[i].imagpart < MT_ZERO) {
              os << " - i" << std::setw(15) << Teuchos::ScalarTraits<MagnitudeType>::magnitude(ritzValues_[i].imagpart);
            } else {
              os << " + i" << std::setw(15) << ritzValues_[i].imagpart;
            }              
            os << std::setw(20) << ritzResiduals_[i] << endl;
          }
        }
      } else {
        os << std::setw(20) << "[ NONE COMPUTED ]" << endl;
      }
    }
    os << endl;
    os <<"================================================================================" << endl;
    os << endl;
  }
  
} // End of namespace Anasazi

#endif

// End of file AnasaziBlockKrylovSchur.hpp