BelosGCRODRSolMgr.hpp

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// @HEADER
// *****************************************************************************
//                 Belos: Block Linear Solvers Package
//
// Copyright 2004 NTESS and the Belos contributors.
// SPDX-License-Identifier: BSD-3-Clause
// *****************************************************************************
// @HEADER
 
#ifndef BELOS_GCRODR_SOLMGR_HPP
#define BELOS_GCRODR_SOLMGR_HPP

 
#include "BelosConfigDefs.hpp"
#include "BelosOrthoManagerFactory.hpp"
#include "BelosSolverManager.hpp"
#include "BelosLinearProblem.hpp"
#include "BelosTypes.hpp"
 
#include "BelosGCRODRIter.hpp"
#include "BelosBlockFGmresIter.hpp"
#include "BelosStatusTestMaxIters.hpp"
#include "BelosStatusTestGenResNorm.hpp"
#include "BelosStatusTestCombo.hpp"
#include "BelosStatusTestOutputFactory.hpp"
#include "BelosOutputManager.hpp"
#include "Teuchos_BLAS.hpp" // includes Teuchos_ConfigDefs.hpp
#include "Teuchos_LAPACK.hpp"
#include "Teuchos_as.hpp"
 
#ifdef BELOS_TEUCHOS_TIME_MONITOR
#  include "Teuchos_TimeMonitor.hpp"
#endif // BELOS_TEUCHOS_TIME_MONITOR
 
#if defined(HAVE_TEUCHOSCORE_CXX11)
#  include <type_traits>
#  if defined(HAVE_TEUCHOS_COMPLEX)
#include "Kokkos_Complex.hpp"
#  endif
#endif // defined(HAVE_TEUCHOSCORE_CXX11)

 
namespace Belos {



  class GCRODRSolMgrLinearProblemFailure : public BelosError {
    public:
      GCRODRSolMgrLinearProblemFailure(const std::string& what_arg) : BelosError(what_arg) {}
  };

  class GCRODRSolMgrOrthoFailure : public BelosError {
    public:
      GCRODRSolMgrOrthoFailure(const std::string& what_arg) : BelosError(what_arg) {}
  };

  class GCRODRSolMgrLAPACKFailure : public BelosError {
    public:
      GCRODRSolMgrLAPACKFailure(const std::string& what_arg) : BelosError(what_arg) {}
  };

  class GCRODRSolMgrRecyclingFailure : public BelosError {
    public:
      GCRODRSolMgrRecyclingFailure(const std::string& what_arg) : BelosError(what_arg) {}
  };


 
  template<class ScalarType, class MV, class OP,
           const bool lapackSupportsScalarType =
           Belos::Details::LapackSupportsScalar<ScalarType>::value>
  class GCRODRSolMgr :
    public Details::SolverManagerRequiresLapack<ScalarType,MV,OP>
  {
    static const bool requiresLapack =
      Belos::Details::LapackSupportsScalar<ScalarType>::value;
    typedef Details::SolverManagerRequiresLapack<ScalarType, MV, OP,
                                                 requiresLapack> base_type;
 
  public:
    GCRODRSolMgr () :
      base_type ()
    {}
    GCRODRSolMgr (const Teuchos::RCP<LinearProblem<ScalarType,MV,OP> >& problem,
                  const Teuchos::RCP<Teuchos::ParameterList>& pl) :
      base_type ()
    {}
    virtual ~GCRODRSolMgr () {}
  };

  template<class ScalarType, class MV, class OP>
  class GCRODRSolMgr<ScalarType, MV, OP, true> :
    public Details::SolverManagerRequiresLapack<ScalarType, MV, OP, true>
  {
 
#if defined(HAVE_TEUCHOSCORE_CXX11)
#  if defined(HAVE_TEUCHOS_COMPLEX)
      #if defined(HAVE_TEUCHOS_LONG_DOUBLE) 
        static_assert (std::is_same<ScalarType, std::complex<float> >::value ||
                       std::is_same<ScalarType, std::complex<double> >::value ||
                       std::is_same<ScalarType, Kokkos::complex<double> >::value ||
                       std::is_same<ScalarType, float>::value ||
                       std::is_same<ScalarType, double>::value ||
                       std::is_same<ScalarType, long double>::value,
                       "Belos::GCRODRSolMgr: ScalarType must be one of the four "
                       "types (S,D,C,Z) supported by LAPACK or long double (largely not impl'd).");
       #else
        static_assert (std::is_same<ScalarType, std::complex<float> >::value ||
                       std::is_same<ScalarType, std::complex<double> >::value ||
                       std::is_same<ScalarType, Kokkos::complex<double> >::value ||
                       std::is_same<ScalarType, float>::value ||
                       std::is_same<ScalarType, double>::value,
                       "Belos::GCRODRSolMgr: ScalarType must be one of the four "
                       "types (S,D,C,Z) supported by LAPACK.");
       #endif 
#  else
      #if defined(HAVE_TEUCHOS_LONG_DOUBLE) 
        static_assert (std::is_same<ScalarType, float>::value ||
                       std::is_same<ScalarType, double>::value ||
                       std::is_same<ScalarType, long double>::value,
                       "Belos::GCRODRSolMgr: ScalarType must be float, double or long double.  "
                       "Complex arithmetic support is currently disabled.  To "
                       "enable it, set Teuchos_ENABLE_COMPLEX=ON.");
      #else
        static_assert (std::is_same<ScalarType, float>::value ||
                       std::is_same<ScalarType, double>::value,
                       "Belos::GCRODRSolMgr: ScalarType must be float or double.  "
                       "Complex arithmetic support is currently disabled.  To "
                       "enable it, set Teuchos_ENABLE_COMPLEX=ON.");
      #endif  
#  endif // defined(HAVE_TEUCHOS_COMPLEX)
#endif // defined(HAVE_TEUCHOSCORE_CXX11)
 
  private:
    typedef MultiVecTraits<ScalarType,MV> MVT;
    typedef OperatorTraits<ScalarType,MV,OP> OPT;
    typedef Teuchos::ScalarTraits<ScalarType> SCT;
    typedef typename Teuchos::ScalarTraits<ScalarType>::magnitudeType MagnitudeType;
    typedef Teuchos::ScalarTraits<MagnitudeType> MT;
    typedef OrthoManagerFactory<ScalarType, MV, OP> ortho_factory_type;
 
  public:


     GCRODRSolMgr();

    GCRODRSolMgr (const Teuchos::RCP<LinearProblem<ScalarType,MV,OP> > &problem,
                  const Teuchos::RCP<Teuchos::ParameterList> &pl);

    virtual ~GCRODRSolMgr() {};

    Teuchos::RCP<SolverManager<ScalarType, MV, OP> > clone () const override {
      return Teuchos::rcp(new GCRODRSolMgr<ScalarType,MV,OP,true>);
    }




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

    Teuchos::RCP<const Teuchos::ParameterList> getValidParameters() const override;

    Teuchos::RCP<const Teuchos::ParameterList> getCurrentParameters() const override {
      return params_;
    }

    Teuchos::Array<Teuchos::RCP<Teuchos::Time> > getTimers() const {
      return Teuchos::tuple(timerSolve_);
    }

    MagnitudeType achievedTol() const override {
      return achievedTol_;
    }

    int getNumIters() const override {
      return numIters_;
    }

    bool isLOADetected() const override { return false; }




    void setProblem( const Teuchos::RCP<LinearProblem<ScalarType,MV,OP> > &problem ) override {
      problem_ = problem;
    }

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




    void reset( const ResetType type ) override {
      if ((type & Belos::Problem) && !Teuchos::is_null(problem_)) {
        bool set = problem_->setProblem();
        if  (!set)
          throw "Could not set problem.";
      }
      else if (type & Belos::RecycleSubspace) {
        keff = 0;
      }
    }




    ReturnType solve() override;



    std::string description() const override;

 
  private:
 
    // Called by all constructors; Contains init instructions common to all constructors
    void init();
 
    // Initialize solver state storage
    void initializeStateStorage();
 
    // Compute updated recycle space given existing recycle space and newly generated Krylov space
    void buildRecycleSpace2(Teuchos::RCP<GCRODRIter<ScalarType,MV,OP> > gcrodr_iter);
 
    //  Computes harmonic eigenpairs of projected matrix created during the priming solve.
    //  HH is the projected problem from the initial cycle of Gmres, it is (at least) of dimension m+1 x m.
    //  PP contains the harmonic eigenvectors corresponding to the recycledBlocks eigenvalues of smallest magnitude.
    //  The return value is the number of vectors needed to be stored, recycledBlocks or recycledBlocks+1.
    int getHarmonicVecs1(int m,
                         const Teuchos::SerialDenseMatrix<int,ScalarType>& HH,
                         Teuchos::SerialDenseMatrix<int,ScalarType>& PP);
 
    //  Computes harmonic eigenpairs of projected matrix created during one cycle.
    //  HH is the total block projected problem from the GCRO-DR algorithm, it is (at least) of dimension keff+m+1 x keff+m.
    //  VV is the Krylov vectors from the projected GMRES algorithm, which has (at least) m+1 vectors.
    //  PP contains the harmonic eigenvectors corresponding to the recycledBlocks eigenvalues of smallest magnitude.
    //  The return value is the number of vectors needed to be stored, recycledBlocks or recycledBlocks+1.
    int getHarmonicVecs2(int keff, int m,
                         const Teuchos::SerialDenseMatrix<int,ScalarType>& HH,
                         const Teuchos::RCP<const MV>& VV,
                         Teuchos::SerialDenseMatrix<int,ScalarType>& PP);
 
    // Sort list of n floating-point numbers and return permutation vector
    void sort(std::vector<MagnitudeType>& dlist, int n, std::vector<int>& iperm);
 
    // Lapack interface
    Teuchos::LAPACK<int,ScalarType> lapack;
 
    // Linear problem.
    Teuchos::RCP<LinearProblem<ScalarType,MV,OP> > problem_;
 
    // Output manager.
    Teuchos::RCP<OutputManager<ScalarType> > printer_;
    Teuchos::RCP<std::ostream> outputStream_;
 
    // Status test.
    Teuchos::RCP<StatusTest<ScalarType,MV,OP> > sTest_;
    Teuchos::RCP<StatusTestMaxIters<ScalarType,MV,OP> > maxIterTest_;
    Teuchos::RCP<StatusTest<ScalarType,MV,OP> > convTest_;
    Teuchos::RCP<StatusTestGenResNorm<ScalarType,MV,OP> > expConvTest_, impConvTest_;
    Teuchos::RCP<StatusTestOutput<ScalarType,MV,OP> > outputTest_;

    Teuchos::RCP<MatOrthoManager<ScalarType,MV,OP> > ortho_;
 
    // Current parameter list.
    Teuchos::RCP<Teuchos::ParameterList> params_;
 
    // Default solver values.
    static constexpr double orthoKappa_default_ = 0.0;
    static constexpr int maxRestarts_default_ = 100;
    static constexpr int maxIters_default_ = 1000;
    static constexpr int numBlocks_default_ = 50;
    static constexpr int blockSize_default_ = 1;
    static constexpr int recycledBlocks_default_ = 5;
    static constexpr int verbosity_default_ = Belos::Errors;
    static constexpr int outputStyle_default_ = Belos::General;
    static constexpr int outputFreq_default_ = -1;
    static constexpr const char * impResScale_default_ = "Norm of Preconditioned Initial Residual";
    static constexpr const char * expResScale_default_ = "Norm of Initial Residual";
    static constexpr const char * label_default_ = "Belos";
    static constexpr const char * orthoType_default_ = "ICGS";
 
    // Current solver values.
    MagnitudeType convTol_, orthoKappa_, achievedTol_;
    int maxRestarts_, maxIters_, numIters_;
    int verbosity_, outputStyle_, outputFreq_;
    std::string orthoType_;
    std::string impResScale_, expResScale_;

    // Solver State Storage
    //
    // The number of blocks and recycle blocks (m and k, respectively)
    int numBlocks_, recycledBlocks_;
    // Current size of recycled subspace
    int keff;
    //
    // Residual vector
    Teuchos::RCP<MV> r_;
    //
    // Search space
    Teuchos::RCP<MV> V_;
    //
    // Recycled subspace and its image
    Teuchos::RCP<MV> U_, C_;
    //
    // Updated recycle space and its image
    Teuchos::RCP<MV> U1_, C1_;
    //
    // Storage used in constructing harmonic Ritz values/vectors
    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > H2_;
    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > H_;
    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > B_;
    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > PP_;
    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > HP_;
    std::vector<ScalarType> tau_;
    std::vector<ScalarType> work_;
    Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > R_;
    std::vector<int> ipiv_;
 
    // Timers.
    std::string label_;
    Teuchos::RCP<Teuchos::Time> timerSolve_;
 
    // Internal state variables.
    bool isSet_;
 
    // Have we generated or regenerated a recycle space yet this solve?
    bool builtRecycleSpace_;
  };
 
 
// Empty Constructor
template<class ScalarType, class MV, class OP>
GCRODRSolMgr<ScalarType,MV,OP,true>::GCRODRSolMgr():
  achievedTol_(0.0),
  numIters_(0)
{
  init ();
}
 
 
// Basic Constructor
template<class ScalarType, class MV, class OP>
GCRODRSolMgr<ScalarType,MV,OP,true>::
GCRODRSolMgr(const Teuchos::RCP<LinearProblem<ScalarType,MV,OP> >& problem,
             const Teuchos::RCP<Teuchos::ParameterList>& pl):
  achievedTol_(0.0),
  numIters_(0)
{
  // Initialize local pointers to null, and initialize local variables
  // to default values.
  init ();
 
  TEUCHOS_TEST_FOR_EXCEPTION(
    problem == Teuchos::null, std::invalid_argument,
    "Belos::GCRODRSolMgr constructor: The solver manager's "
    "constructor needs the linear problem argument 'problem' "
    "to be non-null.");
  problem_ = problem;
 
  // Set the parameters using the list that was passed in.  If null,
  // we defer initialization until a non-null list is set (by the
  // client calling setParameters(), or by calling solve() -- in
  // either case, a null parameter list indicates that default
  // parameters should be used).
  if (! pl.is_null ()) {
    setParameters (pl);
  }
}
 
// Common instructions executed in all constructors
template<class ScalarType, class MV, class OP>
void GCRODRSolMgr<ScalarType,MV,OP,true>::init () {
  outputStream_ = Teuchos::rcpFromRef(std::cout);
  convTol_ = DefaultSolverParameters::convTol;
  orthoKappa_ = orthoKappa_default_;
  maxRestarts_ = maxRestarts_default_;
  maxIters_ = maxIters_default_;
  numBlocks_ = numBlocks_default_;
  recycledBlocks_ = recycledBlocks_default_;
  verbosity_ = verbosity_default_;
  outputStyle_ = outputStyle_default_;
  outputFreq_ = outputFreq_default_;
  orthoType_ = orthoType_default_;
  impResScale_ = impResScale_default_;
  expResScale_ = expResScale_default_;
  label_ = label_default_;
  isSet_ = false;
  builtRecycleSpace_ = false;
  keff = 0;
  r_ = Teuchos::null;
  V_ = Teuchos::null;
  U_ = Teuchos::null;
  C_ = Teuchos::null;
  U1_ = Teuchos::null;
  C1_ = Teuchos::null;
  PP_ = Teuchos::null;
  HP_ = Teuchos::null;
  H2_ = Teuchos::null;
  R_ = Teuchos::null;
  H_ = Teuchos::null;
  B_ = Teuchos::null;
}
 
template<class ScalarType, class MV, class OP>
void
GCRODRSolMgr<ScalarType,MV,OP,true>::
setParameters (const Teuchos::RCP<Teuchos::ParameterList> &params)
{
  using Teuchos::isParameterType;
  using Teuchos::getParameter;
  using Teuchos::null;
  using Teuchos::ParameterList;
  using Teuchos::parameterList;
  using Teuchos::RCP;
  using Teuchos::rcp;
  using Teuchos::rcp_dynamic_cast;
  using Teuchos::rcpFromRef;
  using Teuchos::Exceptions::InvalidParameter;
  using Teuchos::Exceptions::InvalidParameterName;
  using Teuchos::Exceptions::InvalidParameterType;
 
  // The default parameter list contains all parameters that
  // GCRODRSolMgr understands, and none that it doesn't understand.
  RCP<const ParameterList> defaultParams = getValidParameters();
 
  // Create the internal parameter list if one doesn't already exist.
  //
  // (mfh 28 Feb 2011, 10 Mar 2011) At the time this code was written,
  // ParameterList did not have validators or validateParameters().
  // This is why the code below carefully validates the parameters one
  // by one and fills in defaults.  This code could be made a lot
  // shorter by using validators.  To do so, we would have to define
  // appropriate validators for all the parameters.  (This would more
  // or less just move all that validation code out of this routine
  // into to getValidParameters().)
  //
  // For an analogous reason, GCRODRSolMgr defines default parameter
  // values as class data, as well as in the default ParameterList.
  // This redundancy could be removed by defining the default
  // parameter values only in the default ParameterList (which
  // documents each parameter as well -- handy!).
  if (params_.is_null()) {
    params_ = parameterList (*defaultParams);
  } else {
    // A common case for setParameters() is for it to be called at the
    // beginning of the solve() routine.  This follows the Belos
    // pattern of delaying initialization until the last possible
    // moment (when the user asks Belos to perform the solve).  In
    // this common case, we save ourselves a deep copy of the input
    // parameter list.
    if (params_ != params) {
      // Make a deep copy of the input parameter list.  This allows
      // the caller to modify or change params later, without
      // affecting the behavior of this solver.  This solver will then
      // only change its internal parameters if setParameters() is
      // called again.
      params_ = parameterList (*params);
    }
 
    // Fill in any missing parameters and their default values.  Also,
    // throw an exception if the parameter list has any misspelled or
    // "extra" parameters.  If you don't like this behavior, you'll
    // want to replace the line of code below with your desired
    // validation scheme.  Note that Teuchos currently only implements
    // two options:
    //
    // 1. validateParameters() requires that params_ has all the
    //    parameters that the default list has, and none that it
    //    doesn't have.
    //
    // 2. validateParametersAndSetDefaults() fills in missing
    //    parameters in params_ using the default list, but requires
    //    that any parameter provided in params_ is also in the
    //    default list.
    //
    // Here is an easy way to ignore any "extra" or misspelled
    // parameters: Make a deep copy of the default list, fill in any
    // "missing" parameters from the _input_ list, and then validate
    // the input list using the deep copy of the default list.  We
    // show this in code:
    //
    // RCP<ParameterList> defaultCopy = parameterList (*getValidParameters ());
    // defaultCopy->validateParametersAndSetDefaults (params);
    // params->validateParametersAndSetDefaults (defaultCopy);
    //
    // This method is not entirely robust, because the input list may
    // have incorrect validators set for existing parameters in the
    // default list.  This would then cause "validation" of the
    // default list to throw an exception.  As a result, we've chosen
    // for now to be intolerant of misspellings and "extra" parameters
    // in the input list.
    params_->validateParametersAndSetDefaults (*defaultParams);
  }
 
  // Check for maximum number of restarts.
  if (params->isParameter ("Maximum Restarts")) {
    maxRestarts_ = params->get("Maximum Restarts", maxRestarts_default_);
 
    // Update parameter in our list.
    params_->set ("Maximum Restarts", maxRestarts_);
  }
 
  // Check for maximum number of iterations
  if (params->isParameter ("Maximum Iterations")) {
    maxIters_ = params->get ("Maximum Iterations", maxIters_default_);
 
    // Update parameter in our list and in status test.
    params_->set ("Maximum Iterations", maxIters_);
    if (! maxIterTest_.is_null())
      maxIterTest_->setMaxIters (maxIters_);
  }
 
  // Check for the maximum number of blocks.
  if (params->isParameter ("Num Blocks")) {
    numBlocks_ = params->get ("Num Blocks", numBlocks_default_);
    TEUCHOS_TEST_FOR_EXCEPTION(numBlocks_ <= 0, std::invalid_argument,
                       "Belos::GCRODRSolMgr: The \"Num Blocks\" parameter must "
                       "be strictly positive, but you specified a value of "
                       << numBlocks_ << ".");
    // Update parameter in our list.
    params_->set ("Num Blocks", numBlocks_);
  }
 
  // Check for the maximum number of blocks.
  if (params->isParameter ("Num Recycled Blocks")) {
    recycledBlocks_ = params->get ("Num Recycled Blocks",
                                   recycledBlocks_default_);
    TEUCHOS_TEST_FOR_EXCEPTION(recycledBlocks_ <= 0, std::invalid_argument,
                       "Belos::GCRODRSolMgr: The \"Num Recycled Blocks\" "
                       "parameter must be strictly positive, but you specified "
                       "a value of " << recycledBlocks_ << ".");
    TEUCHOS_TEST_FOR_EXCEPTION(recycledBlocks_ >= numBlocks_, std::invalid_argument,
                       "Belos::GCRODRSolMgr: The \"Num Recycled Blocks\" "
                       "parameter must be less than the \"Num Blocks\" "
                       "parameter, but you specified \"Num Recycled Blocks\" "
                       "= " << recycledBlocks_ << " and \"Num Blocks\" = "
                       << numBlocks_ << ".");
    // Update parameter in our list.
    params_->set("Num Recycled Blocks", recycledBlocks_);
  }
 
  // Check to see if the timer label changed.  If it did, update it in
  // the parameter list, and create a new timer with that label (if
  // Belos was compiled with timers enabled).
  if (params->isParameter ("Timer Label")) {
    std::string tempLabel = params->get ("Timer Label", label_default_);
 
    // Update parameter in our list and solver timer
    if (tempLabel != label_) {
      label_ = tempLabel;
      params_->set ("Timer Label", label_);
      std::string solveLabel = label_ + ": GCRODRSolMgr total solve time";
#ifdef BELOS_TEUCHOS_TIME_MONITOR
      timerSolve_ = Teuchos::TimeMonitor::getNewCounter (solveLabel);
#endif
      if (ortho_ != Teuchos::null) {
        ortho_->setLabel( label_ );
      }
    }
  }
 
  // Check for a change in verbosity level
  if (params->isParameter ("Verbosity")) {
    if (isParameterType<int> (*params, "Verbosity")) {
      verbosity_ = params->get ("Verbosity", verbosity_default_);
    } else {
      verbosity_ = (int) getParameter<Belos::MsgType> (*params, "Verbosity");
    }
    // Update parameter in our list.
    params_->set ("Verbosity", verbosity_);
    // If the output manager (printer_) is null, then we will
    // instantiate it later with the correct verbosity.
    if (! printer_.is_null())
      printer_->setVerbosity (verbosity_);
  }
 
  // Check for a change in output style
  if (params->isParameter ("Output Style")) {
    if (isParameterType<int> (*params, "Output Style")) {
      outputStyle_ = params->get ("Output Style", outputStyle_default_);
    } else {
      outputStyle_ = (int) getParameter<OutputType> (*params, "Output Style");
    }
 
    // Update parameter in our list.
    params_->set ("Output Style", outputStyle_);
    // We will (re)instantiate the output status test afresh below.
    outputTest_ = null;
  }
 
  // Get the output stream for the output manager.
  //
  // While storing the output stream in the parameter list (either as
  // an RCP or as a nonconst reference) is convenient and safe for
  // programming, it makes it impossible to serialize the parameter
  // list, read it back in from the serialized representation, and get
  // the same output stream as before.  This is because output streams
  // may be arbitrary constructed objects.
  //
  // In case the user tried reading in the parameter list from a
  // serialized representation and the output stream can't be read
  // back in, we set the output stream to point to std::cout.  This
  // ensures reasonable behavior.
  if (params->isParameter ("Output Stream")) {
    try {
      outputStream_ = getParameter<RCP<std::ostream> > (*params, "Output Stream");
    } catch (InvalidParameter&) {
      outputStream_ = rcpFromRef (std::cout);
    }
    // We assume that a null output stream indicates that the user
    // doesn't want to print anything, so we replace it with a "black
    // hole" stream that prints nothing sent to it.  (We can't use a
    // null output stream, since the output manager always sends
    // things it wants to print to the output stream.)
    if (outputStream_.is_null()) {
      outputStream_ = rcp (new Teuchos::oblackholestream);
    }
    // Update parameter in our list.
    params_->set ("Output Stream", outputStream_);
    // If the output manager (printer_) is null, then we will
    // instantiate it later with the correct output stream.
    if (! printer_.is_null()) {
      printer_->setOStream (outputStream_);
    }
  }
 
  // frequency level
  if (verbosity_ & Belos::StatusTestDetails) {
    if (params->isParameter ("Output Frequency")) {
      outputFreq_ = params->get ("Output Frequency", outputFreq_default_);
    }
 
    // Update parameter in out list and output status test.
    params_->set("Output Frequency", outputFreq_);
    if (! outputTest_.is_null())
      outputTest_->setOutputFrequency (outputFreq_);
  }
 
  // Create output manager if we need to, using the verbosity level
  // and output stream that we fetched above.  We do this here because
  // instantiating an OrthoManager using OrthoManagerFactory requires
  // a valid OutputManager.
  if (printer_.is_null()) {
    printer_ = rcp (new OutputManager<ScalarType> (verbosity_, outputStream_));
  }
 
  // Get the orthogonalization manager name ("Orthogonalization").
  //
  // Getting default values for the orthogonalization manager
  // parameters ("Orthogonalization Parameters") requires knowing the
  // orthogonalization manager name.  Save it for later, and also
  // record whether it's different than before.
  OrthoManagerFactory<ScalarType, MV, OP> factory;
  bool changedOrthoType = false;
  if (params->isParameter ("Orthogonalization")) {
    const std::string& tempOrthoType =
      params->get ("Orthogonalization", orthoType_default_);
    // Ensure that the specified orthogonalization type is valid.
    if (! factory.isValidName (tempOrthoType)) {
      std::ostringstream os;
      os << "Belos::GCRODRSolMgr: Invalid orthogonalization name \""
         << tempOrthoType << "\".  The following are valid options "
         << "for the \"Orthogonalization\" name parameter: ";
      factory.printValidNames (os);
      throw std::invalid_argument (os.str());
    }
    if (tempOrthoType != orthoType_) {
      changedOrthoType = true;
      orthoType_ = tempOrthoType;
      // Update parameter in our list.
      params_->set ("Orthogonalization", orthoType_);
    }
  }
 
  // Get any parameters for the orthogonalization ("Orthogonalization
  // Parameters").  If not supplied, the orthogonalization manager
  // factory will supply default values.
  //
  // NOTE (mfh 12 Jan 2011) For the sake of backwards compatibility,
  // if params has an "Orthogonalization Constant" parameter and the
  // DGKS orthogonalization manager is to be used, the value of this
  // parameter will override DGKS's "depTol" parameter.
  //
  // Users must supply the orthogonalization manager parameters as a
  // sublist (supplying it as an RCP<ParameterList> would make the
  // resulting parameter list not serializable).
  RCP<ParameterList> orthoParams;
  { // The nonmember function sublist() returns an RCP<ParameterList>,
    // which is what we want here.
    using Teuchos::sublist;
    // Abbreviation to avoid typos.
    const std::string paramName ("Orthogonalization Parameters");
 
    try {
      orthoParams = sublist (params_, paramName, true);
    } catch (InvalidParameter&) {
      // We didn't get the parameter list from params, so get a
      // default parameter list from the OrthoManagerFactory.  Modify
      // params_ so that it has the default parameter list, and set
      // orthoParams to ensure it's a sublist of params_ (and not just
      // a copy of one).
      params_->set (paramName, factory.getDefaultParameters (orthoType_));
      orthoParams = sublist (params_, paramName, true);
    }
  }
  TEUCHOS_TEST_FOR_EXCEPTION(orthoParams.is_null(), std::logic_error,
                             "Failed to get orthogonalization parameters.  "
                             "Please report this bug to the Belos developers.");
 
  // Instantiate a new MatOrthoManager subclass instance if necessary.
  // If not necessary, then tell the existing instance about the new
  // parameters.
  if (ortho_.is_null() || changedOrthoType) {
    // We definitely need to make a new MatOrthoManager, since either
    // we haven't made one yet, or we've changed orthogonalization
    // methods.  Creating the orthogonalization manager requires that
    // the OutputManager (printer_) already be initialized.
    ortho_ = factory.makeMatOrthoManager (orthoType_, null, printer_,
                                          label_, orthoParams);
  } else {
    // If the MatOrthoManager implements the ParameterListAcceptor
    // mix-in interface, we can propagate changes to its parameters
    // without reinstantiating the MatOrthoManager.
    //
    // We recommend that all MatOrthoManager subclasses implement
    // Teuchos::ParameterListAcceptor, but do not (yet) require this.
    typedef Teuchos::ParameterListAcceptor PLA;
    RCP<PLA> pla = rcp_dynamic_cast<PLA> (ortho_);
    if (pla.is_null()) {
      // Oops, it's not a ParameterListAcceptor.  We have to
      // reinstantiate the MatOrthoManager in order to pass in the
      // possibly new parameters.
      ortho_ = factory.makeMatOrthoManager (orthoType_, null, printer_,
                                            label_, orthoParams);
    } else {
      pla->setParameterList (orthoParams);
    }
  }
 
  // The DGKS orthogonalization accepts a "Orthogonalization Constant"
  // parameter (also called kappa in the code, but not in the
  // parameter list).  If its value is provided in the given parameter
  // list, and its value is positive, use it.  Ignore negative values.
  //
  // NOTE (mfh 12 Jan 2011) This overrides the "depTol" parameter that
  // may have been specified in "Orthogonalization Parameters".  We
  // retain this behavior for backwards compatibility.
  if (params->isParameter ("Orthogonalization Constant")) {
    MagnitudeType orthoKappa = orthoKappa_default_;
    if (params->isType<MagnitudeType> ("Orthogonalization Constant")) {
      orthoKappa = params->get ("Orthogonalization Constant", orthoKappa);
    }
    else {
      orthoKappa = params->get ("Orthogonalization Constant", orthoKappa_default_);
    }
 
    if (orthoKappa > 0) {
      orthoKappa_ = orthoKappa;
      // Update parameter in our list.
      params_->set("Orthogonalization Constant", orthoKappa_);
      // Only DGKS currently accepts this parameter.
      if (orthoType_ == "DGKS" && ! ortho_.is_null()) {
        typedef DGKSOrthoManager<ScalarType, MV, OP> ortho_man_type;
        // This cast should always succeed; it's a bug
        // otherwise.  (If the cast fails, then orthoType_
        // doesn't correspond to the OrthoManager subclass
        // instance that we think we have, so we initialized the
        // wrong subclass somehow.)
        rcp_dynamic_cast<ortho_man_type>(ortho_)->setDepTol (orthoKappa_);
      }
    }
  }
 
  // Convergence
  typedef Belos::StatusTestCombo<ScalarType,MV,OP>  StatusTestCombo_t;
  typedef Belos::StatusTestGenResNorm<ScalarType,MV,OP>  StatusTestResNorm_t;
 
  // Check for convergence tolerance
  if (params->isParameter("Convergence Tolerance")) {
    if (params->isType<MagnitudeType> ("Convergence Tolerance")) {
      convTol_ = params->get ("Convergence Tolerance",
                              static_cast<MagnitudeType> (DefaultSolverParameters::convTol));
    }
    else {
      convTol_ = params->get ("Convergence Tolerance", DefaultSolverParameters::convTol);
    }
 
    // Update parameter in our list and residual tests.
    params_->set ("Convergence Tolerance", convTol_);
    if (! impConvTest_.is_null())
      impConvTest_->setTolerance (convTol_);
    if (! expConvTest_.is_null())
      expConvTest_->setTolerance (convTol_);
  }
 
  // Check for a change in scaling, if so we need to build new residual tests.
  if (params->isParameter ("Implicit Residual Scaling")) {
    std::string tempImpResScale =
      getParameter<std::string> (*params, "Implicit Residual Scaling");
 
    // Only update the scaling if it's different.
    if (impResScale_ != tempImpResScale) {
      ScaleType impResScaleType = convertStringToScaleType (tempImpResScale);
      impResScale_ = tempImpResScale;
 
      // Update parameter in our list and residual tests
      params_->set("Implicit Residual Scaling", impResScale_);
      // NOTE (mfh 28 Feb 2011) StatusTestImpResNorm only lets you
      // call defineScaleForm() once.  The code below attempts to call
      // defineScaleForm(); if the scale form has already been
      // defined, it constructs a new StatusTestImpResNorm instance.
      // StatusTestImpResNorm really should not expose the
      // defineScaleForm() method, since it's serving an
      // initialization purpose; all initialization should happen in
      // the constructor whenever possible.  In that case, the code
      // below could be simplified into a single (re)instantiation.
      if (! impConvTest_.is_null()) {
        try {
          impConvTest_->defineScaleForm (impResScaleType, Belos::TwoNorm);
        }
        catch (StatusTestError&) {
          // Delete the convergence test so it gets constructed again.
          impConvTest_ = null;
          convTest_ = null;
        }
      }
    }
  }
 
  if (params->isParameter("Explicit Residual Scaling")) {
    std::string tempExpResScale =
      getParameter<std::string> (*params, "Explicit Residual Scaling");
 
    // Only update the scaling if it's different.
    if (expResScale_ != tempExpResScale) {
      ScaleType expResScaleType = convertStringToScaleType (tempExpResScale);
      expResScale_ = tempExpResScale;
 
      // Update parameter in our list and residual tests
      params_->set("Explicit Residual Scaling", expResScale_);
      // NOTE (mfh 28 Feb 2011) See note above on the (re)construction
      // of StatusTestImpResNorm.
      if (! expConvTest_.is_null()) {
        try {
          expConvTest_->defineScaleForm (expResScaleType, Belos::TwoNorm);
        }
        catch (StatusTestError&) {
          // Delete the convergence test so it gets constructed again.
          expConvTest_ = null;
          convTest_ = null;
        }
      }
    }
  }
  //
  // Create iteration stopping criteria ("status tests") if we need
  // to, by combining three different stopping criteria.
  //
  // First, construct maximum-number-of-iterations stopping criterion.
  if (maxIterTest_.is_null())
    maxIterTest_ = rcp (new StatusTestMaxIters<ScalarType,MV,OP> (maxIters_));
 
  // Implicit residual test, using the native residual to determine if
  // convergence was achieved.
  if (impConvTest_.is_null()) {
    impConvTest_ = rcp (new StatusTestResNorm_t (convTol_));
    impConvTest_->defineScaleForm (convertStringToScaleType (impResScale_),
                                   Belos::TwoNorm);
  }
 
  // Explicit residual test once the native residual is below the tolerance
  if (expConvTest_.is_null()) {
    expConvTest_ = rcp (new StatusTestResNorm_t (convTol_));
    expConvTest_->defineResForm (StatusTestResNorm_t::Explicit, Belos::TwoNorm);
    expConvTest_->defineScaleForm (convertStringToScaleType (expResScale_),
                                   Belos::TwoNorm);
  }
  // Convergence test first tests the implicit residual, then the
  // explicit residual if the implicit residual test passes.
  if (convTest_.is_null()) {
    convTest_ = rcp (new StatusTestCombo_t (StatusTestCombo_t::SEQ,
                                            impConvTest_,
                                            expConvTest_));
  }
  // Construct the complete iteration stopping criterion:
  //
  // "Stop iterating if the maximum number of iterations has been
  // reached, or if the convergence test passes."
  sTest_ = rcp (new StatusTestCombo_t (StatusTestCombo_t::OR,
                                       maxIterTest_,
                                       convTest_));
  // Create the status test output class.
  // This class manages and formats the output from the status test.
  StatusTestOutputFactory<ScalarType,MV,OP> stoFactory (outputStyle_);
  outputTest_ = stoFactory.create (printer_, sTest_, outputFreq_,
                                   Passed+Failed+Undefined);
 
  // Set the solver string for the output test
  std::string solverDesc = " GCRODR ";
  outputTest_->setSolverDesc( solverDesc );
 
  // Create the timer if we need to.
  if (timerSolve_.is_null()) {
    std::string solveLabel = label_ + ": GCRODRSolMgr total solve time";
#ifdef BELOS_TEUCHOS_TIME_MONITOR
    timerSolve_ = Teuchos::TimeMonitor::getNewCounter(solveLabel);
#endif
  }
 
  // Inform the solver manager that the current parameters were set.
  isSet_ = true;
}
 
 
template<class ScalarType, class MV, class OP>
Teuchos::RCP<const Teuchos::ParameterList>
GCRODRSolMgr<ScalarType,MV,OP,true>::getValidParameters() const
{
  using Teuchos::ParameterList;
  using Teuchos::parameterList;
  using Teuchos::RCP;
 
  static RCP<const ParameterList> validPL;
  if (is_null(validPL)) {
    RCP<ParameterList> pl = parameterList ();
 
    // Set all the valid parameters and their default values.
    pl->set("Convergence Tolerance", static_cast<MagnitudeType>(DefaultSolverParameters::convTol),
      "The relative residual tolerance that needs to be achieved by the\n"
      "iterative solver in order for the linear system to be declared converged.");
    pl->set("Maximum Restarts", static_cast<int>(maxRestarts_default_),
      "The maximum number of cycles allowed for each\n"
      "set of RHS solved.");
    pl->set("Maximum Iterations", static_cast<int>(maxIters_default_),
      "The maximum number of iterations allowed for each\n"
      "set of RHS solved.");
    // mfh 25 Oct 2010: "Block Size" must be 1 because GCRODR is
    // currently not a block method: i.e., it does not work on
    // multiple right-hand sides at once.
    pl->set("Block Size", static_cast<int>(blockSize_default_),
      "Block Size Parameter -- currently must be 1 for GCRODR");
    pl->set("Num Blocks", static_cast<int>(numBlocks_default_),
      "The maximum number of vectors allowed in the Krylov subspace\n"
      "for each set of RHS solved.");
    pl->set("Num Recycled Blocks", static_cast<int>(recycledBlocks_default_),
      "The maximum number of vectors in the recycled subspace." );
    pl->set("Verbosity", static_cast<int>(verbosity_default_),
      "What type(s) of solver information should be outputted\n"
      "to the output stream.");
    pl->set("Output Style", static_cast<int>(outputStyle_default_),
      "What style is used for the solver information outputted\n"
      "to the output stream.");
    pl->set("Output Frequency", static_cast<int>(outputFreq_default_),
      "How often convergence information should be outputted\n"
      "to the output stream.");
    pl->set("Output Stream", Teuchos::rcpFromRef(std::cout),
      "A reference-counted pointer to the output stream where all\n"
      "solver output is sent.");
    pl->set("Implicit Residual Scaling", static_cast<const char *>(impResScale_default_),
      "The type of scaling used in the implicit residual convergence test.");
    pl->set("Explicit Residual Scaling", static_cast<const char *>(expResScale_default_),
      "The type of scaling used in the explicit residual convergence test.");
    pl->set("Timer Label", static_cast<const char *>(label_default_),
      "The string to use as a prefix for the timer labels.");
    {
      OrthoManagerFactory<ScalarType, MV, OP> factory;
      pl->set("Orthogonalization", static_cast<const char *>(orthoType_default_),
              "The type of orthogonalization to use.  Valid options: " +
              factory.validNamesString());
      RCP<const ParameterList> orthoParams =
        factory.getDefaultParameters (orthoType_default_);
      pl->set ("Orthogonalization Parameters", *orthoParams,
               "Parameters specific to the type of orthogonalization used.");
    }
    pl->set("Orthogonalization Constant",static_cast<MagnitudeType>(orthoKappa_default_),
            "When using DGKS orthogonalization: the \"depTol\" constant, used "
            "to determine whether another step of classical Gram-Schmidt is "
            "necessary.  Otherwise ignored.");
    validPL = pl;
  }
  return validPL;
}
 
// initializeStateStorage
template<class ScalarType, class MV, class OP>
void GCRODRSolMgr<ScalarType,MV,OP,true>::initializeStateStorage() {
 
    ScalarType zero = Teuchos::ScalarTraits<ScalarType>::zero();
 
    // Check if there is any multivector to clone from.
    Teuchos::RCP<const MV> rhsMV = problem_->getRHS();
    if (rhsMV == Teuchos::null) {
      // Nothing to do
      return;
    }
    else {
 
      // Initialize the state storage
      TEUCHOS_TEST_FOR_EXCEPTION(static_cast<ptrdiff_t>(numBlocks_) > MVT::GetGlobalLength(*rhsMV),std::invalid_argument,
                         "Belos::GCRODRSolMgr::initializeStateStorage(): Cannot generate a Krylov basis with dimension larger the operator!");
 
      // If the subspace has not been initialized before, generate it using the RHS from lp_.
      if (U_ == Teuchos::null) {
        U_ = MVT::Clone( *rhsMV, recycledBlocks_+1 );
      }
      else {
        // Generate U_ by cloning itself ONLY if more space is needed.
        if (MVT::GetNumberVecs(*U_) < recycledBlocks_+1) {
          Teuchos::RCP<const MV> tmp = U_;
          U_ = MVT::Clone( *tmp, recycledBlocks_+1 );
        }
      }
 
      // If the subspace has not been initialized before, generate it using the RHS from lp_.
      if (C_ == Teuchos::null) {
        C_ = MVT::Clone( *rhsMV, recycledBlocks_+1 );
      }
      else {
        // Generate C_ by cloning itself ONLY if more space is needed.
        if (MVT::GetNumberVecs(*C_) < recycledBlocks_+1) {
          Teuchos::RCP<const MV> tmp = C_;
          C_ = MVT::Clone( *tmp, recycledBlocks_+1 );
        }
      }
 
      // If the subspace has not been initialized before, generate it using the RHS from lp_.
      if (V_ == Teuchos::null) {
        V_ = MVT::Clone( *rhsMV, numBlocks_+1 );
      }
      else {
        // Generate V_ by cloning itself ONLY if more space is needed.
        if (MVT::GetNumberVecs(*V_) < numBlocks_+1) {
          Teuchos::RCP<const MV> tmp = V_;
          V_ = MVT::Clone( *tmp, numBlocks_+1 );
        }
      }
 
      // If the subspace has not been initialized before, generate it using the RHS from lp_.
      if (U1_ == Teuchos::null) {
        U1_ = MVT::Clone( *rhsMV, recycledBlocks_+1 );
      }
      else {
        // Generate U1_ by cloning itself ONLY if more space is needed.
        if (MVT::GetNumberVecs(*U1_) < recycledBlocks_+1) {
          Teuchos::RCP<const MV> tmp = U1_;
          U1_ = MVT::Clone( *tmp, recycledBlocks_+1 );
        }
      }
 
      // If the subspace has not been initialized before, generate it using the RHS from lp_.
      if (C1_ == Teuchos::null) {
        C1_ = MVT::Clone( *rhsMV, recycledBlocks_+1 );
      }
      else {
        // Generate C1_ by cloning itself ONLY if more space is needed.
        if (MVT::GetNumberVecs(*C1_) < recycledBlocks_+1) {
          Teuchos::RCP<const MV> tmp = C1_;
          C1_ = MVT::Clone( *tmp, recycledBlocks_+1 );
        }
      }
 
      // Generate r_ only if it doesn't exist
      if (r_ == Teuchos::null)
        r_ = MVT::Clone( *rhsMV, 1 );
 
      // Size of tau_ will change during computation, so just be sure it starts with appropriate size
      tau_.resize(recycledBlocks_+1);
 
      // Size of work_ will change during computation, so just be sure it starts with appropriate size
      work_.resize(recycledBlocks_+1);
 
      // Size of ipiv_ will change during computation, so just be sure it starts with appropriate size
      ipiv_.resize(recycledBlocks_+1);
 
      // Generate H2_ only if it doesn't exist, otherwise resize it.
      if (H2_ == Teuchos::null)
        H2_ = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( numBlocks_+recycledBlocks_+2, numBlocks_+recycledBlocks_+1 ) );
      else {
        if ( (H2_->numRows() != numBlocks_+recycledBlocks_+2) || (H2_->numCols() != numBlocks_+recycledBlocks_+1) )
          H2_->reshape( numBlocks_+recycledBlocks_+2, numBlocks_+recycledBlocks_+1 );
      }
      H2_->putScalar(zero);
 
      // Generate R_ only if it doesn't exist, otherwise resize it.
      if (R_ == Teuchos::null)
        R_ = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( recycledBlocks_+1, recycledBlocks_+1 ) );
      else {
        if ( (R_->numRows() != recycledBlocks_+1) || (R_->numCols() != recycledBlocks_+1) )
          R_->reshape( recycledBlocks_+1, recycledBlocks_+1 );
      }
      R_->putScalar(zero);
 
      // Generate PP_ only if it doesn't exist, otherwise resize it.
      if (PP_ == Teuchos::null)
        PP_ = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( numBlocks_+recycledBlocks_+2, recycledBlocks_+1 ) );
      else {
        if ( (PP_->numRows() != numBlocks_+recycledBlocks_+2) || (PP_->numCols() != recycledBlocks_+1) )
          PP_->reshape( numBlocks_+recycledBlocks_+2, recycledBlocks_+1 );
      }
 
      // Generate HP_ only if it doesn't exist, otherwise resize it.
      if (HP_ == Teuchos::null)
        HP_ = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( numBlocks_+recycledBlocks_+2, numBlocks_+recycledBlocks_+1 ) );
      else {
        if ( (HP_->numRows() != numBlocks_+recycledBlocks_+2) || (HP_->numCols() != numBlocks_+recycledBlocks_+1) )
          HP_->reshape( numBlocks_+recycledBlocks_+2, numBlocks_+recycledBlocks_+1 );
      }
 
    } // end else
}
 
 
// solve()
template<class ScalarType, class MV, class OP>
ReturnType GCRODRSolMgr<ScalarType,MV,OP,true>::solve() {
  using Teuchos::RCP;
  using Teuchos::rcp;
 
  // Set the current parameters if they were not set before.
  // NOTE:  This may occur if the user generated the solver manager with the default constructor and
  // then didn't set any parameters using setParameters().
  if (!isSet_) { setParameters( params_ ); }
 
  ScalarType one = Teuchos::ScalarTraits<ScalarType>::one();
  ScalarType zero = Teuchos::ScalarTraits<ScalarType>::zero();
  std::vector<int> index(numBlocks_+1);
 
  TEUCHOS_TEST_FOR_EXCEPTION(problem_ == Teuchos::null,GCRODRSolMgrLinearProblemFailure, "Belos::GCRODRSolMgr::solve(): Linear problem is not a valid object.");
 
  TEUCHOS_TEST_FOR_EXCEPTION(!problem_->isProblemSet(),GCRODRSolMgrLinearProblemFailure,"Belos::GCRODRSolMgr::solve(): Linear problem is not ready, setProblem() has not been called.");
 
  // Create indices for the linear systems to be solved.
  int numRHS2Solve = MVT::GetNumberVecs( *(problem_->getRHS()) );
  std::vector<int> currIdx(1);
  currIdx[0] = 0;
 
  // Inform the linear problem of the current linear system to solve.
  problem_->setLSIndex( currIdx );
 
  // Check the number of blocks and change them is necessary.
  ptrdiff_t dim = MVT::GetGlobalLength( *(problem_->getRHS()) );
  if (static_cast<ptrdiff_t>(numBlocks_) > dim) {
    numBlocks_ = Teuchos::as<int>(dim);
    printer_->stream(Warnings) <<
      "Warning! Requested Krylov subspace dimension is larger than operator dimension!" << std::endl <<
      " The maximum number of blocks allowed for the Krylov subspace will be adjusted to " << numBlocks_ << std::endl;
    params_->set("Num Blocks", numBlocks_);
  }
 
  // Assume convergence is achieved, then let any failed convergence set this to false.
  bool isConverged = true;
 
  // Initialize storage for all state variables
  initializeStateStorage();

  // Parameter list
  Teuchos::ParameterList plist;
 
  plist.set("Num Blocks",numBlocks_);
  plist.set("Recycled Blocks",recycledBlocks_);

  // GCRODR solver
 
  RCP<GCRODRIter<ScalarType,MV,OP> > gcrodr_iter;
  gcrodr_iter = rcp( new GCRODRIter<ScalarType,MV,OP>(problem_,printer_,outputTest_,ortho_,plist) );
  // Number of iterations required to generate initial recycle space (if needed)
  int prime_iterations = 0;
 
  // Enter solve() iterations
  {
#ifdef BELOS_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor slvtimer(*timerSolve_);
#endif
 
    while ( numRHS2Solve > 0 ) {
 
      // Set flag indicating recycle space has not been generated this solve
      builtRecycleSpace_ = false;
 
      // Reset the status test.
      outputTest_->reset();

      // Initialize recycled subspace for GCRODR
 
      // If there is a subspace to recycle, recycle it, otherwise generate the initial recycled subspace.
      if (keff > 0) {
        TEUCHOS_TEST_FOR_EXCEPTION(keff < recycledBlocks_,GCRODRSolMgrRecyclingFailure,
                           "Belos::GCRODRSolMgr::solve(): Requested size of recycled subspace is not consistent with the current recycle subspace.");
 
        printer_->stream(Debug) << " Now solving RHS index " << currIdx[0] << " using recycled subspace of dimension " << keff << std::endl << std::endl;
        // Compute image of U_ under the new operator
        index.resize(keff);
        for (int ii=0; ii<keff; ++ii) { index[ii] = ii; }
        RCP<const MV> Utmp  = MVT::CloneView( *U_, index );
        RCP<MV> Ctmp  = MVT::CloneViewNonConst( *C_, index );
        problem_->apply( *Utmp, *Ctmp );
 
        RCP<MV> U1tmp = MVT::CloneViewNonConst( *U1_, index );
 
        // Orthogonalize this block
        // Get a matrix to hold the orthonormalization coefficients.
        Teuchos::SerialDenseMatrix<int,ScalarType> Rtmp( Teuchos::View, *R_, keff, keff );
        int rank = ortho_->normalize(*Ctmp, rcp(&Rtmp,false));
        // Throw an error if we could not orthogonalize this block
        TEUCHOS_TEST_FOR_EXCEPTION(rank != keff,GCRODRSolMgrOrthoFailure,"Belos::GCRODRSolMgr::solve(): Failed to compute orthonormal basis for initial recycled subspace.");
 
        // U_ = U_*R^{-1}
        // First, compute LU factorization of R
        int info = 0;
        ipiv_.resize(Rtmp.numRows());
        lapack.GETRF(Rtmp.numRows(),Rtmp.numCols(),Rtmp.values(),Rtmp.stride(),&ipiv_[0],&info);
        TEUCHOS_TEST_FOR_EXCEPTION(info != 0, GCRODRSolMgrLAPACKFailure,"Belos::GCRODRSolMgr::solve(): LAPACK _GETRF failed to compute an LU factorization.");
 
        // Now, form inv(R)
        int lwork = Rtmp.numRows();
        work_.resize(lwork);
        lapack.GETRI(Rtmp.numRows(),Rtmp.values(),Rtmp.stride(),&ipiv_[0],&work_[0],lwork,&info);
        TEUCHOS_TEST_FOR_EXCEPTION(info != 0, GCRODRSolMgrLAPACKFailure,"Belos::GCRODRSolMgr::solve(): LAPACK _GETRI failed to invert triangular matrix.");
 
        // U_ = U1_; (via a swap)
        MVT::MvTimesMatAddMv( one, *Utmp, Rtmp, zero, *U1tmp );
        std::swap(U_, U1_);
 
        // Must reinitialize after swap
        index.resize(keff);
        for (int ii=0; ii<keff; ++ii) { index[ii] = ii; }
        Ctmp  = MVT::CloneViewNonConst( *C_, index );
        Utmp  = MVT::CloneView( *U_, index );
 
        // Compute C_'*r_
        Teuchos::SerialDenseMatrix<int,ScalarType> Ctr(keff,1);
        problem_->computeCurrPrecResVec( &*r_ );
        MVT::MvTransMv( one, *Ctmp, *r_, Ctr );
 
        // Update solution ( x += U_*C_'*r_ )
        RCP<MV> update = MVT::Clone( *problem_->getCurrLHSVec(), 1 );
        MVT::MvInit( *update, 0.0 );
        MVT::MvTimesMatAddMv( one, *Utmp, Ctr, one, *update );
        problem_->updateSolution( update, true );
 
        // Update residual norm ( r -= C_*C_'*r_ )
        MVT::MvTimesMatAddMv( -one, *Ctmp, Ctr, one, *r_ );
 
        // We recycled space from previous call
        prime_iterations = 0;
 
      }
      else {
 
        // Do one cycle of Gmres to "prime the pump" if there is no subspace to recycle
        printer_->stream(Debug) << " No recycled subspace available for RHS index " << currIdx[0] << std::endl << std::endl;
 
        Teuchos::ParameterList primeList;
 
        // Tell the block solver that the block size is one.
        primeList.set("Num Blocks",numBlocks_);
        primeList.set("Recycled Blocks",0);
 
        //  Create GCRODR iterator object to perform one cycle of GMRES.
        RCP<GCRODRIter<ScalarType,MV,OP> > gcrodr_prime_iter;
        gcrodr_prime_iter = rcp( new GCRODRIter<ScalarType,MV,OP>(problem_,printer_,outputTest_,ortho_,primeList) );
 
        // Create the first block in the current Krylov basis (residual).
        problem_->computeCurrPrecResVec( &*r_ );
        index.resize( 1 ); index[0] = 0;
        RCP<MV> v0 =  MVT::CloneViewNonConst( *V_,  index );
        MVT::SetBlock(*r_,index,*v0); // V(:,0) = r
 
        // Set the new state and initialize the solver.
        GCRODRIterState<ScalarType,MV> newstate;
        index.resize( numBlocks_+1 );
        for (int ii=0; ii<(numBlocks_+1); ++ii) { index[ii] = ii; }
        newstate.V  = MVT::CloneViewNonConst( *V_,  index );
        newstate.U = Teuchos::null;
        newstate.C = Teuchos::null;
        newstate.H = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( Teuchos::View, *H2_, numBlocks_+1, numBlocks_, recycledBlocks_+1, recycledBlocks_+1 ) );
        newstate.B = Teuchos::null;
        newstate.curDim = 0;
        gcrodr_prime_iter->initialize(newstate);
 
        // Perform one cycle of GMRES
        bool primeConverged = false;
        try {
          gcrodr_prime_iter->iterate();
 
          // Check convergence first
          if ( convTest_->getStatus() == Passed ) {
            // we have convergence
            primeConverged = true;
          }
        }
        catch (const GCRODRIterOrthoFailure &e) {
          // Try to recover the most recent least-squares solution
          gcrodr_prime_iter->updateLSQR( gcrodr_prime_iter->getCurSubspaceDim() );
 
          // Check to see if the most recent least-squares solution yielded convergence.
          sTest_->checkStatus( &*gcrodr_prime_iter );
          if (convTest_->getStatus() == Passed)
            primeConverged = true;
        }
        catch (const StatusTestNaNError& e) {
          // A NaN was detected in the solver.  Set the solution to zero and return unconverged.
          achievedTol_ = MT::one();
          Teuchos::RCP<MV> X = problem_->getLHS();
          MVT::MvInit( *X, SCT::zero() );
          printer_->stream(Warnings) << "Belos::GCRODRSolMgr::solve(): Warning! NaN has been detected!" 
                                     << std::endl;
          return Unconverged; 
        }
        catch (const std::exception &e) {
          printer_->stream(Errors) << "Error! Caught exception in GCRODRIter::iterate() at iteration "
                                   << gcrodr_prime_iter->getNumIters() << std::endl
                                   << e.what() << std::endl;
          throw;
        }
        // Record number of iterations in generating initial recycle spacec
        prime_iterations = gcrodr_prime_iter->getNumIters();
 
        // Update the linear problem.
        RCP<MV> update = gcrodr_prime_iter->getCurrentUpdate();
        problem_->updateSolution( update, true );
 
        // Get the state.
        newstate = gcrodr_prime_iter->getState();
        int p = newstate.curDim;
 
        // Compute harmonic Ritz vectors
        // NOTE:  The storage for the harmonic Ritz vectors (PP) is made one column larger
        //        just in case we split a complex conjugate pair.
        // NOTE:  Generate a recycled subspace only if we have enough vectors.  If we converged
        //        too early, move on to the next linear system and try to generate a subspace again.
        if (recycledBlocks_ < p+1) {
          int info = 0;
          RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > PPtmp = rcp (new Teuchos::SerialDenseMatrix<int,ScalarType> ( Teuchos::View, *PP_, p, recycledBlocks_+1 ) );
          // getHarmonicVecs1 assumes PP has recycledBlocks_+1 columns available
          keff = getHarmonicVecs1( p, *newstate.H, *PPtmp );
          // Hereafter, only keff columns of PP are needed
          PPtmp = rcp (new Teuchos::SerialDenseMatrix<int,ScalarType> ( Teuchos::View, *PP_, p, keff ) );
          // Now get views into C, U, V
          index.resize(keff);
          for (int ii=0; ii<keff; ++ii) { index[ii] = ii; }
          RCP<MV> Ctmp  = MVT::CloneViewNonConst( *C_, index );
          RCP<MV> Utmp  = MVT::CloneViewNonConst( *U_, index );
          RCP<MV> U1tmp = MVT::CloneViewNonConst( *U1_, index );
          index.resize(p);
          for (int ii=0; ii < p; ++ii) { index[ii] = ii; }
          RCP<const MV> Vtmp = MVT::CloneView( *V_, index );
 
          // Form U (the subspace to recycle)
          // U = newstate.V(:,1:p) * PP;
          MVT::MvTimesMatAddMv( one, *Vtmp, *PPtmp, zero, *U1tmp );
 
          // Form orthonormalized C and adjust U so that C = A*U
 
          // First, compute [Q, R] = qr(H*P);
 
          // Step #1: Form HP = H*P
          Teuchos::SerialDenseMatrix<int,ScalarType> Htmp( Teuchos::View, *H2_, p+1, p, recycledBlocks_+1,recycledBlocks_+1);
          Teuchos::SerialDenseMatrix<int,ScalarType> HPtmp( Teuchos::View, *HP_, p+1, keff );
          HPtmp.multiply( Teuchos::NO_TRANS, Teuchos::NO_TRANS, one, Htmp, *PPtmp, zero );
 
          // Step #1.5: Perform workspace size query for QR
          // factorization of HP.  On input, lwork must be -1.
          // _GEQRF will put the workspace size in work_[0].
          int lwork = -1;
          tau_.resize (keff);
          lapack.GEQRF (HPtmp.numRows (), HPtmp.numCols (), HPtmp.values (),
                        HPtmp.stride (), &tau_[0], &work_[0], lwork, &info);
          TEUCHOS_TEST_FOR_EXCEPTION(
            info != 0, GCRODRSolMgrLAPACKFailure, "Belos::GCRODRSolMgr::solve:"
            " LAPACK's _GEQRF failed to compute a workspace size.");
 
          // Step #2: Compute QR factorization of HP
          //
          // NOTE (mfh 17 Apr 2014) LAPACK promises that the value of
          // work_[0] after the workspace query will fit in int.  This
          // justifies the cast.  We call real() first because
          // static_cast from std::complex to int doesn't work.
          lwork = std::abs (static_cast<int> (Teuchos::ScalarTraits<ScalarType>::real (work_[0])));
          work_.resize (lwork); // Allocate workspace for the QR factorization
          lapack.GEQRF (HPtmp.numRows (), HPtmp.numCols (), HPtmp.values (),
                        HPtmp.stride (), &tau_[0], &work_[0], lwork, &info);
          TEUCHOS_TEST_FOR_EXCEPTION(
            info != 0, GCRODRSolMgrLAPACKFailure,  "Belos::GCRODRSolMgr::solve:"
            " LAPACK's _GEQRF failed to compute a QR factorization.");
 
          // Step #3: Explicitly construct Q and R factors
          // NOTE:  The upper triangular part of HP is copied into R and HP becomes Q.
          Teuchos::SerialDenseMatrix<int,ScalarType> Rtmp( Teuchos::View, *R_, keff, keff );
          for (int ii = 0; ii < keff; ++ii) {
            for (int jj = ii; jj < keff; ++jj) {
              Rtmp(ii,jj) = HPtmp(ii,jj);
            }
          }
          // NOTE (mfh 17 Apr 2014): Teuchos::LAPACK's wrapper for
          // UNGQR dispatches to the correct Scalar-specific routine.
          // It calls {S,D}ORGQR if Scalar is real, and {C,Z}UNGQR if
          // Scalar is complex.
          lapack.UNGQR (HPtmp.numRows (), HPtmp.numCols (), HPtmp.numCols (),
                        HPtmp.values (), HPtmp.stride (), &tau_[0], &work_[0],
                        lwork, &info);
          TEUCHOS_TEST_FOR_EXCEPTION(
            info != 0, GCRODRSolMgrLAPACKFailure, "Belos::GCRODRSolMgr::solve: "
            "LAPACK's _UNGQR failed to construct the Q factor.");
 
          // Now we have [Q,R] = qr(H*P)
 
          // Now compute C = V(:,1:p+1) * Q
          index.resize (p + 1);
          for (int ii = 0; ii < (p+1); ++ii) {
            index[ii] = ii;
          }
          Vtmp = MVT::CloneView( *V_, index ); // need new view into V (p+1 vectors now; needed p above)
          MVT::MvTimesMatAddMv( one, *Vtmp, HPtmp, zero, *Ctmp );
 
          // Finally, compute U = U*R^{-1}.
          // This unfortuntely requires me to form R^{-1} explicitly and execute U = U * R^{-1}, as
          // backsolve capabilities don't exist in the Belos::MultiVec class
 
          // Step #1: First, compute LU factorization of R
          ipiv_.resize(Rtmp.numRows());
          lapack.GETRF(Rtmp.numRows(),Rtmp.numCols(),Rtmp.values(),Rtmp.stride(),&ipiv_[0],&info);
          TEUCHOS_TEST_FOR_EXCEPTION(
            info != 0, GCRODRSolMgrLAPACKFailure, "Belos::GCRODRSolMgr::solve: "
            "LAPACK's _GETRF failed to compute an LU factorization.");
 
          // FIXME (mfh 17 Apr 2014) We have to compute the explicit
          // inverse of R here because Belos::MultiVecTraits doesn't
          // have a triangular solve (where the triangular matrix is
          // globally replicated and the "right-hand side" is the
          // distributed MultiVector).
 
          // Step #2: Form inv(R)
          lwork = Rtmp.numRows();
          work_.resize(lwork);
          lapack.GETRI(Rtmp.numRows(),Rtmp.values(),Rtmp.stride(),&ipiv_[0],&work_[0],lwork,&info);
          TEUCHOS_TEST_FOR_EXCEPTION(
            info != 0, GCRODRSolMgrLAPACKFailure, "Belos::GCRODRSolMgr::solve: "
            "LAPACK's _GETRI failed to invert triangular matrix.");
 
          // Step #3: Let U = U * R^{-1}
          MVT::MvTimesMatAddMv( one, *U1tmp, Rtmp, zero, *Utmp );
 
          printer_->stream(Debug)
            << " Generated recycled subspace using RHS index " << currIdx[0]
            << " of dimension " << keff << std::endl << std::endl;
 
        }  // if (recycledBlocks_ < p+1)
 
        // Return to outer loop if the priming solve converged, set the next linear system.
        if (primeConverged) {
          // Inform the linear problem that we are finished with this block linear system.
          problem_->setCurrLS();
 
          // Update indices for the linear systems to be solved.
          numRHS2Solve -= 1;
          if (numRHS2Solve > 0) {
            currIdx[0]++;
            problem_->setLSIndex (currIdx); // Set the next indices
          }
          else {
            currIdx.resize (numRHS2Solve);
          }
 
          continue;
        }
      } // if (keff > 0) ...
 
      // Prepare for the Gmres iterations with the recycled subspace.
 
      // Set the current number of recycled blocks and subspace dimension with the GCRO-DR iteration.
      gcrodr_iter->setSize( keff, numBlocks_ );
 
      // Reset the number of iterations.
      gcrodr_iter->resetNumIters(prime_iterations);
 
      // Reset the number of calls that the status test output knows about.
      outputTest_->resetNumCalls();
 
      // Compute the residual after the priming solve, it will be the first block in the current Krylov basis.
      problem_->computeCurrPrecResVec( &*r_ );
      index.resize( 1 ); index[0] = 0;
      RCP<MV> v0 =  MVT::CloneViewNonConst( *V_,  index );
      MVT::SetBlock(*r_,index,*v0); // V(:,0) = r
 
      // Set the new state and initialize the solver.
      GCRODRIterState<ScalarType,MV> newstate;
      index.resize( numBlocks_+1 );
      for (int ii=0; ii<(numBlocks_+1); ++ii) { index[ii] = ii; }
      newstate.V  = MVT::CloneViewNonConst( *V_,  index );
      index.resize( keff );
      for (int ii=0; ii<keff; ++ii) { index[ii] = ii; }
      newstate.C  = MVT::CloneViewNonConst( *C_,  index );
      newstate.U  = MVT::CloneViewNonConst( *U_,  index );
      newstate.B = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( Teuchos::View, *H2_,         keff, numBlocks_,    0, keff ) );
      newstate.H = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( Teuchos::View, *H2_, numBlocks_+1, numBlocks_, keff, keff ) );
      newstate.curDim = 0;
      gcrodr_iter->initialize(newstate);
 
      // variables needed for inner loop
      int numRestarts = 0;
      while(1) {
 
        // tell gcrodr_iter to iterate
        try {
          gcrodr_iter->iterate();

          //
          // check convergence first
          //
          if ( convTest_->getStatus() == Passed ) {
            // we have convergence
            break;  // break from while(1){gcrodr_iter->iterate()}
          }
          //
          // check for maximum iterations
          //
          else if ( maxIterTest_->getStatus() == Passed ) {
            // we don't have convergence
            isConverged = false;
            break;  // break from while(1){gcrodr_iter->iterate()}
          }
          //
          // check for restarting, i.e. the subspace is full
          //
          else if ( gcrodr_iter->getCurSubspaceDim() == gcrodr_iter->getMaxSubspaceDim() ) {
 
            // Update the recycled subspace even if we have hit the maximum number of restarts.
 
            // Update the linear problem.
            RCP<MV> update = gcrodr_iter->getCurrentUpdate();
            problem_->updateSolution( update, true );
 
            buildRecycleSpace2(gcrodr_iter);
 
            printer_->stream(Debug)
              << " Generated new recycled subspace using RHS index "
              << currIdx[0] << " of dimension " << keff << std::endl
              << std::endl;
 
            // NOTE:  If we have hit the maximum number of restarts then quit
            if (numRestarts >= maxRestarts_) {
              isConverged = false;
              break; // break from while(1){gcrodr_iter->iterate()}
            }
            numRestarts++;
 
            printer_->stream(Debug)
              << " Performing restart number " << numRestarts << " of "
              << maxRestarts_ << std::endl << std::endl;
 
            // Create the restart vector (first block in the current Krylov basis)
            problem_->computeCurrPrecResVec( &*r_ );
            index.resize( 1 ); index[0] = 0;
            RCP<MV> v00 =  MVT::CloneViewNonConst( *V_,  index );
            MVT::SetBlock(*r_,index,*v00); // V(:,0) = r
 
            // Set the new state and initialize the solver.
            GCRODRIterState<ScalarType,MV> restartState;
            index.resize( numBlocks_+1 );
            for (int ii=0; ii<(numBlocks_+1); ++ii) { index[ii] = ii; }
            restartState.V  = MVT::CloneViewNonConst( *V_,  index );
            index.resize( keff );
            for (int ii=0; ii<keff; ++ii) { index[ii] = ii; }
            restartState.U  = MVT::CloneViewNonConst( *U_,  index );
            restartState.C  = MVT::CloneViewNonConst( *C_,  index );
            restartState.B = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( Teuchos::View, *H2_, keff, numBlocks_, 0, keff ) );
            restartState.H = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( Teuchos::View, *H2_, numBlocks_+1, numBlocks_, keff, keff ) );
            restartState.curDim = 0;
            gcrodr_iter->initialize(restartState);
 
 
          } // end of restarting

          //
          // we returned from iterate(), but none of our status tests Passed.
          // something is wrong, and it is probably our fault.
          //
 
          else {
            TEUCHOS_TEST_FOR_EXCEPTION(
              true, std::logic_error, "Belos::GCRODRSolMgr::solve: "
              "Invalid return from GCRODRIter::iterate().");
          }
        }
        catch (const GCRODRIterOrthoFailure &e) {
          // Try to recover the most recent least-squares solution
          gcrodr_iter->updateLSQR( gcrodr_iter->getCurSubspaceDim() );
 
          // Check to see if the most recent least-squares solution yielded convergence.
          sTest_->checkStatus( &*gcrodr_iter );
          if (convTest_->getStatus() != Passed)
            isConverged = false;
          break;
        }
        catch (const std::exception& e) {
          printer_->stream(Errors)
            << "Error! Caught exception in GCRODRIter::iterate() at iteration "
            << gcrodr_iter->getNumIters() << std::endl << e.what() << std::endl;
          throw;
        }
      }
 
      // Compute the current solution.
      // Update the linear problem.
      RCP<MV> update = gcrodr_iter->getCurrentUpdate();
      problem_->updateSolution( update, true );
 
      // Inform the linear problem that we are finished with this block linear system.
      problem_->setCurrLS();
 
      // If we didn't build a recycle space this solve but ran at least k iterations,
      // force build of new recycle space
 
      if (!builtRecycleSpace_) {
        buildRecycleSpace2(gcrodr_iter);
        printer_->stream(Debug)
          << " Generated new recycled subspace using RHS index " << currIdx[0]
          << " of dimension " << keff << std::endl << std::endl;
      }
 
      // Update indices for the linear systems to be solved.
      numRHS2Solve -= 1;
      if (numRHS2Solve > 0) {
        currIdx[0]++;
        problem_->setLSIndex (currIdx); // Set the next indices
      }
      else {
        currIdx.resize (numRHS2Solve);
      }
    } // while (numRHS2Solve > 0)
  }
 
  sTest_->print (printer_->stream (FinalSummary)); // print final summary
 
  // print timing information
#ifdef BELOS_TEUCHOS_TIME_MONITOR
  // Calling summarize() can be expensive, so don't call unless the
  // user wants to print out timing details.  summarize() will do all
  // the work even if it's passed a "black hole" output stream.
  if (verbosity_ & TimingDetails)
    Teuchos::TimeMonitor::summarize( printer_->stream(TimingDetails) );
#endif // BELOS_TEUCHOS_TIME_MONITOR
 
  // get iteration information for this solve
  numIters_ = maxIterTest_->getNumIters ();
 
  // Save the convergence test value ("achieved tolerance") for this
  // solve.  This solver (unlike BlockGmresSolMgr) always has two
  // residual norm status tests: an explicit and an implicit test.
  // The master convergence test convTest_ is a SEQ combo of the
  // implicit resp. explicit tests.  If the implicit test never
  // passes, then the explicit test won't ever be executed.  This
  // manifests as expConvTest_->getTestValue()->size() < 1.  We deal
  // with this case by using the values returned by
  // impConvTest_->getTestValue().
  {
    const std::vector<MagnitudeType>* pTestValues = expConvTest_->getTestValue();
    if (pTestValues == NULL || pTestValues->size() < 1) {
      pTestValues = impConvTest_->getTestValue();
    }
    TEUCHOS_TEST_FOR_EXCEPTION(pTestValues == NULL, std::logic_error,
      "Belos::GCRODRSolMgr::solve(): The implicit convergence test's getTestValue() "
      "method returned NULL.  Please report this bug to the Belos developers.");
    TEUCHOS_TEST_FOR_EXCEPTION(pTestValues->size() < 1, std::logic_error,
      "Belos::GCRODRSolMgr::solve(): The implicit convergence test's getTestValue() "
      "method returned a vector of length zero.  Please report this bug to the "
      "Belos developers.");
 
    // FIXME (mfh 12 Dec 2011) Does pTestValues really contain the
    // achieved tolerances for all vectors in the current solve(), or
    // just for the vectors from the last deflation?
    achievedTol_ = *std::max_element (pTestValues->begin(), pTestValues->end());
  }
 
  return isConverged ? Converged : Unconverged; // return from solve()
}
 
//  Given existing recycle space and Krylov space, build new recycle space
template<class ScalarType, class MV, class OP>
void GCRODRSolMgr<ScalarType,MV,OP,true>::buildRecycleSpace2(Teuchos::RCP<GCRODRIter<ScalarType,MV,OP> > gcrodr_iter) {
 
  MagnitudeType one = Teuchos::ScalarTraits<MagnitudeType>::one();
  ScalarType zero = Teuchos::ScalarTraits<ScalarType>::zero();
 
  std::vector<MagnitudeType> d(keff);
  std::vector<ScalarType> dscalar(keff);
  std::vector<int> index(numBlocks_+1);
 
  // Get the state
  GCRODRIterState<ScalarType,MV> oldState = gcrodr_iter->getState();
  int p = oldState.curDim;
 
  // insufficient new information to update recycle space
  if (p<1) return;
 
  // Take the norm of the recycled vectors
  {
    index.resize(keff);
    for (int ii=0; ii<keff; ++ii) { index[ii] = ii; }
    Teuchos::RCP<MV> Utmp  = MVT::CloneViewNonConst( *U_, index );
    d.resize(keff);
    dscalar.resize(keff);
    MVT::MvNorm( *Utmp, d );
    for (int i=0; i<keff; ++i) {
      d[i] = one / d[i];
      dscalar[i] = (ScalarType)d[i];
    }
    MVT::MvScale( *Utmp, dscalar );
  }
 
  // Get view into current "full" upper Hessnburg matrix
  Teuchos::RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > H2tmp = Teuchos::rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>( Teuchos::View, *H2_, p+keff+1, p+keff ) );
 
  // Insert D into the leading keff x keff  block of H2
  for (int i=0; i<keff; ++i) {
    (*H2tmp)(i,i) = d[i];
  }
 
  // Compute the harmoic Ritz pairs for the generalized eigenproblem
  // getHarmonicVecs2 assumes PP has recycledBlocks_+1 columns available
  int keff_new;
  {
    Teuchos::SerialDenseMatrix<int,ScalarType> PPtmp( Teuchos::View, *PP_, p+keff, recycledBlocks_+1 );
    keff_new = getHarmonicVecs2( keff, p, *H2tmp, oldState.V, PPtmp );
  }
 
  // Code to form new U, C
  // U = [U V(:,1:p)] * P; (in two steps)
 
  // U(:,1:keff) = matmul(U(:,1:keff_old),PP(1:keff_old,1:keff)) (step 1)
  Teuchos::RCP<MV> U1tmp;
  {
    index.resize( keff );
    for (int ii=0; ii<keff; ++ii) { index[ii] = ii; }
    Teuchos::RCP<const MV> Utmp  = MVT::CloneView( *U_,  index );
    index.resize( keff_new );
    for (int ii=0; ii<keff_new; ++ii) { index[ii] = ii; }
    U1tmp  = MVT::CloneViewNonConst( *U1_,  index );
    Teuchos::SerialDenseMatrix<int,ScalarType> PPtmp( Teuchos::View, *PP_, keff, keff_new );
    MVT::MvTimesMatAddMv( one, *Utmp, PPtmp, zero, *U1tmp );
  }
 
  // U(:,1:keff) = U(:,1:keff) + matmul(V(:,1:m-k),PP(keff_old+1:m-k+keff_old,1:keff)) (step 2)
  {
    index.resize(p);
    for (int ii=0; ii < p; ii++) { index[ii] = ii; }
    Teuchos::RCP<const MV> Vtmp = MVT::CloneView( *V_, index );
    Teuchos::SerialDenseMatrix<int,ScalarType> PPtmp( Teuchos::View, *PP_, p, keff_new, keff );
    MVT::MvTimesMatAddMv( one, *Vtmp, PPtmp, one, *U1tmp );
  }
 
  // Form HP = H*P
  Teuchos::SerialDenseMatrix<int,ScalarType> HPtmp( Teuchos::View, *HP_, p+keff+1, keff_new );
  {
    Teuchos::SerialDenseMatrix<int,ScalarType> PPtmp( Teuchos::View, *PP_, p+keff, keff_new );
    HPtmp.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,one,*H2tmp,PPtmp,zero);
  }
 
  // Workspace size query for QR factorization of HP (the worksize will be placed in work_[0])
  int info = 0, lwork = -1;
  tau_.resize (keff_new);
  lapack.GEQRF (HPtmp.numRows (), HPtmp.numCols (), HPtmp.values (),
                HPtmp.stride (), &tau_[0], &work_[0], lwork, &info);
  TEUCHOS_TEST_FOR_EXCEPTION(
    info != 0, GCRODRSolMgrLAPACKFailure, "Belos::GCRODRSolMgr::solve: "
    "LAPACK's _GEQRF failed to compute a workspace size.");
 
  // NOTE (mfh 18 Apr 2014) LAPACK promises that the value of work_[0]
  // after the workspace query will fit in int.  This justifies the
  // cast.  We call real() first because static_cast from std::complex
  // to int doesn't work.
  lwork = std::abs (static_cast<int> (Teuchos::ScalarTraits<ScalarType>::real (work_[0])));
  work_.resize (lwork); // Allocate workspace for the QR factorization
  lapack.GEQRF (HPtmp.numRows (), HPtmp.numCols (), HPtmp.values (),
                HPtmp.stride (), &tau_[0], &work_[0], lwork, &info);
  TEUCHOS_TEST_FOR_EXCEPTION(
    info != 0, GCRODRSolMgrLAPACKFailure, "Belos::GCRODRSolMgr::solve: "
    "LAPACK's _GEQRF failed to compute a QR factorization.");
 
  // Explicitly construct Q and R factors
  // NOTE:  The upper triangular part of HP is copied into R and HP becomes Q.
  Teuchos::SerialDenseMatrix<int,ScalarType> Rtmp( Teuchos::View, *R_, keff_new, keff_new );
  for(int i=0;i<keff_new;i++) { for(int j=i;j<keff_new;j++) Rtmp(i,j) = HPtmp(i,j); }
 
  // NOTE (mfh 18 Apr 2014): Teuchos::LAPACK's wrapper for UNGQR
  // dispatches to the correct Scalar-specific routine.  It calls
  // {S,D}ORGQR if Scalar is real, and {C,Z}UNGQR if Scalar is
  // complex.
  lapack.UNGQR (HPtmp.numRows (), HPtmp.numCols (), HPtmp.numCols (),
                HPtmp.values (), HPtmp.stride (), &tau_[0], &work_[0],
                lwork, &info);
  TEUCHOS_TEST_FOR_EXCEPTION(
    info != 0, GCRODRSolMgrLAPACKFailure, "Belos::GCRODRSolMgr::solve: "
    "LAPACK's _UNGQR failed to construct the Q factor.");
 
  // Form orthonormalized C and adjust U accordingly so that C = A*U
  // C = [C V] * Q;
 
  // C(:,1:keff) = matmul(C(:,1:keff_old),QQ(1:keff_old,1:keff))
  {
    Teuchos::RCP<MV> C1tmp;
    {
      index.resize(keff);
      for (int i=0; i < keff; i++) { index[i] = i; }
      Teuchos::RCP<const MV> Ctmp  = MVT::CloneView( *C_,  index );
      index.resize(keff_new);
      for (int i=0; i < keff_new; i++) { index[i] = i; }
      C1tmp  = MVT::CloneViewNonConst( *C1_,  index );
      Teuchos::SerialDenseMatrix<int,ScalarType> PPtmp( Teuchos::View, *HP_, keff, keff_new );
      MVT::MvTimesMatAddMv( one, *Ctmp, PPtmp, zero, *C1tmp );
    }
    // Now compute C += V(:,1:p+1) * Q
    {
      index.resize( p+1 );
      for (int i=0; i < p+1; ++i) { index[i] = i; }
      Teuchos::RCP<const MV> Vtmp = MVT::CloneView( *V_, index );
      Teuchos::SerialDenseMatrix<int,ScalarType> PPtmp( Teuchos::View, *HP_, p+1, keff_new, keff, 0 );
      MVT::MvTimesMatAddMv( one, *Vtmp, PPtmp, one, *C1tmp );
    }
  }
 
  // C_ = C1_; (via a swap)
  std::swap(C_, C1_);
 
  // Finally, compute U_ = U_*R^{-1}
  // First, compute LU factorization of R
  ipiv_.resize(Rtmp.numRows());
  lapack.GETRF(Rtmp.numRows(),Rtmp.numCols(),Rtmp.values(),Rtmp.stride(),&ipiv_[0],&info);
  TEUCHOS_TEST_FOR_EXCEPTION(info != 0,GCRODRSolMgrLAPACKFailure,"Belos::GCRODRSolMgr::solve(): LAPACK _GETRF failed to compute an LU factorization.");
 
  // Now, form inv(R)
  lwork = Rtmp.numRows();
  work_.resize(lwork);
  lapack.GETRI(Rtmp.numRows(),Rtmp.values(),Rtmp.stride(),&ipiv_[0],&work_[0],lwork,&info);
  TEUCHOS_TEST_FOR_EXCEPTION(info != 0, GCRODRSolMgrLAPACKFailure,"Belos::GCRODRSolMgr::solve(): LAPACK _GETRI failed to compute an LU factorization.");
 
  {
    index.resize(keff_new);
    for (int i=0; i < keff_new; i++) { index[i] = i; }
    Teuchos::RCP<MV> Utmp  = MVT::CloneViewNonConst( *U_,  index );
    MVT::MvTimesMatAddMv( one, *U1tmp, Rtmp, zero, *Utmp );
  }
 
  // Set the current number of recycled blocks and subspace dimension with the GCRO-DR iteration.
  if (keff != keff_new) {
    keff = keff_new;
    gcrodr_iter->setSize( keff, numBlocks_ );
    // Important to zero this out before next cyle
    Teuchos::SerialDenseMatrix<int,ScalarType> b1( Teuchos::View, *H2_, recycledBlocks_+2, 1, 0, recycledBlocks_ );
    b1.putScalar(zero);
  }
 
}
 
 
//  Compute the harmonic eigenpairs of the projected, dense system.
template<class ScalarType, class MV, class OP>
int GCRODRSolMgr<ScalarType,MV,OP,true>::getHarmonicVecs1(int m,
                                                     const Teuchos::SerialDenseMatrix<int,ScalarType>& HH,
                                                     Teuchos::SerialDenseMatrix<int,ScalarType>& PP) {
  int i, j;
  bool xtraVec = false;
  ScalarType one = Teuchos::ScalarTraits<ScalarType>::one();
 
  // Real and imaginary eigenvalue components
  std::vector<MagnitudeType> wr(m), wi(m);
 
  // Real and imaginary (right) eigenvectors; Don't zero out matrix when constructing
  Teuchos::SerialDenseMatrix<int,ScalarType> vr(m,m,false);
 
  // Magnitude of harmonic Ritz values
  std::vector<MagnitudeType> w(m);
 
  // Sorted order of harmonic Ritz values, also used for GEEV
  std::vector<int> iperm(m);
 
  // Output info
  int info = 0;
 
  // Set flag indicating recycle space has been generated this solve
  builtRecycleSpace_ = true;
 
  // Solve linear system:  H_m^{-H}*e_m
  Teuchos::SerialDenseMatrix<int, ScalarType> HHt( HH, Teuchos::TRANS );
  Teuchos::SerialDenseVector<int, ScalarType> e_m( m );
  e_m[m-1] = one;
  lapack.GESV(m, 1, HHt.values(), HHt.stride(), &iperm[0], e_m.values(), e_m.stride(), &info);
  TEUCHOS_TEST_FOR_EXCEPTION(info != 0, GCRODRSolMgrLAPACKFailure, "Belos::GCRODRSolMgr::solve(): LAPACK GESV failed to compute a solution.");
 
  // Compute H_m + d*H_m^{-H}*e_m*e_m^H
  ScalarType d = HH(m, m-1) * HH(m, m-1);
  Teuchos::SerialDenseMatrix<int, ScalarType> harmHH( Teuchos::Copy, HH, m, m );
  for( i=0; i<m; ++i )
    harmHH(i, m-1) += d * e_m[i];
 
  // Revise to do query for optimal workspace first
  // Create simple storage for the left eigenvectors, which we don't care about.
  const int ldvl = 1;
  ScalarType* vl = 0;
 
  // Size of workspace and workspace for GEEV
  int lwork = -1;
  std::vector<ScalarType> work(1);
  std::vector<MagnitudeType> rwork(2*m);
 
  // First query GEEV for the optimal workspace size
  lapack.GEEV('N', 'V', m, harmHH.values(), harmHH.stride(), &wr[0], &wi[0],
              vl, ldvl, vr.values(), vr.stride(), &work[0], lwork, &rwork[0], &info);
 
  lwork = std::abs (static_cast<int> (Teuchos::ScalarTraits<ScalarType>::real (work[0])));
  work.resize( lwork );
 
  lapack.GEEV('N', 'V', m, harmHH.values(), harmHH.stride(), &wr[0], &wi[0],
              vl, ldvl, vr.values(), vr.stride(), &work[0], lwork, &rwork[0], &info);
  TEUCHOS_TEST_FOR_EXCEPTION(info != 0, GCRODRSolMgrLAPACKFailure,"Belos::GCRODRSolMgr::solve(): LAPACK GEEV failed to compute eigensolutions.");
 
  // Construct magnitude of each harmonic Ritz value
  for( i=0; i<m; ++i )
    w[i] = Teuchos::ScalarTraits<MagnitudeType>::squareroot( wr[i]*wr[i] + wi[i]*wi[i] );
 
  // Construct magnitude of each harmonic Ritz value
  this->sort(w, m, iperm);
 
  const bool scalarTypeIsComplex = Teuchos::ScalarTraits<ScalarType>::isComplex;
 
  // Select recycledBlocks_ smallest eigenvectors
  for( i=0; i<recycledBlocks_; ++i ) {
    for( j=0; j<m; j++ ) {
      PP(j,i) = vr(j,iperm[i]);
    }
  }
 
  if(!scalarTypeIsComplex) {
 
    // Determine exact size for PP (i.e., determine if we need to store an additional vector)
    if (wi[iperm[recycledBlocks_-1]] != 0.0) {
      int countImag = 0;
      for ( i=0; i<recycledBlocks_; ++i ) {
        if (wi[iperm[i]] != 0.0)
          countImag++;
      }
      // Check to see if this count is even or odd:
      if (countImag % 2)
        xtraVec = true;
    }
 
    if (xtraVec) { // we need to store one more vector
      if (wi[iperm[recycledBlocks_-1]] > 0.0) { // I picked the "real" component
        for( j=0; j<m; ++j ) {   // so get the "imag" component
          PP(j,recycledBlocks_) = vr(j,iperm[recycledBlocks_-1]+1);
        }
      }
      else { //  I picked the "imag" component
        for( j=0; j<m; ++j ) {   // so get the "real" component
          PP(j,recycledBlocks_) = vr(j,iperm[recycledBlocks_-1]-1);
        }
      }
    }
 
  }
 
  // Return whether we needed to store an additional vector
  if (xtraVec) {
    return recycledBlocks_+1;
  }
  else {
    return recycledBlocks_;
  }
 
}
 
//  Compute the harmonic eigenpairs of the projected, dense system.
template<class ScalarType, class MV, class OP>
int GCRODRSolMgr<ScalarType,MV,OP,true>::getHarmonicVecs2(int keffloc, int m,
                                                     const Teuchos::SerialDenseMatrix<int,ScalarType>& HH,
                                                     const Teuchos::RCP<const MV>& VV,
                                                     Teuchos::SerialDenseMatrix<int,ScalarType>& PP) {
  int i, j;
  int m2 = HH.numCols();
  bool xtraVec = false;
  ScalarType one = Teuchos::ScalarTraits<ScalarType>::one();
  ScalarType zero = Teuchos::ScalarTraits<ScalarType>::zero();
  std::vector<int> index;
 
  // Real and imaginary eigenvalue components
  std::vector<MagnitudeType> wr(m2), wi(m2);
 
  // Magnitude of harmonic Ritz values
  std::vector<MagnitudeType> w(m2);
 
  // Real and imaginary (right) eigenvectors; Don't zero out matrix when constructing
  Teuchos::SerialDenseMatrix<int,ScalarType> vr(m2,m2,false);
 
  // Sorted order of harmonic Ritz values
  std::vector<int> iperm(m2);
 
  // Set flag indicating recycle space has been generated this solve
  builtRecycleSpace_ = true;
 
  // Form matrices for generalized eigenproblem
 
  // B = H2' * H2; Don't zero out matrix when constructing
  Teuchos::SerialDenseMatrix<int,ScalarType> B(m2,m2,false);
  B.multiply(Teuchos::TRANS,Teuchos::NO_TRANS,one,HH,HH,zero);
 
  // A_tmp = | C'*U        0 |
  //         | V_{m+1}'*U  I |
  Teuchos::SerialDenseMatrix<int,ScalarType> A_tmp( keffloc+m+1, keffloc+m );
 
  // A_tmp(1:keffloc,1:keffloc) = C' * U;
  index.resize(keffloc);
  for (i=0; i<keffloc; ++i) { index[i] = i; }
  Teuchos::RCP<const MV> Ctmp  = MVT::CloneView( *C_, index );
  Teuchos::RCP<const MV> Utmp  = MVT::CloneView( *U_, index );
  Teuchos::SerialDenseMatrix<int,ScalarType> A11( Teuchos::View, A_tmp, keffloc, keffloc );
  MVT::MvTransMv( one, *Ctmp, *Utmp, A11 );
 
  // A_tmp(keffloc+1:m-k+keffloc+1,1:keffloc) = V' * U;
  Teuchos::SerialDenseMatrix<int,ScalarType> A21( Teuchos::View, A_tmp, m+1, keffloc, keffloc );
  index.resize(m+1);
  for (i=0; i < m+1; i++) { index[i] = i; }
  Teuchos::RCP<const MV> Vp = MVT::CloneView( *VV, index );
  MVT::MvTransMv( one, *Vp, *Utmp, A21 );
 
  // A_tmp(keffloc+1:m-k+keffloc,keffloc+1:m-k+keffloc) = eye(m-k);
  for( i=keffloc; i<keffloc+m; i++ ) {
    A_tmp(i,i) = one;
  }
 
  // A = H2' * A_tmp;
  Teuchos::SerialDenseMatrix<int,ScalarType> A( m2, A_tmp.numCols() );
  A.multiply( Teuchos::TRANS, Teuchos::NO_TRANS, one, HH, A_tmp, zero );
 
  // Compute k smallest harmonic Ritz pairs
  // SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
  //                   ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
  //                   IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
  //                   RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
  // MLP: 'SCALING' in DGGEVX generates incorrect eigenvalues. Therefore, only permuting
  char balanc='P', jobvl='N', jobvr='V', sense='N';
  int ld = A.numRows();
  int lwork = 6*ld;
  int ldvl = ld, ldvr = ld;
  int info = 0,ilo = 0,ihi = 0;
  MagnitudeType abnrm = 0.0, bbnrm = 0.0;
  ScalarType *vl = 0; // This is never referenced by dggevx if jobvl == 'N'
  std::vector<ScalarType> beta(ld);
  std::vector<ScalarType> work(lwork);
  std::vector<MagnitudeType> rwork(lwork);
  std::vector<MagnitudeType> lscale(ld), rscale(ld);
  std::vector<MagnitudeType> rconde(ld), rcondv(ld);
  std::vector<int> iwork(ld+6);
  int *bwork = 0; // If sense == 'N', bwork is never referenced
  //lapack.GGEVX(balanc, jobvl, jobvr, sense, ld, A.values(), ld, B.values(), ld, &wr[0], &wi[0],
  //             &beta[0], vl, ldvl, vr.values(), ldvr, &ilo, &ihi, &lscale[0], &rscale[0],
  //             &abnrm, &bbnrm, &rconde[0], &rcondv[0], &work[0], lwork, &iwork[0], bwork, &info);
  lapack.GGEVX(balanc, jobvl, jobvr, sense, ld, A.values(), ld, B.values(), ld, &wr[0], &wi[0],
               &beta[0], vl, ldvl, vr.values(), ldvr, &ilo, &ihi, &lscale[0], &rscale[0],
               &abnrm, &bbnrm, &rconde[0], &rcondv[0], &work[0], lwork, &rwork[0],
               &iwork[0], bwork, &info);
  TEUCHOS_TEST_FOR_EXCEPTION(info != 0, GCRODRSolMgrLAPACKFailure, "Belos::GCRODRSolMgr::solve(): LAPACK GGEVX failed to compute eigensolutions.");
 
  // Construct magnitude of each harmonic Ritz value
  // NOTE : Forming alpha/beta *should* be okay here, given assumptions on construction of matrix pencil above
  for( i=0; i<ld; i++ ) {
    w[i] = Teuchos::ScalarTraits<MagnitudeType>::squareroot (wr[i]*wr[i] + wi[i]*wi[i]) /
      Teuchos::ScalarTraits<ScalarType>::magnitude (beta[i]);
  }
 
  // Construct magnitude of each harmonic Ritz value
  this->sort(w,ld,iperm);
 
  const bool scalarTypeIsComplex = Teuchos::ScalarTraits<ScalarType>::isComplex;
 
  // Select recycledBlocks_ smallest eigenvectors
  for( i=0; i<recycledBlocks_; i++ ) {
    for( j=0; j<ld; j++ ) {
      PP(j,i) = vr(j,iperm[ld-recycledBlocks_+i]);
    }
  }
 
  if(!scalarTypeIsComplex) {
 
    // Determine exact size for PP (i.e., determine if we need to store an additional vector)
    if (wi[iperm[ld-recycledBlocks_]] != 0.0) {
      int countImag = 0;
      for ( i=ld-recycledBlocks_; i<ld; i++ ) {
        if (wi[iperm[i]] != 0.0)
          countImag++;
      }
      // Check to see if this count is even or odd:
      if (countImag % 2)
        xtraVec = true;
    }
 
    if (xtraVec) { // we need to store one more vector
      if (wi[iperm[ld-recycledBlocks_]] > 0.0) { // I picked the "real" component
        for( j=0; j<ld; j++ ) {   // so get the "imag" component
          PP(j,recycledBlocks_) = vr(j,iperm[ld-recycledBlocks_]+1);
        }
      }
      else { // I picked the "imag" component
        for( j=0; j<ld; j++ ) {   // so get the "real" component
          PP(j,recycledBlocks_) = vr(j,iperm[ld-recycledBlocks_]-1);
        }
      }
    }
 
  }
 
  // Return whether we needed to store an additional vector
  if (xtraVec) {
    return recycledBlocks_+1;
  }
  else {
    return recycledBlocks_;
  }
 
}
 
 
// This method sorts list of n floating-point numbers and return permutation vector
template<class ScalarType, class MV, class OP>
void GCRODRSolMgr<ScalarType,MV,OP,true>::sort(std::vector<MagnitudeType>& dlist, int n, std::vector<int>& iperm) {
  int l, r, j, i, flag;
  int    RR2;
  MagnitudeType dRR, dK;
 
  // Initialize the permutation vector.
  for(j=0;j<n;j++)
    iperm[j] = j;
 
  if (n <= 1) return;
 
  l    = n / 2 + 1;
  r    = n - 1;
  l    = l - 1;
  dRR  = dlist[l - 1];
  dK   = dlist[l - 1];
 
  RR2 = iperm[l - 1];
  while (r != 0) {
    j = l;
    flag = 1;
 
    while (flag == 1) {
      i = j;
      j = j + j;
 
      if (j > r + 1)
        flag = 0;
      else {
        if (j < r + 1)
          if (dlist[j] > dlist[j - 1]) j = j + 1;
 
        if (dlist[j - 1] > dK) {
          dlist[i - 1] = dlist[j - 1];
          iperm[i - 1] = iperm[j - 1];
        }
        else {
          flag = 0;
        }
      }
    }
    dlist[i - 1] = dRR;
    iperm[i - 1] = RR2;
 
    if (l == 1) {
      dRR  = dlist [r];
      RR2 = iperm[r];
      dK = dlist[r];
      dlist[r] = dlist[0];
      iperm[r] = iperm[0];
      r = r - 1;
    }
    else {
      l   = l - 1;
      dRR  = dlist[l - 1];
      RR2  = iperm[l - 1];
      dK   = dlist[l - 1];
    }
  }
  dlist[0] = dRR;
  iperm[0] = RR2;
}
 
 
template<class ScalarType, class MV, class OP>
std::string GCRODRSolMgr<ScalarType,MV,OP,true>::description () const {
  std::ostringstream out;
  out << "Belos::GCRODRSolMgr<...,"<<Teuchos::ScalarTraits<ScalarType>::name()<<">";
  out << "{";
  out << "Ortho Type: \"" << orthoType_ << "\"";
  out << ", Num Blocks: " <<numBlocks_;
  out << ", Num Recycle Blocks: " << recycledBlocks_;
  out << ", Max Restarts: " << maxRestarts_;
  out << "}";
  return out.str ();
}
 
} // namespace Belos
 
#endif /* BELOS_GCRODR_SOLMGR_HPP */