BelosPseudoBlockGmresIter.hpp

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// @HEADER
// *****************************************************************************
//                 Belos: Block Linear Solvers Package
//
// Copyright 2004-2016 NTESS and the Belos contributors.
// SPDX-License-Identifier: BSD-3-Clause
// *****************************************************************************
// @HEADER
 
#ifndef BELOS_PSEUDO_BLOCK_GMRES_ITER_HPP
#define BELOS_PSEUDO_BLOCK_GMRES_ITER_HPP
 
#include "BelosConfigDefs.hpp"
#include "BelosTypes.hpp"
#include "BelosIteration.hpp"
#include "BelosGmresIteration.hpp"
 
#include "BelosLinearProblem.hpp"
#include "BelosMatOrthoManager.hpp"
#include "BelosOutputManager.hpp"
#include "BelosStatusTest.hpp"
#include "BelosOperatorTraits.hpp"
#include "BelosMultiVecTraits.hpp"
#include "BelosDenseMatTraits.hpp"
 
#include "Teuchos_BLAS.hpp"
#include "Teuchos_ScalarTraits.hpp"
#include "Teuchos_ParameterList.hpp"
#include "Teuchos_TimeMonitor.hpp"
 
namespace Belos {
 
  template<class ScalarType, class MV, class OP, class DM>  
  class PseudoBlockGmresIter : virtual public Iteration<ScalarType,MV,OP,DM> {
    
  public:
    
    //
    // Convenience typedefs
    //
    typedef MultiVecTraits<ScalarType,MV,DM> MVT;
    typedef OperatorTraits<ScalarType,MV,OP> OPT;
    typedef DenseMatTraits<ScalarType,DM>    DMT;
    typedef Teuchos::ScalarTraits<ScalarType> SCT;
    typedef typename SCT::magnitudeType MagnitudeType;
    
 
    
    PseudoBlockGmresIter( const Teuchos::RCP<LinearProblem<ScalarType,MV,OP,DM> > &problem,
        const Teuchos::RCP<OutputManager<ScalarType> > &printer,
        const Teuchos::RCP<StatusTest<ScalarType,MV,OP,DM> > &tester,
        const Teuchos::RCP<MatOrthoManager<ScalarType,MV,OP,DM> > &ortho,
        Teuchos::ParameterList &params );
    
    virtual ~PseudoBlockGmresIter() {};
    
    
 
    
    void iterate();
    
    void initialize(const PseudoBlockGmresIterState<ScalarType,MV,DM> & newstate);
    
    void initialize()
    {
      PseudoBlockGmresIterState<ScalarType,MV,DM> empty;
      initialize(empty);
    }
    
    PseudoBlockGmresIterState<ScalarType,MV,DM> getState() const {
      PseudoBlockGmresIterState<ScalarType,MV,DM> state;
      state.curDim = curDim_;
      state.V.resize(numRHS_);
      state.H.resize(numRHS_);
      state.Z.resize(numRHS_);
      state.sn.resize(numRHS_);
      state.cs.resize(numRHS_);  
      for (int i=0; i<numRHS_; ++i) {
  state.V[i] = V_[i];
  state.H[i] = H_[i];
  state.Z[i] = Z_[i];
        state.sn[i] = sn_[i];
        state.cs[i] = cs_[i];
      }
      return state;
    }
    
    
    
 
    
    int getNumIters() const { return iter_; }
    
    void resetNumIters( int iter = 0 ) { iter_ = iter; }
    
    Teuchos::RCP<const MV> getNativeResiduals( std::vector<MagnitudeType> *norms ) const;
    
 
    Teuchos::RCP<MV> getCurrentUpdate() const;
    
 
    void updateLSQR( int dim = -1 );
    
    int getCurSubspaceDim() const { 
      if (!initialized_) return 0;
      return curDim_;
    };
    
    int getMaxSubspaceDim() const { return numBlocks_; }
    
    
    
 
    
    const LinearProblem<ScalarType,MV,OP,DM>& getProblem() const { return *lp_; }
    
    int getBlockSize() const { return 1; }
    
    void setBlockSize(int blockSize) { 
      TEUCHOS_TEST_FOR_EXCEPTION(blockSize!=1,std::invalid_argument,
       "Belos::PseudoBlockGmresIter::setBlockSize(): Cannot use a block size that is not one.");
    }
    
    int getNumBlocks() const { return numBlocks_; }
    
    void setNumBlocks(int numBlocks);
    
    bool isInitialized() { return initialized_; }
    
    
  private:
    
    //
    // Classes inputed through constructor that define the linear problem to be solved.
    //
    const Teuchos::RCP<LinearProblem<ScalarType,MV,OP,DM> >       lp_;
    const Teuchos::RCP<OutputManager<ScalarType> >             om_;
    const Teuchos::RCP<StatusTest<ScalarType,MV,OP,DM> >       stest_;
    const Teuchos::RCP<OrthoManager<ScalarType,MV,DM> >        ortho_;
    
    //
    // Algorithmic parameters
    //  
    // numRHS_ is the current number of linear systems being solved.
    int numRHS_;
    // numBlocks_ is the size of the allocated space for the Krylov basis, in blocks.
    int numBlocks_; 
    
    // Storage for QR factorization of the least squares system.
    std::vector<Teuchos::RCP<std::vector<ScalarType> > > sn_;
    std::vector<Teuchos::RCP<std::vector<MagnitudeType> > > cs_;
    
    // Pointers to a work vector used to improve aggregate performance.
    Teuchos::RCP<MV> U_vec_, AU_vec_;    
 
    // Pointers to the current right-hand side and solution multivecs being solved for.
    Teuchos::RCP<MV> cur_block_rhs_, cur_block_sol_;
 
    // 
    // 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_;
    
    // Current subspace dimension, and number of iterations performed.
    int curDim_, iter_;
 
    // 
    // State Storage
    //
    std::vector<Teuchos::RCP<MV> > V_;
    //
    // Projected matrices
    // H_ : Projected matrix from the Krylov factorization AV = VH + FE^T
    //
    std::vector<Teuchos::RCP<DM> > H_;
    // 
    // QR decomposition of Projected matrices for solving the least squares system HY = B.
    // R_: Upper triangular reduction of H
    // Z_: Q applied to right-hand side of the least squares system
    std::vector<Teuchos::RCP<DM> > R_;
    std::vector<Teuchos::RCP<DM> > Z_;  
  };
  
  // Constructor.
  template<class ScalarType, class MV, class OP, class DM>
  PseudoBlockGmresIter<ScalarType,MV,OP,DM>::PseudoBlockGmresIter(const Teuchos::RCP<LinearProblem<ScalarType,MV,OP,DM> > &problem,
                     const Teuchos::RCP<OutputManager<ScalarType> > &printer,
                     const Teuchos::RCP<StatusTest<ScalarType,MV,OP,DM> > &tester,
                     const Teuchos::RCP<MatOrthoManager<ScalarType,MV,OP,DM> > &ortho,
                     Teuchos::ParameterList &params ):
    lp_(problem),
    om_(printer),
    stest_(tester),
    ortho_(ortho),
    numRHS_(0),
    numBlocks_(0),
    initialized_(false),
    curDim_(0),
    iter_(0)
  {
    // Get the maximum number of blocks allowed for each Krylov subspace
    TEUCHOS_TEST_FOR_EXCEPTION(!params.isParameter("Num Blocks"), std::invalid_argument,
                       "Belos::PseudoBlockGmresIter::constructor: mandatory parameter 'Num Blocks' is not specified.");
    int nb = Teuchos::getParameter<int>(params, "Num Blocks");
 
    setNumBlocks( nb );
  }
  
  // Set the block size and make necessary adjustments.
  template <class ScalarType, class MV, class OP, class DM>
  void PseudoBlockGmresIter<ScalarType,MV,OP,DM>::setNumBlocks (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, std::invalid_argument, "Belos::PseudoBlockGmresIter::setNumBlocks was passed a non-positive argument.");
 
    numBlocks_ = numBlocks;
    curDim_ = 0;
 
    initialized_ = false;
  }
 
  // Get the current update from this subspace.
  template <class ScalarType, class MV, class OP, class DM>
  Teuchos::RCP<MV> PseudoBlockGmresIter<ScalarType,MV,OP,DM>::getCurrentUpdate() const
  {
    //
    // If this is the first iteration of the Arnoldi factorization, 
    // there is no update, so return Teuchos::null. 
    //
    Teuchos::RCP<MV> currentUpdate = Teuchos::null;
    if (curDim_==0) { 
      return currentUpdate; 
    } else {
      currentUpdate = MVT::Clone(*(V_[0]), numRHS_);
      std::vector<int> index(1), index2(curDim_);
      for (int i=0; i<curDim_; ++i) {
        index2[i] = i;
      }
      const ScalarType one = Teuchos::ScalarTraits<ScalarType>::one();
      const ScalarType zero = Teuchos::ScalarTraits<ScalarType>::zero();
      Teuchos::BLAS<int,ScalarType> blas;
      
      for (int i=0; i<numRHS_; ++i) {
        index[0] = i;
        Teuchos::RCP<MV> cur_block_copy_vec = MVT::CloneViewNonConst( *currentUpdate, index );
        //
        //  Make a view and then copy the RHS of the least squares problem.  DON'T OVERWRITE IT!
        //
        Teuchos::RCP<DM> y = DMT::SubviewCopy(*Z_[i], curDim_, 1);
        DMT::SyncDeviceToHost( *y );
        DMT::SyncDeviceToHost( *H_[i] );
        //
        //  Solve the least squares problem and compute current solutions.
        //
        blas.TRSM( Teuchos::LEFT_SIDE, Teuchos::UPPER_TRI, Teuchos::NO_TRANS,
             Teuchos::NON_UNIT_DIAG, curDim_, 1, one,  
       DMT::GetConstRawHostPtr(*H_[i]), DMT::GetStride(*H_[i]), 
                   DMT::GetRawHostPtr(*y), DMT::GetStride(*y) );
 
        DMT::SyncHostToDevice( *y );  
        DMT::SyncHostToDevice( *H_[i] );  
  Teuchos::RCP<const MV> Vjp1 = MVT::CloneView( *V_[i], index2 );
  MVT::MvTimesMatAddMv( one, *Vjp1, *y, zero, *cur_block_copy_vec );
      }
    }
    return currentUpdate;
  }
  
 
  // Get the native residuals stored in this iteration.  
  // Note:  No residual vector will be returned by Gmres.
  template <class ScalarType, class MV, class OP, class DM>
  Teuchos::RCP<const MV> 
  PseudoBlockGmresIter<ScalarType,MV,OP,DM>::
  getNativeResiduals (std::vector<MagnitudeType> *norms) const 
  {
    if (norms)
      { // Resize the incoming std::vector if necessary.  The type
  // cast avoids the compiler warning resulting from a signed /
  // unsigned integer comparison.
  if (static_cast<int> (norms->size()) < numRHS_)
    norms->resize (numRHS_); 
 
  for (int j = 0; j < numRHS_; ++j) 
    {
            DMT::SyncDeviceToHost( *Z_[j] );
      const ScalarType curNativeResid = DMT::ValueConst(*Z_[j],curDim_,0);
      (*norms)[j] = SCT::magnitude (curNativeResid);
    }
    }
    return Teuchos::null;
  }
 
  
  template <class ScalarType, class MV, class OP, class DM>
  void 
  PseudoBlockGmresIter<ScalarType,MV,OP,DM>::
  initialize (const PseudoBlockGmresIterState<ScalarType,MV,DM> & newstate)
  {
    using Teuchos::RCP;
 
    // (Re)set the number of right-hand sides, by interrogating the
    // current LinearProblem to solve.
    this->numRHS_ = MVT::GetNumberVecs (*(lp_->getCurrLHSVec()));
 
    // NOTE:  In PseudoBlockGmresIter, V and Z are required!!!  
    // Inconsistent multivectors widths and lengths will not be tolerated, and
    // will be treated with exceptions.
    //
    std::string errstr ("Belos::PseudoBlockGmresIter::initialize(): "
      "Specified multivectors must have a consistent "
      "length and width.");
 
    // Check that newstate has V and Z arrays with nonzero length.
    TEUCHOS_TEST_FOR_EXCEPTION((int)newstate.V.size()==0 || (int)newstate.Z.size()==0, 
           std::invalid_argument,
                       "Belos::PseudoBlockGmresIter::initialize(): "
           "V and/or Z was not specified in the input state; "
           "the V and/or Z arrays have length zero.");
 
    // In order to create basis multivectors, we have to clone them
    // from some already existing multivector.  We require that at
    // least one of the right-hand side B and left-hand side X in the
    // LinearProblem be non-null.  Thus, we can clone from either B or
    // X.  We prefer to close from B, since B is in the range of the
    // operator A and the basis vectors should also be in the range of
    // A (the first basis vector is a scaled residual vector).
    // However, if B is not given, we will try our best by cloning
    // from X.
    RCP<const MV> lhsMV = lp_->getLHS();
    RCP<const MV> rhsMV = lp_->getRHS();
 
    // If the right-hand side is null, we make do with the left-hand
    // side, otherwise we use the right-hand side.
    RCP<const MV> vectorInBasisSpace = rhsMV.is_null() ? lhsMV : rhsMV;
    //RCP<const MV> tmp = ( (rhsMV!=Teuchos::null)? rhsMV: lhsMV );
 
    TEUCHOS_TEST_FOR_EXCEPTION(vectorInBasisSpace.is_null(), 
           std::invalid_argument,
                       "Belos::PseudoBlockGmresIter::initialize(): "
           "The linear problem to solve does not specify multi"
           "vectors from which we can clone basis vectors.  The "
           "right-hand side(s), left-hand side(s), or both should "
           "be nonnull.");
 
    // Check the new dimension is not more that the maximum number of
    // allowable blocks.
    TEUCHOS_TEST_FOR_EXCEPTION(newstate.curDim > numBlocks_+1, 
           std::invalid_argument, 
           errstr);
    curDim_ = newstate.curDim;
 
    // Initialize the state storage.  If the subspace has not be
    // initialized before, generate it using the right-hand side or
    // left-hand side from the LinearProblem lp_ to solve.
    V_.resize(numRHS_);
    for (int i=0; i<numRHS_; ++i) {
      // Create a new vector if we need to.  We "need to" if the
      // current vector V_[i] is null, or if it doesn't have enough
      // columns.
      if (V_[i].is_null() || MVT::GetNumberVecs(*V_[i]) < numBlocks_ + 1) {
        V_[i] = MVT::Clone (*vectorInBasisSpace, numBlocks_ + 1);
      }
      // Check that the newstate vector newstate.V[i] has dimensions
      // consistent with those of V_[i].
      TEUCHOS_TEST_FOR_EXCEPTION( MVT::GetGlobalLength(*newstate.V[i]) != MVT::GetGlobalLength(*V_[i]),
                          std::invalid_argument, errstr );
      TEUCHOS_TEST_FOR_EXCEPTION( MVT::GetNumberVecs(*newstate.V[i]) < newstate.curDim,
                          std::invalid_argument, errstr );
      //
      // If newstate.V[i] and V_[i] are not identically the same
      // vector, then copy newstate.V[i] into V_[i].  
      //
      int lclDim = MVT::GetNumberVecs(*newstate.V[i]);
      if (newstate.V[i] != V_[i]) {
        // Only copy over the first block and print a warning.
        if (curDim_ == 0 && lclDim > 1) {
          om_->stream(Warnings) 
      << "Belos::PseudoBlockGmresIter::initialize(): the solver was "
      << "initialized with a kernel of " << lclDim 
      << std::endl 
      << "The block size however is only " << 1 
      << std::endl
      << "The last " << lclDim - 1 << " vectors will be discarded." 
      << std::endl;
        }
        std::vector<int> nevind (curDim_ + 1);
        for (int j = 0; j < curDim_ + 1; ++j) 
    nevind[j] = j;
 
        RCP<const MV> newV = MVT::CloneView (*newstate.V[i], nevind);
        RCP<MV> lclV = MVT::CloneViewNonConst( *V_[i], nevind );
        MVT::Assign(*newV, *lclV);
 
        // Done with local pointers
        lclV = Teuchos::null;
      }
    }
 
 
    // Check size of Z
    Z_.resize(numRHS_);
    for (int i=0; i<numRHS_; ++i) {
      // Create a vector if we need to.
      if (Z_[i] == Teuchos::null) {
  Z_[i] = DMT::Create();
      }
      if (DMT::GetNumRows(*Z_[i]) < numBlocks_+1) {
  DMT::Reshape(*Z_[i], numBlocks_+1, 1, false); 
      }
      
      // Check that the newstate vector is consistent.
      TEUCHOS_TEST_FOR_EXCEPTION(DMT::GetNumRows(*newstate.Z[i]) < curDim_, std::invalid_argument, errstr);
      
      // Put data into Z_, make sure old information is not still hanging around.
      if (newstate.Z[i] != Z_[i]) {
  if (curDim_==0)
    DMT::PutScalar(*Z_[i]);
  
        Teuchos::RCP<const DM> newZ = DMT::SubviewConst(*newstate.Z[i],curDim_+1,1);
        Teuchos::RCP<DM> lclZ = DMT::Subview(*Z_[i],curDim_+1,1);
        DMT::Assign(*lclZ,*newZ);
  
        // Done with local pointers
  lclZ = Teuchos::null;
      }
    }
 
 
    // Check size of H
    H_.resize(numRHS_);
    for (int i=0; i<numRHS_; ++i) {
      // Create a matrix if we need to.
      if (H_[i] == Teuchos::null) {
  H_[i] = DMT::Create();
      }
      if (DMT::GetNumRows(*H_[i]) < numBlocks_+1 || DMT::GetNumCols(*H_[i]) < numBlocks_) {
  DMT::Reshape(*H_[i], numBlocks_+1, numBlocks_, false);
      }
      
      // Put data into H_ if it exists, make sure old information is not still hanging around.
      if ((int)newstate.H.size() == numRHS_) {
  
  // Check that the newstate matrix is consistent.
  TEUCHOS_TEST_FOR_EXCEPTION((DMT::GetNumRows(*newstate.H[i]) < curDim_ || DMT::GetNumCols(*newstate.H[i]) < curDim_), std::invalid_argument, 
         "Belos::PseudoBlockGmresIter::initialize(): Specified Hessenberg matrices must have a consistent size to the current subspace dimension");
  
  if (newstate.H[i] != H_[i]) {
    
    Teuchos::RCP<const DM> newH = DMT::SubviewConst(*newstate.H[i],curDim_+1, curDim_);
    Teuchos::RCP<DM> lclH = DMT::Subview(*H_[i],curDim_+1, curDim_);
    DMT::Assign(*lclH,*newH);
    
    // Done with local pointers
    lclH = Teuchos::null;
  }
      }
    }      
    
    // Reinitialize storage for least squares solve
    //
    cs_.resize(numRHS_);
    sn_.resize(numRHS_);
      
    // Copy over rotation angles if they exist
    if ((int)newstate.cs.size() == numRHS_ && (int)newstate.sn.size() == numRHS_) {
      for (int i=0; i<numRHS_; ++i) {
        if (cs_[i] != newstate.cs[i])
          cs_[i] = Teuchos::rcp( new std::vector<MagnitudeType>(*newstate.cs[i]) );
        if (sn_[i] != newstate.sn[i])
          sn_[i] = Teuchos::rcp( new std::vector<ScalarType>(*newstate.sn[i]) );
      }
    } 
      
    // Resize or create the vectors as necessary
    for (int i=0; i<numRHS_; ++i) {
      if (cs_[i] == Teuchos::null) 
        cs_[i] = Teuchos::rcp( new std::vector<MagnitudeType>(numBlocks_+1) );
      else
        cs_[i]->resize(numBlocks_+1);
      if (sn_[i] == Teuchos::null)
        sn_[i] = Teuchos::rcp( new std::vector<ScalarType>(numBlocks_+1) );
      else
        sn_[i]->resize(numBlocks_+1);
    }
 
    // the solver is initialized
    initialized_ = true;
      
  }
 
 
  // Iterate until the status test informs us we should stop.
  template <class ScalarType, class MV, class OP, class DM>
  void PseudoBlockGmresIter<ScalarType,MV,OP,DM>::iterate()
  {
    //
    // Allocate/initialize data structures
    //
    if (initialized_ == false) {
      initialize();
    }
 
    // Compute the current search dimension. 
    int searchDim = numBlocks_;
    //
    // Associate each initial block of V_[i] with U_vec[i]
    // Reset the index vector (this might have been changed if there was a restart)
    //
    std::vector<int> index(1);
    std::vector<int> index2(1);
    index[0] = curDim_;
    Teuchos::RCP<MV> U_vec = MVT::Clone( *V_[0], numRHS_ );
 
    // Create AU_vec to hold A*U_vec.
    Teuchos::RCP<MV> AU_vec = MVT::Clone( *V_[0], numRHS_ );
 
    for (int i=0; i<numRHS_; ++i) {
      index2[0] = i;
      Teuchos::RCP<const MV> tmp_vec = MVT::CloneView( *V_[i], index );
      Teuchos::RCP<MV> U_vec_view = MVT::CloneViewNonConst( *U_vec, index2 );
      MVT::Assign( *tmp_vec, *U_vec_view );
    }
    
    // iterate until the status test tells us to stop.
    //
    // also break if our basis is full
    //
    while (stest_->checkStatus(this) != Passed && curDim_ < searchDim) {
 
      iter_++;
      //
      // Apply the operator to _work_vector
      //
      lp_->apply( *U_vec, *AU_vec );
      //
      //
      // Resize index.
      //
      int num_prev = curDim_+1;
      index.resize( num_prev );
      for (int i=0; i<num_prev; ++i) { 
  index[i] = i; 
      }
      //
      // Orthogonalize next Krylov vector for each right-hand side.
      //
      for (int i=0; i<numRHS_; ++i) {
  //
  // Get previous Krylov vectors.
  //
  Teuchos::RCP<const MV> V_prev = MVT::CloneView( *V_[i], index );
  Teuchos::Array< Teuchos::RCP<const MV> > V_array( 1, V_prev );
  //
  // Get a view of the new candidate std::vector.
  //
  index2[0] = i;
  Teuchos::RCP<MV> V_new = MVT::CloneViewNonConst( *AU_vec, index2 );
  //
  // Get a view of the current part of the upper-hessenberg matrix.
  //
  Teuchos::RCP<DM> h_new = DMT::Subview(*H_[i], num_prev, 1, 0, curDim_); 
  Teuchos::Array< Teuchos::RCP<DM> > h_array( 1, h_new );
  
  Teuchos::RCP<DM> r_new = DMT::Subview(*H_[i], 1, 1, num_prev, curDim_);
  //
  // Orthonormalize the new block of the Krylov expansion
  // NOTE:  Rank deficiencies are not checked because this is a single-std::vector Krylov method.
  //
  ortho_->projectAndNormalize( *V_new, h_array, r_new, V_array );
  //
  // NOTE:  V_new is a copy of the iter+1 vector in V_[i], so the normalized vector has to be
  // be copied back in when V_new is changed.  
  //
  index2[0] = curDim_+1;
  Teuchos::RCP<MV> tmp_vec = MVT::CloneViewNonConst( *V_[i], index2 );
  MVT::Assign( *V_new, *tmp_vec );
      }
      // 
      // Now _AU_vec is the new _U_vec, so swap these two vectors.
      // NOTE: This alleviates the need for allocating a vector for AU_vec each iteration.
      // 
      Teuchos::RCP<MV> tmp_AU_vec = U_vec;
      U_vec = AU_vec;
      AU_vec = tmp_AU_vec;
      //
      // V has been extended, and H has been extended. 
      //
      // Update the QR factorization of the upper Hessenberg matrix
      //
      updateLSQR();
      //
      // Update basis dim and release all pointers.
      //
      curDim_ += 1;
      //        
    } // end while (statusTest == false)
   
  }
 
  // Update the least squares solution for each right-hand side.
  template<class ScalarType, class MV, class OP, class DM>
  void PseudoBlockGmresIter<ScalarType,MV,OP,DM>::updateLSQR( int dim )
  {
    // Get correct dimension based on input "dim"
    // Remember that ortho failures result in an exit before updateLSQR() is called.
    // Therefore, it is possible that dim == curDim_.
    int curDim = curDim_;
    if (dim >= curDim_ && dim < getMaxSubspaceDim()) {
      curDim = dim;
    }
    
    int i, j;
    const ScalarType zero = Teuchos::ScalarTraits<ScalarType>::zero();
 
    Teuchos::BLAS<int, ScalarType> blas;
    
    for (i=0; i<numRHS_; ++i) {
      //
      // Update the least-squares QR for each linear system.
      //
      // QR factorization of Least-Squares system with Givens rotations
      //
      DMT::SyncDeviceToHost(*H_[i]);
      DMT::SyncDeviceToHost(*Z_[i]);
      // 
      for (j=0; j<curDim; j++) {
  //
  // Apply previous Givens rotations to new column of Hessenberg matrix
  //
  blas.ROT( 1, &(DMT::Value(*H_[i],j,curDim)), 1, &(DMT::Value(*H_[i],j+1, curDim)), 
                  1, &(*cs_[i])[j], &(*sn_[i])[j] );
      }
      //
      // Calculate new Givens rotation
      //
      blas.ROTG( &(DMT::Value(*H_[i],curDim,curDim)), &(DMT::Value(*H_[i],curDim+1,curDim)), 
                 &(*cs_[i])[curDim], &(*sn_[i])[curDim] );
      DMT::Value(*H_[i],curDim+1,curDim) = zero;
      //
      // Update RHS w/ new transformation
      //
      blas.ROT( 1, &(DMT::Value(*Z_[i],curDim,0)), 1, &(DMT::Value(*Z_[i],curDim+1,0)), 
                1, &(*cs_[i])[curDim], &(*sn_[i])[curDim] );
      //
      DMT::SyncHostToDevice(*H_[i]);
      DMT::SyncHostToDevice(*Z_[i]);
      // 
    }
 
  } // end updateLSQR()
 
} // end Belos namespace
 
#endif /* BELOS_PSEUDO_BLOCK_GMRES_ITER_HPP */