BelosGmresPolyOp.hpp

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#ifndef BELOS_GMRESPOLYOP_HPP
#define BELOS_GMRESPOLYOP_HPP

#include "BelosConfigDefs.hpp"
#include "BelosTypes.hpp"

#include "BelosOperator.hpp"
#include "BelosMultiVec.hpp"
#include "BelosOperatorTraits.hpp"
#include "BelosMultiVecTraits.hpp"
#include "BelosLinearProblem.hpp"

#include "BelosGmresIteration.hpp"
#include "BelosBlockGmresIter.hpp"
#include "BelosOrthoManagerFactory.hpp"

#include "BelosStatusTestMaxIters.hpp"
#include "BelosStatusTestGenResNorm.hpp"
#include "BelosStatusTestImpResNorm.hpp"
#include "BelosStatusTestCombo.hpp"
#include "BelosStatusTestOutputFactory.hpp"

#include "BelosOutputManager.hpp"

#include "Teuchos_BLAS.hpp"
#include "Teuchos_LAPACK.hpp"
#include "Teuchos_as.hpp"
#include "Teuchos_RCP.hpp"
#include "Teuchos_SerialDenseMatrix.hpp"
#include "Teuchos_SerialDenseVector.hpp"
#include "Teuchos_SerialDenseSolver.hpp"
#include "Teuchos_ParameterList.hpp"

#ifdef BELOS_TEUCHOS_TIME_MONITOR
  #include "Teuchos_TimeMonitor.hpp"
#endif // BELOS_TEUCHOS_TIME_MONITOR

namespace Belos {
 
  class GmresPolyOpOrthoFailure : public BelosError {public:
    GmresPolyOpOrthoFailure(const std::string& what_arg) : BelosError(what_arg)
      {}};

  // Create a shell class for the MV, inherited off MultiVec<> that will operate with the GmresPolyOp.
  template <class ScalarType, class MV>
  class GmresPolyMv : public MultiVec< ScalarType > 
  {
    public:

     GmresPolyMv ( const Teuchos::RCP<MV>& mv_in )
       : mv_(mv_in)
     {}
     GmresPolyMv ( const Teuchos::RCP<const MV>& mv_in )
     {
       mv_ = Teuchos::rcp_const_cast<MV>( mv_in );
     }
     Teuchos::RCP<MV> getMV() { return mv_; }
     Teuchos::RCP<const MV> getConstMV() const { return mv_; }

     GmresPolyMv * Clone ( const int numvecs ) const 
     {
       GmresPolyMv * newMV = new GmresPolyMv( MVT::Clone( *mv_, numvecs ) );
       return newMV;
     }
     GmresPolyMv * CloneCopy () const 
     {
       GmresPolyMv * newMV = new GmresPolyMv( MVT::CloneCopy( *mv_ ) );
       return newMV;
     }
     GmresPolyMv * CloneCopy ( const std::vector<int>& index ) const 
     {
       GmresPolyMv * newMV = new GmresPolyMv( MVT::CloneCopy( *mv_, index ) );
       return newMV;
     }
     GmresPolyMv * CloneViewNonConst ( const std::vector<int>& index ) 
     {
       GmresPolyMv * newMV = new GmresPolyMv( MVT::CloneViewNonConst( *mv_, index ) );
       return newMV;
     }
     const GmresPolyMv * CloneView ( const std::vector<int>& index ) const 
     {
       const GmresPolyMv * newMV = new GmresPolyMv( MVT::CloneView( *mv_, index ) );
       return newMV;
     }
     ptrdiff_t GetGlobalLength () const { return MVT::GetGlobalLength( *mv_ ); }
     int GetNumberVecs () const { return MVT::GetNumberVecs( *mv_ ); }
     void MvTimesMatAddMv (const ScalarType alpha,
                           const MultiVec<ScalarType>& A,
                           const Teuchos::SerialDenseMatrix<int,ScalarType>& B, const ScalarType beta) 
     {  
       const GmresPolyMv<ScalarType,MV>& A_in = dynamic_cast<const GmresPolyMv<ScalarType,MV>&>(A);
       MVT::MvTimesMatAddMv( alpha, *(A_in.getConstMV()), B, beta, *mv_ );
     }
     void MvAddMv ( const ScalarType alpha, const MultiVec<ScalarType>& A, const ScalarType beta, const MultiVec<ScalarType>& B ) 
     {
       const GmresPolyMv<ScalarType,MV>& A_in = dynamic_cast<const GmresPolyMv<ScalarType,MV>&>(A);
       const GmresPolyMv<ScalarType,MV>& B_in = dynamic_cast<const GmresPolyMv<ScalarType,MV>&>(B);
       MVT::MvAddMv( alpha, *(A_in.getConstMV()), beta, *(B_in.getConstMV()), *mv_ );
     }
     void MvScale ( const ScalarType alpha ) { MVT::MvScale( *mv_, alpha ); }
     void MvScale ( const std::vector<ScalarType>& alpha ) { MVT::MvScale( *mv_, alpha ); }
     void MvTransMv ( const ScalarType alpha, const MultiVec<ScalarType>& A, Teuchos::SerialDenseMatrix<int,ScalarType>& B) const 
     {
       const GmresPolyMv<ScalarType,MV>& A_in = dynamic_cast<const GmresPolyMv<ScalarType,MV>&>(A);
       MVT::MvTransMv( alpha, *(A_in.getConstMV()), *mv_, B );
     }
     void MvDot ( const MultiVec<ScalarType>& A, std::vector<ScalarType>& b ) const 
     {
       const GmresPolyMv<ScalarType,MV>& A_in = dynamic_cast<const GmresPolyMv<ScalarType,MV>&>(A);
       MVT::MvDot( *(A_in.getConstMV()), *mv_, b );
     }
     void MvNorm ( std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType>& normvec, NormType type = TwoNorm ) const
     {
       MVT::MvNorm( *mv_, normvec, type ); 
     }
     void SetBlock ( const MultiVec<ScalarType>& A, const std::vector<int>& index ) 
     {
       const GmresPolyMv<ScalarType,MV>& A_in = dynamic_cast<const GmresPolyMv<ScalarType,MV>&>(A);
       MVT::SetBlock( *(A_in.getConstMV()), index, *mv_ );
     }
     void MvRandom () { MVT::MvRandom( *mv_ ); }
     void MvInit ( const ScalarType alpha ) { MVT::MvInit( *mv_, alpha ); }
     void MvPrint ( std::ostream& os ) const { MVT::MvPrint( *mv_, os ); }

    private:

      typedef MultiVecTraits<ScalarType,MV> MVT;

      Teuchos::RCP<MV> mv_;

   };
 
  template <class ScalarType, class MV, class OP>
  class GmresPolyOp : public Operator<ScalarType> {
  public:
    

    
    GmresPolyOp( const Teuchos::RCP<LinearProblem<ScalarType,MV,OP> >& problem_in, 
                 const Teuchos::RCP<Teuchos::ParameterList>& params_in
               )
      : problem_(problem_in), 
        params_(params_in),
        LP_(problem_in->getLeftPrec()), 
        RP_(problem_in->getRightPrec())
    {
      setParameters( params_ );

      polyUpdateLabel_ = label_ + ": Hybrid Gmres: Vector Update";
#ifdef BELOS_TEUCHOS_TIME_MONITOR
      timerPolyUpdate_ = Teuchos::TimeMonitor::getNewCounter(polyUpdateLabel_);
#endif // BELOS_TEUCHOS_TIME_MONITOR

      if (polyType_ == "Arnoldi" || polyType_=="Roots")
        generateArnoldiPoly();
      else if (polyType_ == "Gmres")
        generateGmresPoly();
      else
        TEUCHOS_TEST_FOR_EXCEPTION(polyType_!="Arnoldi"&&polyType_!="Gmres"&&polyType_!="Roots",std::invalid_argument,
          "Belos::GmresPolyOp: \"Polynomial Type\" must be either \"Arnoldi\", \"Gmres\", or \"Roots\".");
    }

    GmresPolyOp( const Teuchos::RCP<LinearProblem<ScalarType,MV,OP> >& problem_in )
      : problem_(problem_in)
    {
      // If dimension is zero, it will just apply the operator from problem_in in the Apply method.
      dim_ = 0;
    }

    virtual ~GmresPolyOp() {};


    
    void setParameters( const Teuchos::RCP<Teuchos::ParameterList>& params_in );
 

   
    void generateArnoldiPoly();

    void generateGmresPoly();



    
    void ApplyPoly ( const MV& x, MV& y ) const;
    void ApplyArnoldiPoly ( const MV& x, MV& y ) const;
    void ApplyGmresPoly ( const MV& x, MV& y ) const;
    void ApplyRootsPoly ( const MV& x, MV& y ) const;

    void Apply ( const MultiVec<ScalarType>& x, MultiVec<ScalarType>& y, ETrans /* trans */=NOTRANS ) const
    {
      const GmresPolyMv<ScalarType,MV>& x_in = dynamic_cast<const GmresPolyMv<ScalarType,MV>&>(x);
      GmresPolyMv<ScalarType,MV>& y_in = dynamic_cast<GmresPolyMv<ScalarType,MV>&>(y);
      ApplyPoly( *(x_in.getConstMV()), *(y_in.getMV()) ); 
    }

    int polyDegree() const { return dim_; }

    private:

#ifdef BELOS_TEUCHOS_TIME_MONITOR
    Teuchos::RCP<Teuchos::Time> timerPolyUpdate_;
#endif // BELOS_TEUCHOS_TIME_MONITOR
    std::string polyUpdateLabel_;

    typedef int OT; //Ordinal type 
    typedef MultiVecTraits<ScalarType,MV> MVT;
    typedef Teuchos::ScalarTraits<ScalarType> SCT ;
    typedef typename Teuchos::ScalarTraits<ScalarType>::magnitudeType MagnitudeType;
    typedef Teuchos::ScalarTraits<MagnitudeType> MCT ;

    // Default polynomial parameters
    static constexpr int maxDegree_default_ = 25;
    static constexpr int verbosity_default_ = Belos::Errors;
    static constexpr bool randomRHS_default_ = true;
    static constexpr const char * label_default_ = "Belos";
    static constexpr const char * polyType_default_ = "Roots";
    static constexpr const char * orthoType_default_ = "DGKS";
    static constexpr bool damp_default_ = false;
    static constexpr bool addRoots_default_ = true;

    // Variables for generating the polynomial
    Teuchos::RCP<LinearProblem<ScalarType,MV,OP> > problem_;
    Teuchos::RCP<Teuchos::ParameterList> params_;
    Teuchos::RCP<const OP> LP_, RP_;

    // Output manager.
    Teuchos::RCP<OutputManager<ScalarType> > printer_;
    Teuchos::RCP<std::ostream> outputStream_ = Teuchos::rcpFromRef(std::cout);
 
    // Orthogonalization manager.
    Teuchos::RCP<MatOrthoManager<ScalarType,MV,OP> > ortho_;

    // Current polynomial parameters
    MagnitudeType polyTol_ = DefaultSolverParameters::polyTol;
    int maxDegree_ = maxDegree_default_;
    int verbosity_ = verbosity_default_;
    bool randomRHS_ = randomRHS_default_;
    std::string label_ = label_default_;
    std::string polyType_ = polyType_default_;
    std::string orthoType_ = orthoType_default_;
    int dim_ = 0;
    bool damp_ = damp_default_;
    bool addRoots_ = addRoots_default_;
    
    // Variables for Arnoldi polynomial
    mutable Teuchos::RCP<MV> V_, wL_, wR_;
    Teuchos::SerialDenseMatrix<OT,ScalarType> H_, y_;
    Teuchos::SerialDenseVector<OT,ScalarType> r0_;

    // Variables for Gmres polynomial;
    bool autoDeg = false;
    Teuchos::SerialDenseMatrix< OT, ScalarType > pCoeff_;

    // Variables for Roots polynomial:
    Teuchos::SerialDenseMatrix< OT, MagnitudeType > theta_; 
    
    // Modified Leja sorting function. Takes a serial dense matrix of M harmonic Ritz values and an index
    // of values from 0 to M. Returns the sorted values and sorted index, similar to Matlab.
    void SortModLeja(Teuchos::SerialDenseMatrix< OT, MagnitudeType > &thetaN, std::vector<int> &index) const ;

    //Function determines whether added roots are needed and adds them if option is turned on.
    void ComputeAddedRoots();
  };
  
  template <class ScalarType, class MV, class OP>
  void GmresPolyOp<ScalarType, MV, OP>::setParameters( const Teuchos::RCP<Teuchos::ParameterList>& params_in )
  {
    // Check which Gmres polynomial to use
    if (params_in->isParameter("Polynomial Type")) {
      polyType_ = params_in->get("Polynomial Type", polyType_default_);
    }

    // Check for polynomial convergence tolerance
    if (params_in->isParameter("Polynomial Tolerance")) {
      if (params_in->isType<MagnitudeType> ("Polynomial Tolerance")) {
        polyTol_ = params_in->get ("Polynomial Tolerance",
                                 static_cast<MagnitudeType> (DefaultSolverParameters::polyTol));
      }     
      else {
        polyTol_ = params_in->get ("Polynomial Tolerance", DefaultSolverParameters::polyTol);
      }
    }

    // Check for maximum polynomial degree
    if (params_in->isParameter("Maximum Degree")) {
      maxDegree_ = params_in->get("Maximum Degree", maxDegree_default_);
    }

    // Check for maximum polynomial degree
    if (params_in->isParameter("Random RHS")) {
      randomRHS_ = params_in->get("Random RHS", randomRHS_default_);
    }
  
    // Check for a change in verbosity level
    if (params_in->isParameter("Verbosity")) {
      if (Teuchos::isParameterType<int>(*params_in,"Verbosity")) {
        verbosity_ = params_in->get("Verbosity", verbosity_default_);
      } 
      else {
        verbosity_ = (int)Teuchos::getParameter<Belos::MsgType>(*params_in,"Verbosity");
      }
    }

    if (params_in->isParameter("Orthogonalization")) {
      orthoType_ = params_in->get("Orthogonalization",orthoType_default_);
    }

    // Check for timer label
    if (params_in->isParameter("Timer Label")) {
      label_ = params_in->get("Timer Label", label_default_);
    }
 
    // Output stream
    if (params_in->isParameter("Output Stream")) {
      outputStream_ = Teuchos::getParameter<Teuchos::RCP<std::ostream> >(*params_in,"Output Stream");
    }

    // Check for damped polynomial
    if (params_in->isParameter("Damped Poly")) {
      damp_ = params_in->get("Damped Poly", damp_default_);
    }

    // Check for root-adding
    if (params_in->isParameter("Add Roots")) {
      addRoots_ = params_in->get("Add Roots", addRoots_default_);
    }
  }

  template <class ScalarType, class MV, class OP>
  void GmresPolyOp<ScalarType, MV, OP>::generateGmresPoly()
  {
    Teuchos::RCP< MV > V = MVT::Clone( *problem_->getRHS(), maxDegree_+1 );   

    //Make power basis:
    std::vector<int> index(1,0);
    Teuchos::RCP< MV > V0 = MVT::CloneViewNonConst(*V, index);
    if (randomRHS_)
      MVT::MvRandom( *V0 );
    else
      MVT::Assign( *problem_->getRHS(), *V0 );

    if ( !LP_.is_null() ) {
      Teuchos::RCP< MV > Vtemp = MVT::CloneCopy(*V0);
      problem_->applyLeftPrec( *Vtemp, *V0);
    }
    if ( damp_ ) {
      Teuchos::RCP< MV > Vtemp = MVT::CloneCopy(*V0);
      problem_->apply( *Vtemp, *V0);
    }

    for(int i=0; i< maxDegree_; i++)
    {
      index[0] = i;
      Teuchos::RCP< const MV > Vi = MVT::CloneView(*V, index);
      index[0] = i+1;
      Teuchos::RCP< MV > Vip1 = MVT::CloneViewNonConst(*V, index);
      problem_->apply( *Vi, *Vip1);
    }

    //Consider AV:
    Teuchos::Range1D range( 1, maxDegree_);
    Teuchos::RCP< const MV > AV = MVT::CloneView( *V, range);
   
    //Make lhs (AV)^T(AV)
    Teuchos::SerialDenseMatrix< OT, ScalarType > AVtransAV( maxDegree_, maxDegree_);
    MVT::MvTransMv( SCT::one(), *AV, *AV, AVtransAV);
    //This process adds pDeg*pDeg + pDeg inner products that aren't in the final count.

    Teuchos::LAPACK< OT, ScalarType > lapack;
    int infoInt;
    bool status = true; //Keep adjusting poly deg when true.
   
    dim_ = maxDegree_; 
    Teuchos::SerialDenseMatrix< OT, ScalarType > lhs;
    while( status && dim_ >= 1)
    {
      Teuchos::SerialDenseMatrix< OT, ScalarType > lhstemp(Teuchos::Copy, AVtransAV,  dim_, dim_);
      lapack.POTRF( 'U', dim_, lhstemp.values(), lhstemp.stride(), &infoInt);
      
      if(autoDeg == false)
      { 
        status = false;
        if(infoInt != 0)
        {
          std::cout << "BelosGmresPolyOp.hpp: LAPACK POTRF was not successful!!" << std::endl;
          std::cout << "Error code: " << infoInt << std::endl; 
        } 
      } 
      else
      {
        if(infoInt != 0)
        {//Had bad factor.  Reduce poly degree.
          dim_--;
        } 
        else
        {
          status = false;
        } 
      } 
      if(status == false)
      {
        lhs = lhstemp;
      } 
    } 
    if(dim_ == 0)
    {
      pCoeff_.shape( 1, 1);
      pCoeff_(0,0) = SCT::one(); 
      std::cout << "Poly Degree is zero.  No preconditioner created." << std::endl;
    } 
    else
    {
      pCoeff_.shape( dim_, 1);
      //Get correct submatrix of AV:
      Teuchos::Range1D rangeSub( 1, dim_);
      Teuchos::RCP< const MV > AVsub = MVT::CloneView( *V, rangeSub);
      
      //Compute rhs (AV)^T V0
      MVT::MvTransMv( SCT::one(), *AVsub, *V0, pCoeff_);
      lapack.POTRS( 'U', dim_, 1, lhs.values(), lhs.stride(), pCoeff_.values(), pCoeff_.stride(), &infoInt);
      if(infoInt != 0) 
      {
        std::cout << "BelosGmresPolyOp.hpp: LAPACK POTRS was not successful!!" << std::endl;
        std::cout << "Error code: " << infoInt << std::endl; 
      } 
    } 
  }

  template <class ScalarType, class MV, class OP>
  void GmresPolyOp<ScalarType, MV, OP>::generateArnoldiPoly()
  {
    std::string polyLabel = label_ + ": GmresPolyOp creation";

    // Create a copy of the linear problem that has a zero initial guess and random RHS.
    std::vector<int> idx(1,0);
    Teuchos::RCP<MV> newX  = MVT::Clone( *(problem_->getLHS()), 1 );
    Teuchos::RCP<MV> newB  = MVT::Clone( *(problem_->getRHS()), 1 );
    MVT::MvInit( *newX, SCT::zero() );
    if (randomRHS_) {
      MVT::MvRandom( *newB );
    }
    else {
      MVT::Assign( *(MVT::CloneView(*(problem_->getRHS()), idx)), *newB );
    }
    Teuchos::RCP<LinearProblem<ScalarType,MV,OP> > newProblem =
      Teuchos::rcp( new LinearProblem<ScalarType,MV,OP>( problem_->getOperator(), newX, newB ) );
    newProblem->setInitResVec( newB );
    newProblem->setLeftPrec( problem_->getLeftPrec() );
    newProblem->setRightPrec( problem_->getRightPrec() );
    newProblem->setLabel(polyLabel);
    newProblem->setProblem();
    newProblem->setLSIndex( idx );

    // Create a parameter list for the GMRES iteration.
    Teuchos::ParameterList polyList;
 
    // Tell the block solver that the block size is one.
    polyList.set("Num Blocks",maxDegree_);
    polyList.set("Block Size",1);
    polyList.set("Keep Hessenberg", true);

    // Create output manager.
    printer_ = Teuchos::rcp( new OutputManager<ScalarType>(verbosity_, outputStream_) );

    // Create orthogonalization manager if we need to.  
    if (ortho_.is_null()) {
      params_->set("Orthogonalization", orthoType_);
      Belos::OrthoManagerFactory<ScalarType, MV, OP> factory;
      Teuchos::RCP<Teuchos::ParameterList> paramsOrtho;   // can be null

      ortho_ = factory.makeMatOrthoManager (orthoType_, Teuchos::null, printer_, polyLabel, paramsOrtho);
    }

    // Create a simple status test that either reaches the relative residual tolerance or maximum polynomial size.
    Teuchos::RCP<StatusTestMaxIters<ScalarType,MV,OP> > maxItrTst =
      Teuchos::rcp( new StatusTestMaxIters<ScalarType,MV,OP>( maxDegree_ ) );
  
    // Implicit residual test, using the native residual to determine if convergence was achieved.
    Teuchos::RCP<StatusTestGenResNorm<ScalarType,MV,OP> > convTst =
      Teuchos::rcp( new StatusTestGenResNorm<ScalarType,MV,OP>( polyTol_ ) );
    convTst->defineScaleForm( convertStringToScaleType("Norm of RHS"), Belos::TwoNorm );
  
    // Convergence test that stops the iteration when either are satisfied.
    Teuchos::RCP<StatusTestCombo<ScalarType,MV,OP> > polyTest =
      Teuchos::rcp( new StatusTestCombo<ScalarType,MV,OP>( StatusTestCombo<ScalarType,MV,OP>::OR, maxItrTst, convTst ) );
  
    // Create Gmres iteration object to perform one cycle of Gmres.
    Teuchos::RCP<BlockGmresIter<ScalarType,MV,OP> > gmres_iter;
    gmres_iter = Teuchos::rcp( new BlockGmresIter<ScalarType,MV,OP>(newProblem,printer_,polyTest,ortho_,polyList) );
 
    // Create the first block in the current Krylov basis (residual).
    Teuchos::RCP<MV> V_0 = MVT::CloneCopy( *newB );
    if ( !LP_.is_null() )
      newProblem->applyLeftPrec( *newB, *V_0 );
    if ( damp_ )
    {
      Teuchos::RCP< MV > Vtemp = MVT::CloneCopy(*V_0);
      newProblem->apply( *Vtemp, *V_0 );
    }
  
    // Get a matrix to hold the orthonormalization coefficients.
    r0_.resize(1);

    // Orthonormalize the new V_0
    int rank = ortho_->normalize( *V_0, Teuchos::rcpFromRef(r0_) );
    TEUCHOS_TEST_FOR_EXCEPTION(rank != 1,GmresPolyOpOrthoFailure,
      "Belos::GmresPolyOp::generateArnoldiPoly(): Failed to compute initial block of orthonormal vectors for polynomial generation.");
  
    // Set the new state and initialize the solver.
    GmresIterationState<ScalarType,MV> newstate;
    newstate.V = V_0;
    newstate.z = Teuchos::rcpFromRef( r0_);
    newstate.curDim = 0;
    gmres_iter->initializeGmres(newstate);

    // Perform Gmres iteration
    try {
      gmres_iter->iterate();
    }
    catch (GmresIterationOrthoFailure& e) {
      // Try to recover the most recent least-squares solution
      gmres_iter->updateLSQR( gmres_iter->getCurSubspaceDim() );
    }
    catch (std::exception& e) {
      using std::endl;
      printer_->stream(Errors) << "Error! Caught exception in BlockGmresIter::iterate() at iteration "
        << gmres_iter->getNumIters() << endl << e.what () << endl;
      throw;
    }

    // Get the solution for this polynomial, use in comparison below
    Teuchos::RCP<MV> currX = gmres_iter->getCurrentUpdate();
  
    // Record polynomial info, get current GMRES state
    GmresIterationState<ScalarType,MV> gmresState = gmres_iter->getState();
  
    // If the polynomial has no dimension, the tolerance is too low, return false
    dim_ = gmresState.curDim;
    if (dim_ == 0) {
      return;
    }
    if(polyType_ == "Arnoldi"){
      //  Make a view and then copy the RHS of the least squares problem.
      //
      y_ = Teuchos::SerialDenseMatrix<OT,ScalarType>( Teuchos::Copy, *gmresState.z, dim_, 1 );
      H_ = *gmresState.H;

      //
      // Solve the least squares problem.
      //
      Teuchos::BLAS<OT,ScalarType> blas;
      blas.TRSM( Teuchos::LEFT_SIDE, Teuchos::UPPER_TRI, Teuchos::NO_TRANS,
          Teuchos::NON_UNIT_DIAG, dim_, 1, SCT::one(),
          gmresState.R->values(), gmresState.R->stride(),
          y_.values(), y_.stride() );
    }
    else{ //Generate Roots Poly
    //Find Harmonic Ritz Values to use as polynomial roots:
    
    //Copy of square H used to find poly roots:
    H_ = Teuchos::SerialDenseMatrix<OT,ScalarType>(Teuchos::Copy, *gmresState.H, dim_, dim_);
    //Zero out below subdiagonal of H:
    for(int i=0; i <= dim_-3; i++) {
      for(int k=i+2; k <= dim_-1; k++) {
        H_(k,i) = SCT::zero();
      }
    }
    //Extra copy of H because equilibrate changes the matrix: 
    Teuchos::SerialDenseMatrix<OT,ScalarType> Htemp (Teuchos::Copy, H_, dim_, dim_);

    //View the m+1,m element and last col of H:
    ScalarType Hlast = (*gmresState.H)(dim_,dim_-1);
    Teuchos::SerialDenseMatrix<OT,ScalarType> HlastCol (Teuchos::View, H_, dim_, 1, 0, dim_-1);

    //Set up linear system for H^{-*}e_m:
    Teuchos::SerialDenseMatrix< OT, ScalarType > F(dim_,1), E(dim_,1);
    E.putScalar(SCT::zero());
    E(dim_-1,0) = SCT::one();
      
    Teuchos::SerialDenseSolver< OT, ScalarType > HSolver;
    HSolver.setMatrix( Teuchos::rcpFromRef(Htemp));
    HSolver.solveWithTransposeFlag( Teuchos::CONJ_TRANS );
    HSolver.setVectors( Teuchos::rcpFromRef(F), Teuchos::rcpFromRef(E));
    HSolver.factorWithEquilibration( true );

    //Factor matrix and solve for F = H^{-*}e_m:
    int info = 0;
    info = HSolver.factor();
    if(info != 0){
      std::cout << "Hsolver factor: info = " << info << std::endl;
    }
    info = HSolver.solve();
    if(info != 0){
      std::cout << "Hsolver solve : info = " << info << std::endl;
    }

    //Scale F and adjust H for Harmonic Ritz value eigenproblem:
    F.scale(Hlast*Hlast); 
    HlastCol += F; 

    //Set up for eigenvalue problem to get Harmonic Ritz Values:
    Teuchos::LAPACK< OT, ScalarType > lapack;
    theta_.shape(dim_,2);//1st col for real part, 2nd col for imaginary 

    const int ldv = 1;
    ScalarType* vlr = 0;

    // Size of workspace and workspace for DGEEV
    int lwork = -1;
    std::vector<ScalarType> work(1);
    std::vector<MagnitudeType> rwork(2*dim_);

    //Find workspace size for DGEEV:
    lapack.GEEV('N','N',dim_,H_.values(),H_.stride(),theta_[0],theta_[1],vlr, ldv, vlr, ldv, &work[0], lwork, &rwork[0], &info);
    lwork = std::abs (static_cast<int> (Teuchos::ScalarTraits<ScalarType>::real (work[0])));
    work.resize( lwork );
    // Solve for Harmonic Ritz Values:
    lapack.GEEV('N','N',dim_,H_.values(),H_.stride(),theta_[0],theta_[1],vlr, ldv, vlr, ldv, &work[0], lwork, &rwork[0], &info);

    if(info != 0){
      std::cout << "GEEV solve : info = " << info << std::endl;
    }

    // Set index for sort function, verify roots are non-zero,
    // and sort Harmonic Ritz Values:
    const MagnitudeType tol = 10.0 * Teuchos::ScalarTraits<MagnitudeType>::eps();
    std::vector<int> index(dim_);
    for(int i=0; i<dim_; ++i){ 
      index[i] = i; 
      // Check if real + imag parts of roots < tol. 
      TEUCHOS_TEST_FOR_EXCEPTION(hypot(theta_(i,0),theta_(i,1)) < tol, std::runtime_error, "BelosGmresPolyOp Error: One of the computed polynomial roots is approximately zero.  This will cause a divide by zero error!  Your matrix may be close to singular.  Please select a lower polynomial degree or give a shifted matrix.");
    }
    SortModLeja(theta_,index);

    //Add roots if neded.
    ComputeAddedRoots();

   }
  }
  
  //Function determines whether added roots are needed and adds them if option is turned on.
  template <class ScalarType, class MV, class OP>
  void GmresPolyOp<ScalarType, MV, OP>::ComputeAddedRoots()
  {
    // Store theta (with cols for real and imag parts of Harmonic Ritz Vals) 
    // as one vector of complex numbers to perform arithmetic:
    std::vector<std::complex<MagnitudeType>> cmplxHRitz (dim_);
    for(unsigned int i=0; i<cmplxHRitz.size(); ++i){
      cmplxHRitz[i] = std::complex<MagnitudeType>( theta_(i,0), theta_(i,1) );
    }
   
    // Compute product of factors (pof) to determine added roots: 
    const MagnitudeType one(1.0);
    std::vector<MagnitudeType> pof (dim_,one);
    for(int j=0; j<dim_; ++j) {
      for(int i=0; i<dim_; ++i) {
        if(i!=j) {
          pof[j] = std::abs(pof[j]*(one-(cmplxHRitz[j]/cmplxHRitz[i])));
        }
      }
    }

    // Compute number of extra roots needed:
    std::vector<int> extra (dim_);
    int totalExtra = 0;
    for(int i=0; i<dim_; ++i){
      if (pof[i] > MCT::zero())
        extra[i] = ceil((log10(pof[i])-MagnitudeType(4.0))/MagnitudeType(14.0));
      else
        extra[i] = 0;
      if(extra[i] > 0){
        totalExtra += extra[i];
      }
    }
    if (totalExtra){
      printer_->stream(Warnings) << "Warning: Need to add " << totalExtra << " extra roots." << std::endl;}

    // If requested to add roots, append them to the theta matrix:
    if(addRoots_ && totalExtra>0)
    {
      theta_.reshape(dim_+totalExtra,2);
      // Make a matrix copy for perturbed roots:
      Teuchos::SerialDenseMatrix<OT,MagnitudeType> thetaPert (Teuchos::Copy, theta_, dim_+totalExtra, 2);

      //Add extra eigenvalues to matrix and perturb for sort:
      int count = dim_;
      for(int i=0; i<dim_; ++i){
        for(int j=0; j< extra[i]; ++j){
          theta_(count,0) = theta_(i,0);
          theta_(count,1) = theta_(i,1);
          thetaPert(count,0) = theta_(i,0)+(j+MCT::one())*MagnitudeType(5e-8);
          thetaPert(count,1) = theta_(i,1);
          ++count;
        }
      }

      // Update polynomial degree:
      dim_ += totalExtra;
      if (totalExtra){
        printer_->stream(Warnings) << "New poly degree is: " << dim_ << std::endl;}

      // Create a new index and sort perturbed roots:
      std::vector<int> index2(dim_);
      for(int i=0; i<dim_; ++i){ 
        index2[i] = i; 
      }
      SortModLeja(thetaPert,index2);
      //Apply sorting to non-perturbed roots:
      for(int i=0; i<dim_; ++i)
      { 
        thetaPert(i,0) = theta_(index2[i],0);
        thetaPert(i,1) = theta_(index2[i],1);
      }
      theta_ = thetaPert;

    } 
  }

  // Modified Leja sorting function. Takes a serial dense matrix of M harmonic Ritz values and an index
  // of values from 0 to M. Returns the sorted values and sorted index, similar to Matlab.
  template <class ScalarType, class MV, class OP>
  void GmresPolyOp<ScalarType, MV, OP>::SortModLeja(Teuchos::SerialDenseMatrix< OT, MagnitudeType > &thetaN, std::vector<int> &index) const 
  {
    //Sort theta values via Modified Leja Ordering:

    // Set up blank matrices to track sorting:
    int dimN = index.size();
    std::vector<int> newIndex(dimN);
    Teuchos::SerialDenseMatrix< OT, MagnitudeType > sorted (thetaN.numRows(), thetaN.numCols());
    Teuchos::SerialDenseVector< OT, MagnitudeType > absVal (thetaN.numRows());
    Teuchos::SerialDenseVector< OT, MagnitudeType > prod (thetaN.numRows());

    //Compute all absolute values and find maximum:
    for(int i = 0; i < dimN; i++){
      absVal(i) = hypot(thetaN(i,0), thetaN(i,1)); 
    }
    MagnitudeType * maxPointer = std::max_element(absVal.values(), (absVal.values()+dimN));
    int maxIndex = int (maxPointer- absVal.values());

    //Put largest abs value first in the list:
    sorted(0,0) = thetaN(maxIndex,0);
    sorted(0,1) = thetaN(maxIndex,1);
    newIndex[0] = index[maxIndex];

    int j;  
    // If largest value was complex (for real scalar type) put its conjugate in the next slot.
    if(sorted(0,1)!= SCT::zero() && !SCT::isComplex) 
    {
      sorted(1,0) = thetaN(maxIndex,0);
      sorted(1,1) = -thetaN(maxIndex,1);
      newIndex[1] = index[maxIndex+1];
      j = 2;
    }
    else
    {
      j = 1;
    }

    //Sort remaining values:
    MagnitudeType a, b;
    while( j < dimN )
    {
      //For each value, compute (a log of) a product of differences:
      for(int i = 0; i < dimN; i++) 
      {
        prod(i) = MCT::one();
        for(int k = 0; k < j; k++)
        {
          a = thetaN(i,0) - sorted(k,0);
          b = thetaN(i,1) - sorted(k,1);
          if (a*a + b*b > MCT::zero())
            prod(i) = prod(i) + log10(hypot(a,b));
          else {
            prod(i) = -std::numeric_limits<MagnitudeType>::infinity();
            break;
          }
        }
      }
      
      //Value with largest product goes in the next slot:
      maxPointer = std::max_element(prod.values(), (prod.values()+dimN));
      maxIndex = int (maxPointer- prod.values());
      sorted(j,0) = thetaN(maxIndex,0);
      sorted(j,1) = thetaN(maxIndex,1);
      newIndex[j] = index[maxIndex];

      //If it was complex (and scalar type real) put its conjugate in next slot:
      if(sorted(j,1)!= SCT::zero() && !SCT::isComplex) 
      {
        j++;
        sorted(j,0) = thetaN(maxIndex,0);
        sorted(j,1) = -thetaN(maxIndex,1);
        newIndex[j] = index[maxIndex+1];
      }
      j++;
    }

    //Return sorted values and sorted indices:
    thetaN = sorted;
    index = newIndex;
  } //End Modified Leja ordering

  template <class ScalarType, class MV, class OP>
  void GmresPolyOp<ScalarType, MV, OP>::ApplyPoly( const MV& x, MV& y ) const 
  {
    if (dim_) {
      if (polyType_ == "Arnoldi")
        ApplyArnoldiPoly(x, y);
      else if (polyType_ == "Gmres")
        ApplyGmresPoly(x, y);
      else if (polyType_ == "Roots")
        ApplyRootsPoly(x, y);
    }
    else {
      // Just apply the operator in problem_ to x and return y.
      problem_->applyOp( x, y );
    }  
  }

  template <class ScalarType, class MV, class OP>
  void GmresPolyOp<ScalarType, MV, OP>::ApplyGmresPoly( const MV& x, MV& y ) const 
  {
    Teuchos::RCP<MV> AX = MVT::CloneCopy(x);
    Teuchos::RCP<MV> AX2 = MVT::Clone( x, MVT::GetNumberVecs(x) );

    // Apply left preconditioner.
    if (!LP_.is_null()) {
      Teuchos::RCP<MV> Xtmp = MVT::Clone( x, MVT::GetNumberVecs(x) );
      problem_->applyLeftPrec( *AX, *Xtmp ); // Left precondition x into the first vector 
      AX = Xtmp;
    } 

    {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
      Teuchos::TimeMonitor updateTimer( *timerPolyUpdate_ );
#endif
      MVT::MvAddMv(pCoeff_(0,0), *AX, SCT::zero(), y, y); //y= coeff_i(A^ix)
    }
    for( int i=1; i < dim_; i++)
    {
      Teuchos::RCP<MV> X, Y;
      if ( i%2 )
      {
        X = AX;
        Y = AX2;
      }
      else
      {
        X = AX2;
        Y = AX;
      }
      problem_->apply(*X, *Y);
      {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor updateTimer( *timerPolyUpdate_ );
#endif
        MVT::MvAddMv(pCoeff_(i,0), *Y, SCT::one(), y, y); //y= coeff_i(A^ix) +y
      }
    }

    // Apply right preconditioner.
    if (!RP_.is_null()) {
      Teuchos::RCP<MV> Ytmp = MVT::CloneCopy(y);
      problem_->applyRightPrec( *Ytmp, y );
    }
  }

  template <class ScalarType, class MV, class OP>
  void GmresPolyOp<ScalarType, MV, OP>::ApplyRootsPoly( const MV& x, MV& y ) const 
  {
    MVT::MvInit( y, SCT::zero() ); //Zero out y to take the vector with poly applied.
    Teuchos::RCP<MV> prod = MVT::CloneCopy(x);
    Teuchos::RCP<MV> Xtmp = MVT::Clone( x, MVT::GetNumberVecs(x) );
    Teuchos::RCP<MV> Xtmp2 = MVT::Clone( x, MVT::GetNumberVecs(x) );

    // Apply left preconditioner.
    if (!LP_.is_null()) {
      problem_->applyLeftPrec( *prod, *Xtmp ); // Left precondition x into the first vector 
      prod = Xtmp;
    } 
    
    int i=0;   
    while(i < dim_-1)
    {
      if(theta_(i,1)== SCT::zero() || SCT::isComplex) //Real Harmonic Ritz value or complex scalars
      {
        {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
          Teuchos::TimeMonitor updateTimer( *timerPolyUpdate_ );
#endif
          MVT::MvAddMv(SCT::one(), y, SCT::one()/theta_(i,0), *prod, y); //poly = poly + 1/theta_i * prod
        }
        problem_->apply(*prod, *Xtmp); // temp = A*prod
        {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
          Teuchos::TimeMonitor updateTimer( *timerPolyUpdate_ );
#endif
          MVT::MvAddMv(SCT::one(), *prod, -SCT::one()/theta_(i,0), *Xtmp, *prod); //prod = prod - 1/theta_i * temp
        }
        i++;
      }
      else //Current theta is complex and has a conjugate; combine to preserve real arithmetic 
      {
        MagnitudeType mod = theta_(i,0)*theta_(i,0) + theta_(i,1)*theta_(i,1); //mod = a^2 + b^2
        problem_->apply(*prod, *Xtmp); // temp = A*prod
        {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
          Teuchos::TimeMonitor updateTimer( *timerPolyUpdate_ );
#endif
          MVT::MvAddMv(2*theta_(i,0), *prod, -SCT::one(), *Xtmp, *Xtmp); //temp = 2a*prod-temp 
          MVT::MvAddMv(SCT::one(), y, SCT::one()/mod, *Xtmp, y); //poly = poly + 1/mod*temp
        }
        if( i < dim_-2 )
        {
          problem_->apply(*Xtmp, *Xtmp2); // temp2 = A*temp
          {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
            Teuchos::TimeMonitor updateTimer( *timerPolyUpdate_ );
#endif
            MVT::MvAddMv(SCT::one(), *prod, -SCT::one()/mod, *Xtmp2, *prod); //prod = prod - 1/mod * temp2
          }
        }
        i = i + 2; 
      }
    }
    if(theta_(dim_-1,1)== SCT::zero() || SCT::isComplex) 
    {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
      Teuchos::TimeMonitor updateTimer( *timerPolyUpdate_ );
#endif
      MVT::MvAddMv(SCT::one(), y, SCT::one()/theta_(dim_-1,0), *prod, y); //poly = poly + 1/theta_i * prod
    }

    // Apply right preconditioner.
    if (!RP_.is_null()) {
      Teuchos::RCP<MV> Ytmp = MVT::CloneCopy(y);
      problem_->applyRightPrec( *Ytmp, y );
    }
  }

  template <class ScalarType, class MV, class OP>
  void GmresPolyOp<ScalarType, MV, OP>::ApplyArnoldiPoly( const MV& x, MV& y ) const 
  {
    // Initialize vector storage.
    if (V_.is_null()) {
      V_ = MVT::Clone( x, dim_ );
      if (!LP_.is_null()) {
        wL_ = MVT::Clone( y, 1 );
      }
      if (!RP_.is_null()) {
        wR_ = MVT::Clone( y, 1 );
      }
    }
    //
    // Apply polynomial to x.
    // 
    int n = MVT::GetNumberVecs( x );
    std::vector<int> idxi(1), idxi2, idxj(1);

    // Select vector x[j].
    for (int j=0; j<n; ++j) {

      idxi[0] = 0;
      idxj[0] = j;
      Teuchos::RCP<const MV> x_view = MVT::CloneView( x, idxj );
      Teuchos::RCP<MV> y_view = MVT::CloneViewNonConst( y, idxj );
      if (!LP_.is_null()) {
        Teuchos::RCP<MV> v_curr = MVT::CloneViewNonConst( *V_, idxi );
        problem_->applyLeftPrec( *x_view, *v_curr ); // Left precondition x into the first vector of V
      } else {
        MVT::SetBlock( *x_view, idxi, *V_ );  // Set x as the first vector of V
      }

      for (int i=0; i<dim_-1; ++i) {

        // Get views into the current and next vectors
        idxi2.resize(i+1);
        for (int ii=0; ii<i+1; ++ii) { idxi2[ii] = ii; }
        Teuchos::RCP<const MV> v_prev = MVT::CloneView( *V_, idxi2 );
        // the tricks below with wR_ and wL_ (potentially set to v_curr and v_next) unfortunately imply that 
        // v_curr and v_next must be non-const views.
        Teuchos::RCP<MV> v_curr = MVT::CloneViewNonConst( *V_, idxi );
        idxi[0] = i+1;
        Teuchos::RCP<MV> v_next = MVT::CloneViewNonConst( *V_, idxi );

        //---------------------------------------------
        // Apply operator to next vector
        //---------------------------------------------
        // 1) Apply right preconditioner, if we have one.
        if (!RP_.is_null()) {
          problem_->applyRightPrec( *v_curr, *wR_ );
        } else {
          wR_ = v_curr;
        }
        // 2) Check for left preconditioner, if none exists, point at the next vector.
        if (LP_.is_null()) {
          wL_ = v_next;
        }
        // 3) Apply operator A.
        problem_->applyOp( *wR_, *wL_ );
        // 4) Apply left preconditioner, if we have one.
        if (!LP_.is_null()) {
          problem_->applyLeftPrec( *wL_, *v_next );
        }

        // Compute A*v_curr - v_prev*H(1:i,i)
        Teuchos::SerialDenseMatrix<OT,ScalarType> h(Teuchos::View,H_,i+1,1,0,i);
        {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
          Teuchos::TimeMonitor updateTimer( *timerPolyUpdate_ );
#endif
          MVT::MvTimesMatAddMv( -SCT::one(), *v_prev, h, SCT::one(), *v_next );
        }

        // Scale by H(i+1,i)
        MVT::MvScale( *v_next, SCT::one()/H_(i+1,i) );  
      }

      // Compute output y = V*y_./r0_
      if (!RP_.is_null()) {
        {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
          Teuchos::TimeMonitor updateTimer( *timerPolyUpdate_ );
#endif
          MVT::MvTimesMatAddMv( SCT::one()/r0_(0), *V_, y_, SCT::zero(), *wR_ );
        }
        problem_->applyRightPrec( *wR_, *y_view );
      } 
      else {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor updateTimer( *timerPolyUpdate_ );
#endif
        MVT::MvTimesMatAddMv( SCT::one()/r0_(0), *V_, y_, SCT::zero(), *y_view );
      }
    } // (int j=0; j<n; ++j)
  } // end Apply()
} // end Belos namespace

#endif

// end of file BelosGmresPolyOp.hpp