BelosICGSOrthoManager.hpp

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#ifndef BELOS_ICGS_ORTHOMANAGER_HPP
#define BELOS_ICGS_ORTHOMANAGER_HPP

// #define ORTHO_DEBUG

#include "BelosConfigDefs.hpp"
#include "BelosMultiVecTraits.hpp"
#include "BelosOperatorTraits.hpp"
#include "BelosMatOrthoManager.hpp"

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

namespace Belos {

  template<class ScalarType>
  Teuchos::RCP<const Teuchos::ParameterList> TEUCHOS_DEPRECATED
  getDefaultIcgsParameters()
  {
    using Teuchos::as;
    typedef typename Teuchos::ScalarTraits<ScalarType>::magnitudeType magnitude_type;
    typedef Teuchos::ScalarTraits<magnitude_type> STM;

    // Default parameter values for ICGS orthogonalization.
    // Documentation will be embedded in the parameter list.
    const int defaultMaxNumOrthogPasses = 2;
    const magnitude_type eps = STM::eps();
    const magnitude_type defaultBlkTol = 
      as<magnitude_type> (10) * STM::squareroot (eps);
    const magnitude_type defaultSingTol = as<magnitude_type> (10) * eps;

    Teuchos::RCP<Teuchos::ParameterList> params = 
      Teuchos::parameterList ("ICGS");
    params->set ("maxNumOrthogPasses", defaultMaxNumOrthogPasses,
                 "Maximum number of orthogonalization passes "
                 "(includes the first).  Default is 2, since "
                 "\"twice is enough\" for Krylov methods.");
    params->set ("blkTol", defaultBlkTol, 
                 "Block reorthogonalization threshhold.");
    params->set ("singTol", defaultSingTol, 
                 "Singular block detection threshold.");
    return params;
  }

  template<class ScalarType>
  Teuchos::RCP<const Teuchos::ParameterList> TEUCHOS_DEPRECATED
  getFastIcgsParameters()
  {
    using Teuchos::ParameterList;
    using Teuchos::RCP;
    using Teuchos::rcp;
    using Teuchos::ScalarTraits;
    typedef typename ScalarTraits<ScalarType>::magnitudeType magnitude_type;
    typedef ScalarTraits<magnitude_type> STM;

    RCP<const ParameterList> defaultParams = getDefaultIcgsParameters<ScalarType>();
    // Start with a clone of the default parameters.
    RCP<ParameterList> params = rcp (new ParameterList (*defaultParams));

    const int maxBlkOrtho = 1;
    const magnitude_type blkTol = STM::zero();
    const magnitude_type singTol = STM::zero();

    params->set ("maxNumOrthogPasses", maxBlkOrtho);
    params->set ("blkTol", blkTol);
    params->set ("singTol", singTol);

    return params;
  }

  template<class ScalarType>
  void TEUCHOS_DEPRECATED
  readIcgsParameters (const Teuchos::RCP<const Teuchos::ParameterList>& params,
                      int& maxNumOrthogPasses,
                      typename Teuchos::ScalarTraits<ScalarType>::magnitudeType& blkTol,
                      typename Teuchos::ScalarTraits<ScalarType>::magnitudeType& singTol)
  {
    using Teuchos::ParameterList;
    using Teuchos::RCP;
    typedef typename Teuchos::ScalarTraits<ScalarType>::magnitudeType magnitude_type;
    const magnitude_type zero = Teuchos::ScalarTraits<magnitude_type>::zero();
    RCP<const ParameterList> defaultParams = getDefaultIcgsParameters<ScalarType>();

    // Using temporary variables and fetching all values before
    // setting the output arguments ensures the strong exception
    // guarantee for this function: if an exception is thrown, no
    // externally visible side effects (in this case, setting the
    // output arguments) have taken place.
    int _maxNumOrthogPasses;
    magnitude_type _blkTol, _singTol;
    if (params.is_null()) {
      _maxNumOrthogPasses = defaultParams->get<int> ("maxNumOrthogPasses");
      _blkTol = defaultParams->get<magnitude_type> ("blkTol");
      _singTol = defaultParams->get<magnitude_type> ("singTol");
    } else {
      try {
        _maxNumOrthogPasses = params->get<int> ("maxNumOrthogPasses");
        if (_maxNumOrthogPasses < 1)
          _maxNumOrthogPasses = defaultParams->get<int> ("maxNumOrthogPasses");
      } catch (Teuchos::Exceptions::InvalidParameter&) {
        _maxNumOrthogPasses = defaultParams->get<int> ("maxNumOrthogPasses");
      }

      try {
  _blkTol = params->get<magnitude_type> ("blkTol");
  if (_blkTol < zero)
    _blkTol = defaultParams->get<magnitude_type> ("blkTol");
      } catch (Teuchos::Exceptions::InvalidParameter&) {
  try {
    // People may have used depTol instead of blkTol for this
    // parameter's name, by analogy with DGKS.
    _blkTol = params->get<magnitude_type> ("depTol");
  } catch (Teuchos::Exceptions::InvalidParameter&) {
    _blkTol = defaultParams->get<magnitude_type> ("blkTol");
  }
      }

      try {
        _singTol = params->get<magnitude_type> ("singTol");
        if (_singTol < zero)
          _singTol = defaultParams->get<magnitude_type> ("singTol");
      } catch (Teuchos::Exceptions::InvalidParameter&) {
        _singTol = defaultParams->get<magnitude_type> ("singTol");
      }
    }
    maxNumOrthogPasses = _maxNumOrthogPasses;
    blkTol = _blkTol;
    singTol = _singTol;
  }


  template<class ScalarType, class MV, class OP>
  class ICGSOrthoManager : 
    public MatOrthoManager<ScalarType,MV,OP>, 
    public Teuchos::ParameterListAcceptorDefaultBase
  {
  private:
    typedef typename Teuchos::ScalarTraits<ScalarType>::magnitudeType MagnitudeType;
    typedef typename Teuchos::ScalarTraits<MagnitudeType> MGT;
    typedef Teuchos::ScalarTraits<ScalarType>  SCT;
    typedef MultiVecTraits<ScalarType,MV>      MVT;
    typedef OperatorTraits<ScalarType,MV,OP>   OPT;

  public:


    ICGSOrthoManager( const std::string& label = "Belos",
                      Teuchos::RCP<const OP> Op = Teuchos::null,
                      const int max_ortho_steps = 2,
                      const MagnitudeType blk_tol = 10*MGT::squareroot( MGT::eps() ),
                      const MagnitudeType sing_tol = 10*MGT::eps() )
      : MatOrthoManager<ScalarType,MV,OP>(Op), 
      max_ortho_steps_( max_ortho_steps ),
      blk_tol_( blk_tol ),
      sing_tol_( sing_tol ),
      label_( label )
    {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
      std::string orthoLabel = label_ + ": Orthogonalization";
      timerOrtho_ = Teuchos::TimeMonitor::getNewTimer(orthoLabel);

      std::string updateLabel = label_ + ": Ortho (Update)";
      timerUpdate_ = Teuchos::TimeMonitor::getNewTimer(updateLabel);

      std::string normLabel = label_ + ": Ortho (Norm)";
      timerNorm_ = Teuchos::TimeMonitor::getNewTimer(normLabel);

      std::string ipLabel = label_ + ": Ortho (Inner Product)";
      timerInnerProd_ = Teuchos::TimeMonitor::getNewTimer(ipLabel); 
#endif
    }

    ICGSOrthoManager (const Teuchos::RCP<Teuchos::ParameterList>& plist,
          const std::string& label = "Belos",
                      Teuchos::RCP<const OP> Op = Teuchos::null) :
      MatOrthoManager<ScalarType,MV,OP>(Op), 
      max_ortho_steps_ (2),
      blk_tol_ (10 * MGT::squareroot (MGT::eps())),
      sing_tol_ (10 * MGT::eps()),
      label_ (label)
    {
      setParameterList (plist);

#ifdef BELOS_TEUCHOS_TIME_MONITOR
      std::string orthoLabel = label_ + ": Orthogonalization";
      timerOrtho_ = Teuchos::TimeMonitor::getNewTimer(orthoLabel);

      std::string updateLabel = label_ + ": Ortho (Update)";
      timerUpdate_ = Teuchos::TimeMonitor::getNewTimer(updateLabel);

      std::string normLabel = label_ + ": Ortho (Norm)";
      timerNorm_ = Teuchos::TimeMonitor::getNewTimer(normLabel);

      std::string ipLabel = label_ + ": Ortho (Inner Product)";
      timerInnerProd_ = Teuchos::TimeMonitor::getNewTimer(ipLabel); 
#endif
    }

    ~ICGSOrthoManager() {}



    void 
    setParameterList (const Teuchos::RCP<Teuchos::ParameterList>& plist)
    {
      using Teuchos::Exceptions::InvalidParameterName;
      using Teuchos::ParameterList;
      using Teuchos::parameterList;
      using Teuchos::RCP;

      RCP<const ParameterList> defaultParams = getValidParameters();
      RCP<ParameterList> params;
      if (plist.is_null()) {
  params = parameterList (*defaultParams);
      } else {
  params = plist;
  // Some users might want to specify "blkTol" as "depTol".  Due
  // to this case, we don't invoke
  // validateParametersAndSetDefaults on params.  Instead, we go
  // through the parameter list one parameter at a time and look
  // for alternatives.
      }
  
      // Using temporary variables and fetching all values before
      // setting the output arguments ensures the strong exception
      // guarantee for this function: if an exception is thrown, no
      // externally visible side effects (in this case, setting the
      // output arguments) have taken place.
      int maxNumOrthogPasses;
      MagnitudeType blkTol;
      MagnitudeType singTol;

      try {
  maxNumOrthogPasses = params->get<int> ("maxNumOrthogPasses");
      } catch (InvalidParameterName&) {
  maxNumOrthogPasses = defaultParams->get<int> ("maxNumOrthogPasses");
  params->set ("maxNumOrthogPasses", maxNumOrthogPasses);
      }

      // Handling of the "blkTol" parameter is a special case.  This
      // is because some users may prefer to call this parameter
      // "depTol" for consistency with DGKS.  However, our default
      // parameter list calls this "blkTol", and we don't want the
      // default list's value to override the user's value.  Thus, we
      // first check the user's parameter list for both names, and
      // only then access the default parameter list.
      try {
  blkTol = params->get<MagnitudeType> ("blkTol");
      } catch (InvalidParameterName&) {
  try {
    blkTol = params->get<MagnitudeType> ("depTol");
    // "depTol" is the wrong name, so remove it and replace with
    // "blkTol".  We'll set "blkTol" below.
    params->remove ("depTol");
  } catch (InvalidParameterName&) {
    blkTol = defaultParams->get<MagnitudeType> ("blkTol");
  }
  params->set ("blkTol", blkTol);
      }

      try {
  singTol = params->get<MagnitudeType> ("singTol");
      } catch (InvalidParameterName&) {
  singTol = defaultParams->get<MagnitudeType> ("singTol");
  params->set ("singTol", singTol);
      }

      max_ortho_steps_ = maxNumOrthogPasses;
      blk_tol_ = blkTol;
      sing_tol_ = singTol;

      setMyParamList (params);
    }

    Teuchos::RCP<const Teuchos::ParameterList> 
    getValidParameters () const
    {
      using Teuchos::as;
      using Teuchos::ParameterList;
      using Teuchos::parameterList;
      using Teuchos::RCP;

      if (defaultParams_.is_null()) {
  RCP<ParameterList> params = parameterList ("ICGS");

  // Default parameter values for ICGS orthogonalization.
  // Documentation will be embedded in the parameter list.
  const int defaultMaxNumOrthogPasses = 2;
  const MagnitudeType eps = MGT::eps();
  const MagnitudeType defaultBlkTol = 
    as<MagnitudeType> (10) * MGT::squareroot (eps);
  const MagnitudeType defaultSingTol = as<MagnitudeType> (10) * eps;

  params->set ("maxNumOrthogPasses", defaultMaxNumOrthogPasses,
         "Maximum number of orthogonalization passes (includes the "
         "first).  Default is 2, since \"twice is enough\" for Krylov "
         "methods.");
  params->set ("blkTol", defaultBlkTol, "Block reorthogonalization "
         "threshhold.");
  params->set ("singTol", defaultSingTol, "Singular block detection "
         "threshold.");
  defaultParams_ = params;
      }
      return defaultParams_;
    }

    Teuchos::RCP<const Teuchos::ParameterList> 
    getFastParameters () const 
    {
      using Teuchos::as;
      using Teuchos::ParameterList;
      using Teuchos::parameterList;
      using Teuchos::RCP;

      RCP<const ParameterList> defaultParams = getValidParameters ();
      // Start with a clone of the default parameters.
      RCP<ParameterList> params = parameterList (*defaultParams);

      const int maxBlkOrtho = 1; // No block reorthogonalization
      const MagnitudeType blkTol = MGT::zero();
      const MagnitudeType singTol = MGT::zero();

      params->set ("maxNumOrthogPasses", maxBlkOrtho);
      params->set ("blkTol", blkTol);
      params->set ("singTol", singTol);

      return params;
    }



    void setBlkTol( const MagnitudeType blk_tol ) { 
      // Update the parameter list as well.
      Teuchos::RCP<Teuchos::ParameterList> params = getNonconstParameterList();
      if (! params.is_null()) {
  // If it's null, then we haven't called setParameterList()
  // yet.  It's entirely possible to construct the parameter
  // list on demand, so we don't try to create the parameter
  // list here.
  params->set ("blkTol", blk_tol);
      }
      blk_tol_ = blk_tol; 
    }

    void setSingTol( const MagnitudeType sing_tol ) { 
      // Update the parameter list as well.
      Teuchos::RCP<Teuchos::ParameterList> params = getNonconstParameterList();
      if (! params.is_null()) {
  // If it's null, then we haven't called setParameterList()
  // yet.  It's entirely possible to construct the parameter
  // list on demand, so we don't try to create the parameter
  // list here.
  params->set ("singTol", sing_tol);
      }
      sing_tol_ = sing_tol; 
    }

    MagnitudeType getBlkTol() const { return blk_tol_; } 

    MagnitudeType getSingTol() const { return sing_tol_; } 





    void project ( MV &X, Teuchos::RCP<MV> MX, 
                   Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                   Teuchos::ArrayView<Teuchos::RCP<const MV> > Q) const;


    void project ( MV &X, 
                   Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                   Teuchos::ArrayView<Teuchos::RCP<const MV> > Q) const {
      project(X,Teuchos::null,C,Q);
    }


 
    int normalize ( MV &X, Teuchos::RCP<MV> MX, 
                    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > B) const;


    int normalize ( MV &X, Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > B ) const {
      return normalize(X,Teuchos::null,B);
    }


    int projectAndNormalize ( MV &X, Teuchos::RCP<MV> MX, 
                              Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                              Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > B, 
                              Teuchos::ArrayView<Teuchos::RCP<const MV> > Q) const;

    int projectAndNormalize ( MV &X, 
                              Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                              Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > B, 
                              Teuchos::ArrayView<Teuchos::RCP<const MV> > Q ) const {
      return projectAndNormalize(X,Teuchos::null,C,B,Q);
    }




    typename Teuchos::ScalarTraits<ScalarType>::magnitudeType 
    orthonormError(const MV &X) const {
      return orthonormError(X,Teuchos::null);
    }

    typename Teuchos::ScalarTraits<ScalarType>::magnitudeType 
    orthonormError(const MV &X, Teuchos::RCP<const MV> MX) const;

    typename Teuchos::ScalarTraits<ScalarType>::magnitudeType 
    orthogError(const MV &X1, const MV &X2) const {
      return orthogError(X1,Teuchos::null,X2);
    }

    typename Teuchos::ScalarTraits<ScalarType>::magnitudeType 
    orthogError(const MV &X1, Teuchos::RCP<const MV> MX1, const MV &X2) const;




    void setLabel(const std::string& label);

    const std::string& getLabel() const { return label_; }


  private:
    int max_ortho_steps_;
    MagnitudeType blk_tol_;
    MagnitudeType sing_tol_;

    std::string label_;
#ifdef BELOS_TEUCHOS_TIME_MONITOR
    Teuchos::RCP<Teuchos::Time> timerOrtho_, timerUpdate_, 
      timerNorm_, timerScale_, timerInnerProd_;
#endif // BELOS_TEUCHOS_TIME_MONITOR

    mutable Teuchos::RCP<Teuchos::ParameterList> defaultParams_;
  
    int findBasis(MV &X, Teuchos::RCP<MV> MX, 
                  Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > C, 
                  bool completeBasis, int howMany = -1 ) const;
    
    bool blkOrtho1 ( MV &X, Teuchos::RCP<MV> MX, 
                     Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                     Teuchos::ArrayView<Teuchos::RCP<const MV> > Q) const;

    bool blkOrtho ( MV &X, Teuchos::RCP<MV> MX, 
                    Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                    Teuchos::ArrayView<Teuchos::RCP<const MV> > Q) const;

    int blkOrthoSing ( MV &X, Teuchos::RCP<MV> MX, 
                       Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                       Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > B, 
                       Teuchos::ArrayView<Teuchos::RCP<const MV> > QQ) const;    
  };

  // Set the label for this orthogonalization manager and create new timers if it's changed
  template<class ScalarType, class MV, class OP>
  void ICGSOrthoManager<ScalarType,MV,OP>::setLabel(const std::string& label)
  { 
    if (label != label_) {
      label_ = label;
      std::string orthoLabel = label_ + ": Orthogonalization";
#ifdef BELOS_TEUCHOS_TIME_MONITOR
      timerOrtho_ = Teuchos::TimeMonitor::getNewTimer(orthoLabel);
#endif

      std::string updateLabel = label_ + ": Ortho (Update)";
#ifdef BELOS_TEUCHOS_TIME_MONITOR
      timerUpdate_ = Teuchos::TimeMonitor::getNewTimer(updateLabel);
#endif

      std::string normLabel = label_ + ": Ortho (Norm)";
#ifdef BELOS_TEUCHOS_TIME_MONITOR
      timerNorm_ = Teuchos::TimeMonitor::getNewTimer(normLabel);
#endif

      std::string ipLabel = label_ + ": Ortho (Inner Product)";
#ifdef BELOS_TEUCHOS_TIME_MONITOR
      timerInnerProd_ = Teuchos::TimeMonitor::getNewTimer(ipLabel);
#endif
    }
  } 

  // Compute the distance from orthonormality
  template<class ScalarType, class MV, class OP>
  typename Teuchos::ScalarTraits<ScalarType>::magnitudeType 
  ICGSOrthoManager<ScalarType,MV,OP>::orthonormError(const MV &X, Teuchos::RCP<const MV> MX) const {
    const ScalarType ONE = SCT::one();
    int rank = MVT::GetNumberVecs(X);
    Teuchos::SerialDenseMatrix<int,ScalarType> xTx(rank,rank);
    MatOrthoManager<ScalarType,MV,OP>::innerProd(X,X,MX,xTx);
    for (int i=0; i<rank; i++) {
      xTx(i,i) -= ONE;
    }
    return xTx.normFrobenius();
  }

  // Compute the distance from orthogonality
  template<class ScalarType, class MV, class OP>
  typename Teuchos::ScalarTraits<ScalarType>::magnitudeType 
  ICGSOrthoManager<ScalarType,MV,OP>::orthogError(const MV &X1, Teuchos::RCP<const MV> MX1, const MV &X2) const {
    int r1 = MVT::GetNumberVecs(X1);
    int r2  = MVT::GetNumberVecs(X2);
    Teuchos::SerialDenseMatrix<int,ScalarType> xTx(r2,r1);
    MatOrthoManager<ScalarType,MV,OP>::innerProd(X2,X1,MX1,xTx);
    return xTx.normFrobenius();
  }

  // Find an Op-orthonormal basis for span(X) - span(W)
  template<class ScalarType, class MV, class OP>
  int ICGSOrthoManager<ScalarType, MV, OP>::projectAndNormalize(
                                    MV &X, Teuchos::RCP<MV> MX, 
                                    Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                                    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > B, 
                                    Teuchos::ArrayView<Teuchos::RCP<const MV> > Q ) const 
  {
    using Teuchos::Array;
    using Teuchos::null;
    using Teuchos::is_null;
    using Teuchos::RCP;
    using Teuchos::rcp;
    using Teuchos::SerialDenseMatrix;
    typedef SerialDenseMatrix< int, ScalarType > serial_dense_matrix_type;
    typedef typename Array< RCP< const MV > >::size_type size_type;

#ifdef BELOS_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor orthotimer(*timerOrtho_);
#endif

    ScalarType    ONE  = SCT::one();
    ScalarType    ZERO  = SCT::zero();

    int nq = Q.size();
    int xc = MVT::GetNumberVecs( X );
    int xr = MVT::GetVecLength( X );
    int rank = xc;

    // If the user doesn't want to store the normalization
    // coefficients, allocate some local memory for them.  This will
    // go away at the end of this method.
    if (is_null (B)) {
      B = rcp (new serial_dense_matrix_type (xc, xc));
    }
    // Likewise, if the user doesn't want to store the projection
    // coefficients, allocate some local memory for them.  Also make
    // sure that all the entries of C are the right size.  We're going
    // to overwrite them anyway, so we don't have to worry about the
    // contents (other than to resize them if they are the wrong
    // size).
    if (C.size() < nq)
      C.resize (nq);
    for (size_type k = 0; k < nq; ++k)
      {
        const int numRows = MVT::GetNumberVecs (*Q[k]);
        const int numCols = xc; // Number of vectors in X

        if (is_null (C[k]))
          C[k] = rcp (new serial_dense_matrix_type (numRows, numCols));
        else if (C[k]->numRows() != numRows || C[k]->numCols() != numCols)
        {
          int err = C[k]->reshape (numRows, numCols);
          TEUCHOS_TEST_FOR_EXCEPTION(err != 0, std::runtime_error, 
              "IMGS orthogonalization: failed to reshape "
              "C[" << k << "] (the array of block "
              "coefficients resulting from projecting X "
              "against Q[1:" << nq << "]).");
        }
      }

    /******   DO NOT MODIFY *MX IF _hasOp == false   ******/
    if (this->_hasOp) {
      if (MX == Teuchos::null) {
        // we need to allocate space for MX
        MX = MVT::Clone(X,MVT::GetNumberVecs(X));
        OPT::Apply(*(this->_Op),X,*MX);
      }
    }
    else {
      // Op == I  -->  MX = X (ignore it if the user passed it in)
      MX = Teuchos::rcp( &X, false );
    }

    int mxc = MVT::GetNumberVecs( *MX );
    int mxr = MVT::GetVecLength( *MX );

    // short-circuit
    TEUCHOS_TEST_FOR_EXCEPTION( xc == 0 || xr == 0, std::invalid_argument, "Belos::ICGSOrthoManager::projectAndNormalize(): X must be non-empty" );

    int numbas = 0;
    for (int i=0; i<nq; i++) {
      numbas += MVT::GetNumberVecs( *Q[i] );
    }

    // check size of B
    TEUCHOS_TEST_FOR_EXCEPTION( B->numRows() != xc || B->numCols() != xc, std::invalid_argument, 
                        "Belos::ICGSOrthoManager::projectAndNormalize(): Size of X must be consistant with size of B" );
    // check size of X and MX
    TEUCHOS_TEST_FOR_EXCEPTION( xc<0 || xr<0 || mxc<0 || mxr<0, std::invalid_argument, 
                        "Belos::ICGSOrthoManager::projectAndNormalize(): MVT returned negative dimensions for X,MX" );
    // check size of X w.r.t. MX 
    TEUCHOS_TEST_FOR_EXCEPTION( xc!=mxc || xr!=mxr, std::invalid_argument, 
                        "Belos::ICGSOrthoManager::projectAndNormalize(): Size of X must be consistant with size of MX" );
    // check feasibility
    //TEUCHOS_TEST_FOR_EXCEPTION( numbas+xc > xr, std::invalid_argument, 
    //                    "Belos::ICGSOrthoManager::projectAndNormalize(): Orthogonality constraints not feasible" );

    // Some flags for checking dependency returns from the internal orthogonalization methods
    bool dep_flg = false;

    // Make a temporary copy of X and MX, just in case a block dependency is detected.
    Teuchos::RCP<MV> tmpX, tmpMX;
    tmpX = MVT::CloneCopy(X);
    if (this->_hasOp) {
      tmpMX = MVT::CloneCopy(*MX);
    }

    if (xc == 1) {

      // Use the cheaper block orthogonalization.
      // NOTE: Don't check for dependencies because the update has one vector.
      dep_flg = blkOrtho1( X, MX, C, Q );

      // Normalize the new block X
      {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor normTimer( *timerNorm_ );
#endif
        if ( B == Teuchos::null ) {
          B = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(xc,xc) );
        }
        std::vector<ScalarType> diag(xc);
        MVT::MvDot( X, *MX, diag );
        (*B)(0,0) = SCT::squareroot(SCT::magnitude(diag[0]));
        rank = 1; 
        MVT::MvAddMv( ONE/(*B)(0,0), X, ZERO, X, X );
        if (this->_hasOp) {
          // Update MXj.
          MVT::MvAddMv( ONE/(*B)(0,0), *MX, ZERO, *MX, *MX );
        }
      }

    } 
    else {

      // Use the cheaper block orthogonalization.
      dep_flg = blkOrtho( X, MX, C, Q );

      // If a dependency has been detected in this block, then perform
      // the more expensive nonblock (single vector at a time)
      // orthogonalization.
      if (dep_flg) {
        rank = blkOrthoSing( *tmpX, tmpMX, C, B, Q );

        // Copy tmpX back into X.
        MVT::MvAddMv( ONE, *tmpX, ZERO, *tmpX, X );
        if (this->_hasOp) {
          MVT::MvAddMv( ONE, *tmpMX, ZERO, *tmpMX, *MX );
        }
      } 
      else {
        // There is no dependency, so orthonormalize new block X
        rank = findBasis( X, MX, B, false );
        if (rank < xc) {
          // A dependency was found during orthonormalization of X,
          // rerun orthogonalization using more expensive nonblock
          // orthogonalization.
          rank = blkOrthoSing( *tmpX, tmpMX, C, B, Q );

          // Copy tmpX back into X.
          MVT::MvAddMv( ONE, *tmpX, ZERO, *tmpX, X );
          if (this->_hasOp) {
            MVT::MvAddMv( ONE, *tmpMX, ZERO, *tmpMX, *MX );
          }
        }    
      }
    } // if (xc == 1) {

    // this should not raise an std::exception; but our post-conditions oblige us to check
    TEUCHOS_TEST_FOR_EXCEPTION( rank > xc || rank < 0, std::logic_error, 
                        "Belos::ICGSOrthoManager::projectAndNormalize(): Debug error in rank variable." );

    // Return the rank of X.
    return rank;
  }



  // Find an Op-orthonormal basis for span(X), with rank numvectors(X)
  template<class ScalarType, class MV, class OP>
  int ICGSOrthoManager<ScalarType, MV, OP>::normalize(
                                MV &X, Teuchos::RCP<MV> MX, 
                                Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > B ) const {

#ifdef BELOS_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor orthotimer(*timerOrtho_);
#endif

    // call findBasis, with the instruction to try to generate a basis of rank numvecs(X)
    return findBasis(X, MX, B, true);
  }



  template<class ScalarType, class MV, class OP>
  void ICGSOrthoManager<ScalarType, MV, OP>::project(
                          MV &X, Teuchos::RCP<MV> MX, 
                          Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                          Teuchos::ArrayView<Teuchos::RCP<const MV> > Q) const {
    // For the inner product defined by the operator Op or the identity (Op == 0)
    //   -> Orthogonalize X against each Q[i]
    // Modify MX accordingly
    //
    // Note that when Op is 0, MX is not referenced
    //
    // Parameter variables
    //
    // X  : Vectors to be transformed
    //
    // MX : Image of the block of vectors X by the mass matrix
    //
    // Q  : Bases to orthogonalize against. These are assumed orthonormal, mutually and independently.
    //
    
#ifdef BELOS_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor orthotimer(*timerOrtho_);
#endif
    
    int xc = MVT::GetNumberVecs( X );
    int xr = MVT::GetVecLength( X );
    int nq = Q.size();
    std::vector<int> qcs(nq);
    // short-circuit
    if (nq == 0 || xc == 0 || xr == 0) {
      return;
    }
    int qr = MVT::GetVecLength ( *Q[0] );
    // if we don't have enough C, expand it with null references
    // if we have too many, resize to throw away the latter ones
    // if we have exactly as many as we have Q, this call has no effect
    C.resize(nq);


    /******   DO NOT MODIFY *MX IF _hasOp == false   ******/
    if (this->_hasOp) {
      if (MX == Teuchos::null) {
        // we need to allocate space for MX
        MX = MVT::Clone(X,MVT::GetNumberVecs(X));
        OPT::Apply(*(this->_Op),X,*MX);
      }
    }
    else {
      // Op == I  -->  MX = X (ignore it if the user passed it in)
      MX = Teuchos::rcp( &X, false );
    }
    int mxc = MVT::GetNumberVecs( *MX );
    int mxr = MVT::GetVecLength( *MX );

    // check size of X and Q w.r.t. common sense
    TEUCHOS_TEST_FOR_EXCEPTION( xc<0 || xr<0 || mxc<0 || mxr<0, std::invalid_argument, 
                        "Belos::ICGSOrthoManager::project(): MVT returned negative dimensions for X,MX" );
    // check size of X w.r.t. MX and Q
    TEUCHOS_TEST_FOR_EXCEPTION( xc!=mxc || xr!=mxr || xr!=qr, std::invalid_argument, 
                        "Belos::ICGSOrthoManager::project(): Size of X not consistant with MX,Q" );

    // tally up size of all Q and check/allocate C
    int baslen = 0;
    for (int i=0; i<nq; i++) {
      TEUCHOS_TEST_FOR_EXCEPTION( MVT::GetVecLength( *Q[i] ) != qr, std::invalid_argument, 
                          "Belos::ICGSOrthoManager::project(): Q lengths not mutually consistant" );
      qcs[i] = MVT::GetNumberVecs( *Q[i] );
      TEUCHOS_TEST_FOR_EXCEPTION( qr < qcs[i], std::invalid_argument, 
                          "Belos::ICGSOrthoManager::project(): Q has less rows than columns" );
      baslen += qcs[i];

      // check size of C[i]
      if ( C[i] == Teuchos::null ) {
        C[i] = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(qcs[i],xc) );
      }
      else {
        TEUCHOS_TEST_FOR_EXCEPTION( C[i]->numRows() != qcs[i] || C[i]->numCols() != xc , std::invalid_argument, 
                           "Belos::ICGSOrthoManager::project(): Size of Q not consistant with size of C" );
      }
    }

    // Use the cheaper block orthogonalization, don't check for rank deficiency.
    blkOrtho( X, MX, C, Q );

  }  

  // Find an Op-orthonormal basis for span(X), with the option of extending the subspace so that 
  // the rank is numvectors(X)
  template<class ScalarType, class MV, class OP>
  int ICGSOrthoManager<ScalarType, MV, OP>::findBasis(
                MV &X, Teuchos::RCP<MV> MX, 
                Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > B,
                bool completeBasis, int howMany ) const {
    // For the inner product defined by the operator Op or the identity (Op == 0)
    //   -> Orthonormalize X 
    // Modify MX accordingly
    //
    // Note that when Op is 0, MX is not referenced
    //
    // Parameter variables
    //
    // X  : Vectors to be orthonormalized
    //
    // MX : Image of the multivector X under the operator Op
    //
    // Op  : Pointer to the operator for the inner product
    //
    //

#ifdef BELOS_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor normTimer( *timerNorm_ );
#endif

    const ScalarType ONE  = SCT::one();
    const MagnitudeType ZERO = SCT::magnitude(SCT::zero());

    int xc = MVT::GetNumberVecs( X );
    int xr = MVT::GetVecLength( X );

    if (howMany == -1) {
      howMany = xc;
    }

    /*******************************************************
     *  If _hasOp == false, we will not reference MX below *
     *******************************************************/

    // if Op==null, MX == X (via pointer)
    // Otherwise, either the user passed in MX or we will allocated and compute it
    if (this->_hasOp) {
      if (MX == Teuchos::null) {
        // we need to allocate space for MX
        MX = MVT::Clone(X,xc);
        OPT::Apply(*(this->_Op),X,*MX);
      }
    }

    /* if the user doesn't want to store the coefficienets, 
     * allocate some local memory for them 
     */
    if ( B == Teuchos::null ) {
      B = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(xc,xc) );
    }

    int mxc = (this->_hasOp) ? MVT::GetNumberVecs( *MX ) : xc;
    int mxr = (this->_hasOp) ? MVT::GetVecLength( *MX )  : xr;

    // check size of C, B
    TEUCHOS_TEST_FOR_EXCEPTION( xc == 0 || xr == 0, std::invalid_argument, 
                        "Belos::ICGSOrthoManager::findBasis(): X must be non-empty" );
    TEUCHOS_TEST_FOR_EXCEPTION( B->numRows() != xc || B->numCols() != xc, std::invalid_argument, 
                        "Belos::ICGSOrthoManager::findBasis(): Size of X not consistant with size of B" );
    TEUCHOS_TEST_FOR_EXCEPTION( xc != mxc || xr != mxr, std::invalid_argument, 
                        "Belos::ICGSOrthoManager::findBasis(): Size of X not consistant with size of MX" );
    TEUCHOS_TEST_FOR_EXCEPTION( xc > xr, std::invalid_argument, 
                        "Belos::ICGSOrthoManager::findBasis(): Size of X not feasible for normalization" );
    TEUCHOS_TEST_FOR_EXCEPTION( howMany < 0 || howMany > xc, std::invalid_argument, 
                        "Belos::ICGSOrthoManager::findBasis(): Invalid howMany parameter" );

    /* xstart is which column we are starting the process with, based on howMany
     * columns before xstart are assumed to be Op-orthonormal already
     */
    int xstart = xc - howMany;

    for (int j = xstart; j < xc; j++) {

      // numX is 
      // * number of currently orthonormal columns of X
      // * the index of the current column of X
      int numX = j;
      bool addVec = false;

      // Get a view of the vector currently being worked on.
      std::vector<int> index(1);
      index[0] = numX;
      Teuchos::RCP<MV> Xj = MVT::CloneViewNonConst( X, index );
      Teuchos::RCP<MV> MXj;
      if ((this->_hasOp)) {
        // MXj is a view of the current vector in MX
        MXj = MVT::CloneViewNonConst( *MX, index );
      }
      else {
        // MXj is a pointer to Xj, and MUST NOT be modified
        MXj = Xj;
      }

      // Get a view of the previous vectors.
      std::vector<int> prev_idx( numX );
      Teuchos::RCP<const MV> prevX, prevMX;

      if (numX > 0) {
        for (int i=0; i<numX; i++) {
          prev_idx[i] = i;
        }
        prevX = MVT::CloneView( X, prev_idx );
        if (this->_hasOp) {
          prevMX = MVT::CloneView( *MX, prev_idx );
        }
      } 

      // Make storage for these Gram-Schmidt iterations.
      Teuchos::SerialDenseMatrix<int,ScalarType> product(numX, 1);
      std::vector<ScalarType> oldDot( 1 ), newDot( 1 );
      //
      // Save old MXj vector and compute Op-norm
      //
      Teuchos::RCP<MV> oldMXj = MVT::CloneCopy( *MXj ); 
      MVT::MvDot( *Xj, *MXj, oldDot );
      // Xj^H Op Xj should be real and positive, by the hermitian positive definiteness of Op
      TEUCHOS_TEST_FOR_EXCEPTION( SCT::real(oldDot[0]) < ZERO, OrthoError, 
          "Belos::ICGSOrthoManager::findBasis(): Negative definiteness discovered in inner product" );

      if (numX > 0) {

        Teuchos::SerialDenseMatrix<int,ScalarType> P2(numX,1);

        for (int i=0; i<max_ortho_steps_; ++i) {

          // product <- prevX^T MXj
          {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
            Teuchos::TimeMonitor innerProdTimer( *timerInnerProd_ );
#endif
            MatOrthoManager<ScalarType,MV,OP>::innerProd(*prevX,*Xj,MXj,P2);
          }

          // Xj <- Xj - prevX prevX^T MXj   
          //     = Xj - prevX product
          {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
            Teuchos::TimeMonitor updateTimer( *timerUpdate_ );
#endif
            MVT::MvTimesMatAddMv( -ONE, *prevX, P2, ONE, *Xj );
          }

          // Update MXj
          if (this->_hasOp) {
            // MXj <- Op*Xj_new
            //      = Op*(Xj_old - prevX prevX^T MXj)
            //      = MXj - prevMX product
#ifdef BELOS_TEUCHOS_TIME_MONITOR
            Teuchos::TimeMonitor updateTimer( *timerUpdate_ );
#endif
            MVT::MvTimesMatAddMv( -ONE, *prevMX, P2, ONE, *MXj );
          }

          // Set coefficients
          if ( i==0 )
            product = P2;
          else
            product += P2;
        }

      } // if (numX > 0)

      // Compute Op-norm with old MXj
      MVT::MvDot( *Xj, *oldMXj, newDot );

      // Check to see if the new vector is dependent.
      if (completeBasis) {
        //
        // We need a complete basis, so add random vectors if necessary
        //
        if ( SCT::magnitude(newDot[0]) < SCT::magnitude(sing_tol_*oldDot[0]) ) {

          // Add a random vector and orthogonalize it against previous vectors in block.
          addVec = true;
#ifdef ORTHO_DEBUG
          std::cout << "Belos::ICGSOrthoManager::findBasis() --> Random for column " << numX << std::endl;
#endif
          //
          Teuchos::RCP<MV> tempXj = MVT::Clone( X, 1 );
          Teuchos::RCP<MV> tempMXj;
          MVT::MvRandom( *tempXj );
          if (this->_hasOp) {
            tempMXj = MVT::Clone( X, 1 );
            OPT::Apply( *(this->_Op), *tempXj, *tempMXj );
          } 
          else {
            tempMXj = tempXj;
          }
          MVT::MvDot( *tempXj, *tempMXj, oldDot );
          //
          for (int num_orth=0; num_orth<max_ortho_steps_; num_orth++){
            {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
              Teuchos::TimeMonitor innerProdTimer( *timerInnerProd_ );
#endif
              MatOrthoManager<ScalarType,MV,OP>::innerProd(*prevX,*tempXj,tempMXj,product);
            }
            {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
              Teuchos::TimeMonitor updateTimer( *timerUpdate_ );
#endif
              MVT::MvTimesMatAddMv( -ONE, *prevX, product, ONE, *tempXj );
            }
            if (this->_hasOp) {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
              Teuchos::TimeMonitor updateTimer( *timerUpdate_ );
#endif
              MVT::MvTimesMatAddMv( -ONE, *prevMX, product, ONE, *tempMXj );
            }
          }
          // Compute new Op-norm
          MVT::MvDot( *tempXj, *tempMXj, newDot );
          //
          if ( SCT::magnitude(newDot[0]) >= SCT::magnitude(oldDot[0]*sing_tol_) ) {
            // Copy vector into current column of _basisvecs
            MVT::MvAddMv( ONE, *tempXj, ZERO, *tempXj, *Xj );
            if (this->_hasOp) {
              MVT::MvAddMv( ONE, *tempMXj, ZERO, *tempMXj, *MXj );
            }
          }
          else {
            return numX;
          } 
        }
      }
      else {
        //
        // We only need to detect dependencies.
        //
        if ( SCT::magnitude(newDot[0]) < SCT::magnitude(oldDot[0]*blk_tol_) ) {
          return numX;
        }
      }

      // If we haven't left this method yet, then we can normalize the new vector Xj.
      // Normalize Xj.
      // Xj <- Xj / std::sqrt(newDot)
      ScalarType diag = SCT::squareroot(SCT::magnitude(newDot[0]));
      {
        MVT::MvAddMv( ONE/diag, *Xj, ZERO, *Xj, *Xj );
        if (this->_hasOp) {
          // Update MXj.
          MVT::MvAddMv( ONE/diag, *MXj, ZERO, *MXj, *MXj );
        }
      }

      // If we've added a random vector, enter a zero in the j'th diagonal element.
      if (addVec) {
        (*B)(j,j) = ZERO;
      }
      else {
        (*B)(j,j) = diag;
      }

      // Save the coefficients, if we are working on the original vector and not a randomly generated one
      if (!addVec) {
        for (int i=0; i<numX; i++) {
          (*B)(i,j) = product(i,0);
        }
      }

    } // for (j = 0; j < xc; ++j)

    return xc;
  }

  // Routine to compute the block orthogonalization
  template<class ScalarType, class MV, class OP>
  bool 
  ICGSOrthoManager<ScalarType, MV, OP>::blkOrtho1 ( MV &X, Teuchos::RCP<MV> MX, 
                                                    Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                                                    Teuchos::ArrayView<Teuchos::RCP<const MV> > Q) const
  {
    int nq = Q.size();
    int xc = MVT::GetNumberVecs( X );
    const ScalarType ONE  = SCT::one();

    std::vector<int> qcs( nq );
    for (int i=0; i<nq; i++) {
      qcs[i] = MVT::GetNumberVecs( *Q[i] );
    }

    // Perform the Gram-Schmidt transformation for a block of vectors

    Teuchos::Array<Teuchos::RCP<MV> > MQ(nq);
    // Define the product Q^T * (Op*X)
    for (int i=0; i<nq; i++) {
      // Multiply Q' with MX
      {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor innerProdTimer( *timerInnerProd_ );
#endif
        MatOrthoManager<ScalarType,MV,OP>::innerProd(*Q[i],X,MX,*C[i]);
      }
      // Multiply by Q and subtract the result in X
      {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor updateTimer( *timerUpdate_ );
#endif
        MVT::MvTimesMatAddMv( -ONE, *Q[i], *C[i], ONE, X );
      }

      // Update MX, with the least number of applications of Op as possible
      if (this->_hasOp) {
        if (xc <= qcs[i]) {
          OPT::Apply( *(this->_Op), X, *MX);
        }
        else {
          // this will possibly be used again below; don't delete it
          MQ[i] = MVT::Clone( *Q[i], qcs[i] );
          OPT::Apply( *(this->_Op), *Q[i], *MQ[i] );
          MVT::MvTimesMatAddMv( -ONE, *MQ[i], *C[i], ONE, *MX );
        }
      }
    }

    // Do as many steps of classical Gram-Schmidt as required by max_ortho_steps_
    for (int j = 1; j < max_ortho_steps_; ++j) {
      
      for (int i=0; i<nq; i++) {
        Teuchos::SerialDenseMatrix<int,ScalarType> C2(*C[i]);

        // Apply another step of classical Gram-Schmidt
        {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
          Teuchos::TimeMonitor innerProdTimer( *timerInnerProd_ );
#endif
          MatOrthoManager<ScalarType,MV,OP>::innerProd(*Q[i],X,MX,C2);
        }
        *C[i] += C2;
        {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
          Teuchos::TimeMonitor updateTimer( *timerUpdate_ );
#endif
          MVT::MvTimesMatAddMv( -ONE, *Q[i], C2, ONE, X );
        }

        // Update MX, with the least number of applications of Op as possible
        if (this->_hasOp) {
          if (MQ[i].get()) {
            // MQ was allocated and computed above; use it
            MVT::MvTimesMatAddMv( -ONE, *MQ[i], C2, ONE, *MX );
          }
          else if (xc <= qcs[i]) {
            // MQ was not allocated and computed above; it was cheaper to use X before and it still is
            OPT::Apply( *(this->_Op), X, *MX);
          }
        }
      } // for (int i=0; i<nq; i++)
    } // for (int j = 0; j < max_ortho_steps; ++j)

    return false;
  }

  // Routine to compute the block orthogonalization
  template<class ScalarType, class MV, class OP>
  bool 
  ICGSOrthoManager<ScalarType, MV, OP>::blkOrtho ( MV &X, Teuchos::RCP<MV> MX, 
                                                   Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                                                   Teuchos::ArrayView<Teuchos::RCP<const MV> > Q) const
  {
    int nq = Q.size();
    int xc = MVT::GetNumberVecs( X );
    bool dep_flg = false;
    const ScalarType ONE  = SCT::one();

    std::vector<int> qcs( nq );
    for (int i=0; i<nq; i++) {
      qcs[i] = MVT::GetNumberVecs( *Q[i] );
    }

    // Perform the Gram-Schmidt transformation for a block of vectors
    
    // Compute the initial Op-norms
    std::vector<ScalarType> oldDot( xc );
    MVT::MvDot( X, *MX, oldDot );

    Teuchos::Array<Teuchos::RCP<MV> > MQ(nq);
    // Define the product Q^T * (Op*X)
    for (int i=0; i<nq; i++) {
      // Multiply Q' with MX
      {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor innerProdTimer( *timerInnerProd_ );
#endif
        MatOrthoManager<ScalarType,MV,OP>::innerProd(*Q[i],X,MX,*C[i]);
      }
      // Multiply by Q and subtract the result in X
      {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
        Teuchos::TimeMonitor updateTimer( *timerUpdate_ );
#endif
        MVT::MvTimesMatAddMv( -ONE, *Q[i], *C[i], ONE, X );
      }
      // Update MX, with the least number of applications of Op as possible
      if (this->_hasOp) {
        if (xc <= qcs[i]) {
          OPT::Apply( *(this->_Op), X, *MX);
        }
        else {
          // this will possibly be used again below; don't delete it
          MQ[i] = MVT::Clone( *Q[i], qcs[i] );
          OPT::Apply( *(this->_Op), *Q[i], *MQ[i] );
          MVT::MvTimesMatAddMv( -ONE, *MQ[i], *C[i], ONE, *MX );
        }
      }
    }

    // Do as many steps of classical Gram-Schmidt as required by max_ortho_steps_
    for (int j = 1; j < max_ortho_steps_; ++j) {
      
      for (int i=0; i<nq; i++) {
        Teuchos::SerialDenseMatrix<int,ScalarType> C2(*C[i]);

        // Apply another step of classical Gram-Schmidt
        {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
          Teuchos::TimeMonitor innerProdTimer( *timerInnerProd_ );
#endif
          MatOrthoManager<ScalarType,MV,OP>::innerProd(*Q[i],X,MX,C2);
        }
        *C[i] += C2;
        {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
          Teuchos::TimeMonitor updateTimer( *timerUpdate_ );
#endif
          MVT::MvTimesMatAddMv( -ONE, *Q[i], C2, ONE, X );
        }

        // Update MX, with the least number of applications of Op as possible
        if (this->_hasOp) {
          if (MQ[i].get()) {
            // MQ was allocated and computed above; use it
            MVT::MvTimesMatAddMv( -ONE, *MQ[i], C2, ONE, *MX );
          }
          else if (xc <= qcs[i]) {
            // MQ was not allocated and computed above; it was cheaper to use X before and it still is
            OPT::Apply( *(this->_Op), X, *MX);
          }
        }
      } // for (int i=0; i<nq; i++)
    } // for (int j = 0; j < max_ortho_steps; ++j)

    // Compute new Op-norms
    std::vector<ScalarType> newDot(xc);
    MVT::MvDot( X, *MX, newDot );

    // Check to make sure the new block of vectors are not dependent on previous vectors
    for (int i=0; i<xc; i++){
      if (SCT::magnitude(newDot[i]) < SCT::magnitude(oldDot[i] * blk_tol_)) {
        dep_flg = true;
        break;
      }
    } // end for (i=0;...)

    return dep_flg;
  }
  
  template<class ScalarType, class MV, class OP>
  int
  ICGSOrthoManager<ScalarType, MV, OP>::blkOrthoSing ( MV &X, Teuchos::RCP<MV> MX, 
                                                       Teuchos::Array<Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > > C, 
                                                       Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > B, 
                                                       Teuchos::ArrayView<Teuchos::RCP<const MV> > QQ) const
  {
    Teuchos::Array<Teuchos::RCP<const MV> > Q (QQ);

    const ScalarType ONE  = SCT::one();
    const ScalarType ZERO  = SCT::zero();
    
    int nq = Q.size();
    int xc = MVT::GetNumberVecs( X );
    std::vector<int> indX( 1 );
    std::vector<ScalarType> oldDot( 1 ), newDot( 1 );

    std::vector<int> qcs( nq );
    for (int i=0; i<nq; i++) {
      qcs[i] = MVT::GetNumberVecs( *Q[i] );
    }

    // Create pointers for the previous vectors of X that have already been orthonormalized.
    Teuchos::RCP<const MV> lastQ;
    Teuchos::RCP<MV> Xj, MXj;
    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > lastC;

    // Perform the Gram-Schmidt transformation for each vector in the block of vectors.
    for (int j=0; j<xc; j++) {
      
      bool dep_flg = false;
      
      // Get a view of the previously orthogonalized vectors and B, add it to the arrays.
      if (j > 0) {
        std::vector<int> index( j );
        for (int ind=0; ind<j; ind++) {
          index[ind] = ind;
        }
        lastQ = MVT::CloneView( X, index );

        // Add these views to the Q and C arrays.
        Q.push_back( lastQ );
        C.push_back( B );
        qcs.push_back( MVT::GetNumberVecs( *lastQ ) );
      }

      // Get a view of the current vector in X to orthogonalize.
      indX[0] = j;
      Xj = MVT::CloneViewNonConst( X, indX );
      if (this->_hasOp) {
        MXj = MVT::CloneViewNonConst( *MX, indX );
      }
      else {
        MXj = Xj;
      }

      // Compute the initial Op-norms
      MVT::MvDot( *Xj, *MXj, oldDot );

      Teuchos::Array<Teuchos::RCP<MV> > MQ(Q.size());
      // Define the product Q^T * (Op*X)
      for (int i=0; i<Q.size(); i++) {

        // Get a view of the current serial dense matrix
        Teuchos::SerialDenseMatrix<int,ScalarType> tempC( Teuchos::View, *C[i], qcs[i], 1, 0, j );

        // Multiply Q' with MX
        MatOrthoManager<ScalarType,MV,OP>::innerProd(*Q[i],*Xj,MXj,tempC);
        // Multiply by Q and subtract the result in Xj
        MVT::MvTimesMatAddMv( -ONE, *Q[i], tempC, ONE, *Xj );

        // Update MXj, with the least number of applications of Op as possible
        if (this->_hasOp) {
          if (xc <= qcs[i]) {
            OPT::Apply( *(this->_Op), *Xj, *MXj);
          }
          else {
            // this will possibly be used again below; don't delete it
            MQ[i] = MVT::Clone( *Q[i], qcs[i] );
            OPT::Apply( *(this->_Op), *Q[i], *MQ[i] );
            MVT::MvTimesMatAddMv( -ONE, *MQ[i], tempC, ONE, *MXj );
          }
        }
      }

      // Do any additional steps of classical Gram-Schmidt orthogonalization 
      for (int num_ortho_steps=1; num_ortho_steps < max_ortho_steps_; ++num_ortho_steps) {

        for (int i=0; i<Q.size(); i++) {
          Teuchos::SerialDenseMatrix<int,ScalarType> tempC( Teuchos::View, *C[i], qcs[i], 1, 0, j );
          Teuchos::SerialDenseMatrix<int,ScalarType> C2( qcs[i], 1 );

          // Apply another step of classical Gram-Schmidt
          MatOrthoManager<ScalarType,MV,OP>::innerProd(*Q[i],*Xj,MXj,C2);
          tempC += C2;
          MVT::MvTimesMatAddMv( -ONE, *Q[i], C2, ONE, *Xj );

          // Update MXj, with the least number of applications of Op as possible
          if (this->_hasOp) {
            if (MQ[i].get()) {
              // MQ was allocated and computed above; use it
              MVT::MvTimesMatAddMv( -ONE, *MQ[i], C2, ONE, *MXj );
            }
            else if (xc <= qcs[i]) {
              // MQ was not allocated and computed above; it was cheaper to use X before and it still is
              OPT::Apply( *(this->_Op), *Xj, *MXj);
            }
          }
        } // for (int i=0; i<Q.size(); i++)

      } // for (int num_ortho_steps=1; num_ortho_steps < max_ortho_steps_; ++num_ortho_steps)

      // Check for linear dependence.
      if (SCT::magnitude(newDot[0]) < SCT::magnitude(oldDot[0]*sing_tol_)) {
        dep_flg = true;
      }

      // Normalize the new vector if it's not dependent
      if (!dep_flg) {
        ScalarType diag = SCT::squareroot(SCT::magnitude(newDot[0]));

        MVT::MvAddMv( ONE/diag, *Xj, ZERO, *Xj, *Xj );
        if (this->_hasOp) {
          // Update MXj.
          MVT::MvAddMv( ONE/diag, *MXj, ZERO, *MXj, *MXj );
        }

        // Enter value on diagonal of B.
        (*B)(j,j) = diag;
      }
      else {
        // Create a random vector and orthogonalize it against all previous columns of Q.
        Teuchos::RCP<MV> tempXj = MVT::Clone( X, 1 );
        Teuchos::RCP<MV> tempMXj;
        MVT::MvRandom( *tempXj );
        if (this->_hasOp) {
          tempMXj = MVT::Clone( X, 1 );
          OPT::Apply( *(this->_Op), *tempXj, *tempMXj );
        } 
        else {
          tempMXj = tempXj;
        }
        MVT::MvDot( *tempXj, *tempMXj, oldDot );
        //
        for (int num_orth=0; num_orth<max_ortho_steps_; num_orth++) {

          for (int i=0; i<Q.size(); i++) {
            Teuchos::SerialDenseMatrix<int,ScalarType> product( qcs[i], 1 );

            // Apply another step of classical Gram-Schmidt
            MatOrthoManager<ScalarType,MV,OP>::innerProd(*Q[i],*tempXj,tempMXj,product);
            MVT::MvTimesMatAddMv( -ONE, *Q[i], product, ONE, *tempXj );

            // Update MXj, with the least number of applications of Op as possible
            if (this->_hasOp) {
              if (MQ[i].get()) {
                // MQ was allocated and computed above; use it
                MVT::MvTimesMatAddMv( -ONE, *MQ[i], product, ONE, *tempMXj );
              }
              else if (xc <= qcs[i]) {
                // MQ was not allocated and computed above; it was cheaper to use X before and it still is
                OPT::Apply( *(this->_Op), *tempXj, *tempMXj);
              }
            }
          } // for (int i=0; i<nq; i++)

        }

        // Compute the Op-norms after the correction step.
        MVT::MvDot( *tempXj, *tempMXj, newDot );

        // Copy vector into current column of Xj
        if ( SCT::magnitude(newDot[0]) >= SCT::magnitude(oldDot[0]*sing_tol_) ) {
          ScalarType diag = SCT::squareroot(SCT::magnitude(newDot[0]));

          // Enter value on diagonal of B.
          (*B)(j,j) = ZERO;

          // Copy vector into current column of _basisvecs
          MVT::MvAddMv( ONE/diag, *tempXj, ZERO, *tempXj, *Xj );
          if (this->_hasOp) {
            MVT::MvAddMv( ONE/diag, *tempMXj, ZERO, *tempMXj, *MXj );
          }
        }
        else {
          return j;
        } 
      } // if (!dep_flg)

      // Remove the vectors from array
      if (j > 0) {
        Q.resize( nq );
        C.resize( nq );
        qcs.resize( nq );
      }

    } // for (int j=0; j<xc; j++)

    return xc;
  }

} // namespace Belos

#endif // BELOS_ICGS_ORTHOMANAGER_HPP