AnasaziTraceMinBase.hpp

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//                 Anasazi: Block Eigensolvers Package
//                 Copyright (2004) Sandia Corporation
//
// Under terms of Contract DE-AC04-94AL85000, there is a non-exclusive
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// TODO: Modify code so R is not computed unless needed

#ifndef ANASAZI_TRACEMIN_BASE_HPP
#define ANASAZI_TRACEMIN_BASE_HPP

#include "AnasaziBasicSort.hpp"
#include "AnasaziConfigDefs.hpp"
#include "AnasaziEigensolver.hpp"
#include "AnasaziMatOrthoManager.hpp"
#include "AnasaziMultiVecTraits.hpp"
#include "AnasaziOperatorTraits.hpp"
#include "AnasaziSaddleOperator.hpp"
#include "AnasaziSaddleContainer.hpp"
#include "AnasaziSolverUtils.hpp"
#include "AnasaziTraceMinRitzOp.hpp"
#include "AnasaziTraceMinTypes.hpp"
#include "AnasaziTypes.hpp"

#include "Teuchos_ParameterList.hpp"
#include "Teuchos_ScalarTraits.hpp"
#include "Teuchos_SerialDenseMatrix.hpp"
#include "Teuchos_TimeMonitor.hpp"

using Teuchos::RCP;
using Teuchos::rcp;

namespace Anasazi {
namespace Experimental {



  template <class ScalarType, class MV>
  struct TraceMinBaseState {
    int curDim;
    RCP<const MV> V;
    RCP<const MV> KV;
    RCP<const MV> MopV;
    RCP<const MV> X; 
    RCP<const MV> KX; 
    RCP<const MV> MX;
    RCP<const MV> R;
    RCP<const std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> > T;
    RCP<const Teuchos::SerialDenseMatrix<int,ScalarType> > KK;
    RCP<const Teuchos::SerialDenseMatrix<int,ScalarType> > RV;
    bool isOrtho;
    int NEV;
    ScalarType largestSafeShift;
    RCP< const std::vector<ScalarType> > ritzShifts;
    TraceMinBaseState() : curDim(0), V(Teuchos::null), KV(Teuchos::null), MopV(Teuchos::null),
                          X(Teuchos::null), KX(Teuchos::null), MX(Teuchos::null), R(Teuchos::null), 
                          T(Teuchos::null), KK(Teuchos::null), RV(Teuchos::null), isOrtho(false), 
                          NEV(0), largestSafeShift(Teuchos::ScalarTraits<ScalarType>::zero()), 
                          ritzShifts(Teuchos::null) {}
  };




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

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

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

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




    void iterate();

    void initialize(TraceMinBaseState<ScalarType,MV> newstate);

    void initialize();

    bool isInitialized() const;

    TraceMinBaseState<ScalarType,MV> getState() const;
    




    int getNumIters() const;

    void resetNumIters();

    RCP<const MV> getRitzVectors();

    std::vector<Value<ScalarType> > getRitzValues();


    std::vector<int> getRitzIndex();


    std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> getResNorms();


    std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> getRes2Norms();


    std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> getRitzRes2Norms();

    int getCurSubspaceDim() const;

    int getMaxSubspaceDim() const;





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

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

    const Eigenproblem<ScalarType,MV,OP>& getProblem() const;

    void setBlockSize(int blockSize);

    int getBlockSize() const;

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

    Teuchos::Array<RCP<const MV> > getAuxVecs() const;




    void setSize(int blockSize, int numBlocks);



    void currentStatus(std::ostream &os);


  protected:
    //
    // Convenience typedefs
    //
    typedef SolverUtils<ScalarType,MV,OP>      Utils;
    typedef MultiVecTraits<ScalarType,MV>      MVT;
    typedef MultiVecTraitsExt<ScalarType,MV>   MVText;
    typedef OperatorTraits<ScalarType,MV,OP>   OPT;
    typedef Teuchos::ScalarTraits<ScalarType>  SCT;
    typedef typename SCT::magnitudeType        MagnitudeType;
    const MagnitudeType ONE;  
    const MagnitudeType ZERO; 
    const MagnitudeType NANVAL;
    //
    // Classes inputed through constructor that define the eigenproblem to be solved.
    //
    const RCP<Eigenproblem<ScalarType,MV,OP> >     problem_;
    const RCP<SortManager<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> > sm_;
    const RCP<OutputManager<ScalarType> >          om_;
    RCP<StatusTest<ScalarType,MV,OP> >             tester_;
    const RCP<MatOrthoManager<ScalarType,MV,OP> >  orthman_;
    //
    // Information obtained from the eigenproblem
    //
    RCP<const OP> Op_;
    RCP<const OP> MOp_;
    RCP<const OP> Prec_;
    bool hasM_;
    //
    // Internal timers
    // TODO: Fix the timers
    //
    RCP<Teuchos::Time> timerOp_, timerMOp_, timerSaddle_, timerSortEval_, timerDS_,
                                timerLocal_, timerCompRes_, timerOrtho_, timerInit_;
    //
    // Internal structs
    // TODO: Fix the checks
    //
    struct CheckList {
      bool checkV, checkX, checkMX, 
           checkKX, checkQ, checkKK;
      CheckList() : checkV(false),checkX(false),
                    checkMX(false),checkKX(false),
                    checkQ(false),checkKK(false) {};
    };
    //
    // Internal methods
    //
    std::string accuracyCheck(const CheckList &chk, const std::string &where) const;

    //
    // Counters
    //
    int count_ApplyOp_, count_ApplyM_;

    //
    // Algorithmic parameters.
    //
    // blockSize_ is the solver block size; it controls the number of vectors added to the basis on each iteration.
    int blockSize_;
    // numBlocks_ is the size of the allocated space for the basis, in blocks.
    int numBlocks_; 
    
    // 
    // Current solver state
    //
    // initialized_ specifies that the basis vectors have been initialized and the iterate() routine
    // is capable of running; _initialize is controlled  by the initialize() member method
    // For the implications of the state of initialized_, please see documentation for initialize()
    bool initialized_;
    //
    // curDim_ reflects how much of the current basis is valid 
    // NOTE: 0 <= curDim_ <= blockSize_*numBlocks_
    // this also tells us how many of the values in theta_ are valid Ritz values
    int curDim_;
    //
    // State Multivecs
    // V is the current basis
    // X is the current set of eigenvectors
    // R is the residual
    RCP<MV> X_, KX_, MX_, KV_, MV_, R_, V_;
    //
    // Projected matrices
    //
    RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > KK_, ritzVecs_;
    // 
    // auxiliary vectors
    Teuchos::Array<RCP<const MV> > auxVecs_, MauxVecs_;
    int numAuxVecs_;
    //
    // Number of iterations that have been performed.
    int iter_;
    // 
    // Current eigenvalues, residual norms
    std::vector<MagnitudeType> theta_, Rnorms_, R2norms_;
    // 
    // are the residual norms current with the residual?
    bool Rnorms_current_, R2norms_current_;

    // TraceMin specific variables
    RCP<TraceMinRitzOp<ScalarType,MV,OP> > ritzOp_;    // Operator which incorporates Ritz shifts
    enum SaddleSolType saddleSolType_;                          // Type of saddle point solver being used
    bool previouslyLeveled_;                                    // True if the trace already leveled off
    MagnitudeType previousTrace_;                               // Trace of X'KX at the previous iteration
    bool posSafeToShift_, negSafeToShift_;                      // Whether there are multiple clusters
    MagnitudeType largestSafeShift_;                            // The largest shift considered to be safe - generally the biggest converged eigenvalue
    int NEV_;                                                   // Number of eigenvalues we seek - used in computation of trace
    std::vector<ScalarType> ritzShifts_;                        // Current Ritz shifts

    // This is only used if we're using the Schur complement solver
    RCP<MV> Z_;

    // User provided TraceMin parameters
    enum WhenToShiftType whenToShift_;                          // What triggers a Ritz shift
    enum HowToShiftType howToShift_;                            // Strategy for choosing the Ritz shifts
    bool useMultipleShifts_;                                    // Whether to use one Ritz shift or many
    bool considerClusters_;                                     // Don't shift if there is only one cluster
    bool projectAllVecs_;                                       // Use all vectors in projected minres, or just 1
    bool projectLockedVecs_;                                    // Project locked vectors too in minres; does nothing if projectAllVecs = false
    bool computeAllRes_;                                        // Compute all residuals, or just blocksize ones - helps make Ritz shifts safer
    bool useRHSR_;                                              // Use -R as the RHS of projected minres rather than AX
    MagnitudeType traceThresh_;

    // Variables that are only used if we're shifting when the residual gets small
    // TODO: These are currently unused
    std::string shiftNorm_;                                     // Measure 2-norm or M-norm of residual
    MagnitudeType shiftThresh_;                                 // How small must the residual be?
    bool useRelShiftThresh_;                                    // Are we scaling the shift threshold by the eigenvalues?

    // TraceMin specific functions
    // Returns the trace of KK = X'KX
    ScalarType getTrace() const;
    // Returns true if the change in trace is very small (or was very small at one point)
    // TODO: Check whether I want to return true if the trace WAS small
    bool traceLeveled();
    // Get the residuals of each cluster of eigenvalues
    // TODO: Figure out whether I want to use these for all eigenvalues or just the first
    std::vector<ScalarType> getClusterResids();
    // Computes the Ritz shifts, which is not a trivial task
    // TODO: Make it safer for indefinite matrices maybe?
    void computeRitzShifts(const std::vector<ScalarType>& clusterResids);
    // Compute the tolerance for the inner solves
    // TODO: Come back to this and make sure it does what I want it to
    std::vector<ScalarType> computeTol();
    // Solves a saddle point problem
    void solveSaddlePointProblem(RCP<MV> Delta);
    // Solves a saddle point problem by using unpreconditioned projected minres
    void solveSaddleProj(RCP<MV> Delta) const;
    // Solves a saddle point problem by using preconditioned projected...Krylov...something
    // TODO: Fix this.  Replace minres with gmres or something.
    void solveSaddleProjPrec(RCP<MV> Delta) const;
    // Solves a saddle point problem by explicitly forming the inexact Schur complement
    void solveSaddleSchur (RCP<MV> Delta) const;
    // Solves a saddle point problem with a block diagonal preconditioner
    void solveSaddleBDPrec (RCP<MV> Delta) const; 
    // Computes KK = X'KX
    void computeKK();
    // Computes the eigenpairs of KK
    void computeRitzPairs();
    // Computes X = V evecs
    void computeX();
    // Updates KX and MX without a matvec by MAGIC (or by multiplying KV and MV by evecs)
    void updateKXMX();
    // Updates the residual
    void updateResidual();
    // Adds Delta to the basis
    // TraceMin and TraceMinDavidson each do this differently, which is why this is pure virtual
    virtual void addToBasis(const RCP<const MV> Delta) =0;
  };

  //
  // Implementations
  //


  // Constructor
  // TODO: Allow the users to supply their own linear solver
  // TODO: Add additional checking for logic errors (like trying to use gmres with multiple shifts)
  template <class ScalarType, class MV, class OP>
  TraceMinBase<ScalarType,MV,OP>::TraceMinBase(
        const RCP<Eigenproblem<ScalarType,MV,OP> >    &problem, 
        const RCP<SortManager<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> > &sorter,
        const RCP<OutputManager<ScalarType> >         &printer,
        const RCP<StatusTest<ScalarType,MV,OP> >      &tester,
        const RCP<MatOrthoManager<ScalarType,MV,OP> > &ortho,
        Teuchos::ParameterList &params
        ) :
    ONE(Teuchos::ScalarTraits<MagnitudeType>::one()),
    ZERO(Teuchos::ScalarTraits<MagnitudeType>::zero()),
    NANVAL(Teuchos::ScalarTraits<MagnitudeType>::nan()),
    // problem, tools
    problem_(problem), 
    sm_(sorter),
    om_(printer),
    tester_(tester),
    orthman_(ortho),
    // timers, counters
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
    timerOp_(Teuchos::TimeMonitor::getNewTimer("Anasazi: TraceMinBase::Operation Op*x")),
    timerMOp_(Teuchos::TimeMonitor::getNewTimer("Anasazi: TraceMinBase::Operation M*x")),
    timerSaddle_(Teuchos::TimeMonitor::getNewTimer("Anasazi: TraceMinBase::Solving saddle point problem")),
    timerSortEval_(Teuchos::TimeMonitor::getNewTimer("Anasazi: TraceMinBase::Sorting eigenvalues")),
    timerDS_(Teuchos::TimeMonitor::getNewTimer("Anasazi: TraceMinBase::Direct solve")),
    timerLocal_(Teuchos::TimeMonitor::getNewTimer("Anasazi: TraceMinBase::Local update")),
    timerCompRes_(Teuchos::TimeMonitor::getNewTimer("Anasazi: TraceMinBase::Computing residuals")),
    timerOrtho_(Teuchos::TimeMonitor::getNewTimer("Anasazi: TraceMinBase::Orthogonalization")),
    timerInit_(Teuchos::TimeMonitor::getNewTimer("Anasazi: TraceMinBase::Initialization")),
#endif
    count_ApplyOp_(0),
    count_ApplyM_(0),
    // internal data
    blockSize_(0),
    numBlocks_(0),
    initialized_(false),
    curDim_(0),
    auxVecs_( Teuchos::Array<RCP<const MV> >(0) ), 
    MauxVecs_( Teuchos::Array<RCP<const MV> >(0) ),
    numAuxVecs_(0),
    iter_(0),
    Rnorms_current_(false),
    R2norms_current_(false),
    // TraceMin variables
    previouslyLeveled_(false),
    previousTrace_(ZERO),
    posSafeToShift_(false),
    negSafeToShift_(false),
    largestSafeShift_(ZERO),
    Z_(Teuchos::null)
  {     
    TEUCHOS_TEST_FOR_EXCEPTION(problem_ == Teuchos::null,std::invalid_argument,
                       "Anasazi::TraceMinBase::constructor: user passed null problem pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(sm_ == Teuchos::null,std::invalid_argument,
                       "Anasazi::TraceMinBase::constructor: user passed null sort manager pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(om_ == Teuchos::null,std::invalid_argument,
                       "Anasazi::TraceMinBase::constructor: user passed null output manager pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(tester_ == Teuchos::null,std::invalid_argument,
                       "Anasazi::TraceMinBase::constructor: user passed null status test pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(orthman_ == Teuchos::null,std::invalid_argument,
                       "Anasazi::TraceMinBase::constructor: user passed null orthogonalization manager pointer.");
    TEUCHOS_TEST_FOR_EXCEPTION(problem_->isHermitian() == false, std::invalid_argument,
                       "Anasazi::TraceMinBase::constructor: problem is not hermitian.");

    // get the problem operators
    Op_   = problem_->getOperator();
    MOp_  = problem_->getM();
    Prec_ = problem_->getPrec();
    hasM_ = (MOp_ != Teuchos::null);

    // Set the saddle point solver parameters
    saddleSolType_ = params.get("Saddle Solver Type", PROJECTED_KRYLOV_SOLVER);
    TEUCHOS_TEST_FOR_EXCEPTION(saddleSolType_ != PROJECTED_KRYLOV_SOLVER && saddleSolType_ != SCHUR_COMPLEMENT_SOLVER && saddleSolType_ != BD_PREC_MINRES, std::invalid_argument,
           "Anasazi::TraceMin::constructor: Invalid value for \"Saddle Solver Type\"; valid options are PROJECTED_KRYLOV_SOLVER, SCHUR_COMPLEMENT_SOLVER, and BD_PREC_MINRES.");    

    // Set the Ritz shift parameters
    whenToShift_ = params.get("When To Shift", ALWAYS_SHIFT);
    TEUCHOS_TEST_FOR_EXCEPTION(whenToShift_ != NEVER_SHIFT && whenToShift_ != SHIFT_WHEN_TRACE_LEVELS && whenToShift_ != SHIFT_WHEN_RESID_SMALL && whenToShift_ != ALWAYS_SHIFT, std::invalid_argument,
           "Anasazi::TraceMin::constructor: Invalid value for \"When To Shift\"; valid options are \"NEVER_SHIFT\", \"SHIFT_WHEN_TRACE_LEVELS\", \"SHIFT_WHEN_RESID_SMALL\", and \"ALWAYS_SHIFT\".");  

    traceThresh_ = params.get("Trace Threshold", 2e-2);
    TEUCHOS_TEST_FOR_EXCEPTION(traceThresh_ < 0, std::invalid_argument,
           "Anasazi::TraceMin::constructor: Invalid value for \"Trace Threshold\"; Must be positive.");  

    howToShift_ = params.get("How To Choose Shift", ADJUSTED_RITZ_SHIFT);
    TEUCHOS_TEST_FOR_EXCEPTION(howToShift_ != LARGEST_CONVERGED_SHIFT && howToShift_ != ADJUSTED_RITZ_SHIFT && howToShift_ != RITZ_VALUES_SHIFT, std::invalid_argument,
           "Anasazi::TraceMin::constructor: Invalid value for \"How To Choose Shift\"; valid options are \"LARGEST_CONVERGED_SHIFT\", \"ADJUSTED_RITZ_SHIFT\", \"RITZ_VALUES_SHIFT\".");

    useMultipleShifts_ = params.get("Use Multiple Shifts", true);

    shiftThresh_ = params.get("Shift Threshold", 1e-2);
    useRelShiftThresh_ = params.get("Relative Shift Threshold", true);
    shiftNorm_ = params.get("Shift Norm", "2");
    TEUCHOS_TEST_FOR_EXCEPTION(shiftNorm_ != "2" && shiftNorm_ != "M", std::invalid_argument,
           "Anasazi::TraceMin::constructor: Invalid value for \"Shift Norm\"; valid options are \"2\", \"M\".");

    considerClusters_ = params.get("Consider Clusters", true);

    projectAllVecs_ = params.get("Project All Vectors", true);
    projectLockedVecs_ = params.get("Project Locked Vectors", true);
    useRHSR_ = params.get("Use Residual as RHS", true);
    computeAllRes_ = params.get("Compute All Residuals", true);

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

    NEV_ = problem_->getNEV();

    // Create the Ritz shift operator
    ritzOp_ = rcp( new TraceMinRitzOp<ScalarType,MV,OP>(Op_,MOp_,Prec_) );
  
  // Set the maximum number of inner iterations
  int innerMaxIts = params.get("Maximum Krylov Iterations", 1000);
  ritzOp_->setMaxIts(innerMaxIts);
  }


  // Destructor
  template <class ScalarType, class MV, class OP>
  TraceMinBase<ScalarType,MV,OP>::~TraceMinBase() {}


  // Set the block size
  // This simply calls setSize(), modifying the block size while retaining the number of blocks.
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::setBlockSize (int blockSize) 
  {
    TEUCHOS_TEST_FOR_EXCEPTION(blockSize < 1, std::invalid_argument, "Anasazi::TraceMinBase::setSize(blocksize,numblocks): blocksize must be strictly positive.");
    setSize(blockSize,numBlocks_);
  }


  // Return the current auxiliary vectors
  template <class ScalarType, class MV, class OP>
  Teuchos::Array<RCP<const MV> > TraceMinBase<ScalarType,MV,OP>::getAuxVecs() const {
    return auxVecs_;
  }


  // return the current block size
  template <class ScalarType, class MV, class OP>
  int TraceMinBase<ScalarType,MV,OP>::getBlockSize() const {
    return(blockSize_); 
  }


  // return eigenproblem
  template <class ScalarType, class MV, class OP>
  const Eigenproblem<ScalarType,MV,OP>& TraceMinBase<ScalarType,MV,OP>::getProblem() const { 
    return(*problem_); 
  }


  // return max subspace dim
  template <class ScalarType, class MV, class OP>
  int TraceMinBase<ScalarType,MV,OP>::getMaxSubspaceDim() const {
    return blockSize_*numBlocks_;
  }

  // return current subspace dim
  template <class ScalarType, class MV, class OP>
  int TraceMinBase<ScalarType,MV,OP>::getCurSubspaceDim() const {
    if (!initialized_) return 0;
    return curDim_;
  }


  // return ritz residual 2-norms
  template <class ScalarType, class MV, class OP>
  std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> 
  TraceMinBase<ScalarType,MV,OP>::getRitzRes2Norms() {
    return getRes2Norms();
  }


  // return ritz index
  template <class ScalarType, class MV, class OP>
  std::vector<int> TraceMinBase<ScalarType,MV,OP>::getRitzIndex() {
    std::vector<int> ret(curDim_,0);
    return ret;
  }


  // return ritz values
  template <class ScalarType, class MV, class OP>
  std::vector<Value<ScalarType> > TraceMinBase<ScalarType,MV,OP>::getRitzValues() { 
    std::vector<Value<ScalarType> > ret(curDim_);
    for (int i=0; i<curDim_; ++i) {
      ret[i].realpart = theta_[i];
      ret[i].imagpart = ZERO;
    }
    return ret;
  }


  // return pointer to ritz vectors
  template <class ScalarType, class MV, class OP>
  RCP<const MV> TraceMinBase<ScalarType,MV,OP>::getRitzVectors() {
    return X_;
  }


  // reset number of iterations
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::resetNumIters() { 
    iter_=0; 
  }


  // return number of iterations
  template <class ScalarType, class MV, class OP>
  int TraceMinBase<ScalarType,MV,OP>::getNumIters() const { 
    return(iter_); 
  }


  // return state pointers
  template <class ScalarType, class MV, class OP>
  TraceMinBaseState<ScalarType,MV> TraceMinBase<ScalarType,MV,OP>::getState() const {
    TraceMinBaseState<ScalarType,MV> state;
    state.curDim = curDim_;
    state.V = V_;
    state.X = X_;
    if (Op_ != Teuchos::null) {
      state.KV = KV_;
      state.KX = KX_;
    }
    else {
      state.KV = Teuchos::null;
      state.KX = Teuchos::null;
    }
    if (hasM_) {
      state.MopV = MV_;
      state.MX = MX_;
    }
    else {
      state.MopV = Teuchos::null;
      state.MX = Teuchos::null;
    }
    state.R = R_;
    state.KK = KK_;
    state.RV = ritzVecs_;
    if (curDim_ > 0) {
      state.T = rcp(new std::vector<MagnitudeType>(theta_.begin(),theta_.begin()+curDim_) );
    } else {
      state.T = rcp(new std::vector<MagnitudeType>(0));
    }
    state.ritzShifts = rcp(new std::vector<MagnitudeType>(ritzShifts_.begin(),ritzShifts_.begin()+blockSize_) );
    state.isOrtho = true;
    state.largestSafeShift = largestSafeShift_;

    return state;
  }


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

    // as a data member, this would be redundant and require synchronization with
    // blockSize_ and numBlocks_; we'll use a constant here.
    const int searchDim = blockSize_*numBlocks_;

    // Update obtained from solving saddle point problem
    RCP<MV> Delta = MVT::Clone(*X_,blockSize_);

    // iterate until the status test tells us to stop.
    // also break if our basis is full
    while (tester_->checkStatus(this) != Passed && (numBlocks_ == 1 || curDim_ < searchDim)) {

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

      ++iter_;

      // Solve the saddle point problem
      solveSaddlePointProblem(Delta);

      // Insert Delta at the end of V
      addToBasis(Delta);

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

      // Compute KK := V'KV
      computeKK();

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

      // Compute the Ritz pairs
      computeRitzPairs();

      // Compute X := V RitzVecs
      computeX();

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

      // Compute KX := KV RitzVecs and MX := MV RitzVecs (if necessary)
      updateKXMX();

      if (om_->isVerbosity( Debug ) ) {
        // Check almost everything here
        CheckList chk;
        chk.checkKX = true;
        chk.checkMX = true;
        om_->print( Debug, accuracyCheck(chk, ": after updateKXMX()") );
      }

      // Update the residual vectors
      updateResidual(); 
    } // end while (statusTest == false)

  } // end of iterate()


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


  /* Initialize the state of the solver
   * 
   * POST-CONDITIONS:
   *
   * V_ is orthonormal, orthogonal to auxVecs_, for first curDim_ vectors
   * theta_ contains Ritz w.r.t. V_(1:curDim_)
   * X is Ritz vectors w.r.t. V_(1:curDim_)
   * KX = Op*X
   * MX = M*X if hasM_
   * R = KX - MX*diag(theta_)
   *
   */
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::initialize(TraceMinBaseState<ScalarType,MV> newstate)
  {
    // TODO: Check for bad input
    // NOTE: memory has been allocated by setBlockSize(). Use setBlock below; do not Clone
    // NOTE: Overall time spent in this routine is counted to timerInit_; portions will also be counted towards other primitives

#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor inittimer( *timerInit_ );
#endif

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

    // in TraceMinBase, V is primary
    // the order of dependence follows like so.
    // --init->               V,KK
    //    --ritz analysis->   theta,X  
    //       --op apply->     KX,MX  
    //          --compute->   R
    // 
    // if the user specifies all data for a level, we will accept it.
    // otherwise, we will generate the whole level, and all subsequent levels.
    //
    // the data members are ordered based on dependence, and the levels are
    // partitioned according to the amount of work required to produce the
    // items in a level.
    //
    // inconsistent multivectors widths and lengths will not be tolerated, and
    // will be treated with exceptions.
    //
    // for multivector pointers in newstate which point directly (as opposed to indirectly, via a view) to 
    // multivectors in the solver, the copy will not be affected.

    // set up V and KK: get them from newstate if user specified them
    // otherwise, set them manually
    RCP<MV> lclV, lclKV, lclMV;
    RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > lclKK, lclRV;

    // If V was supplied by the user...
    if (newstate.V != Teuchos::null) {
      om_->stream(Debug) << "Copying V from the user\n";

      TEUCHOS_TEST_FOR_EXCEPTION( MVText::GetGlobalLength(*newstate.V) != MVText::GetGlobalLength(*V_), std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Vector length of V not correct." );
      TEUCHOS_TEST_FOR_EXCEPTION( newstate.curDim < blockSize_, std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Rank of new state must be at least blockSize().");
      TEUCHOS_TEST_FOR_EXCEPTION( newstate.curDim > blockSize_*numBlocks_, std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Rank of new state must be less than getMaxSubspaceDim().");

      TEUCHOS_TEST_FOR_EXCEPTION( MVT::GetNumberVecs(*newstate.V) < newstate.curDim, std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Multivector for basis in new state must be as large as specified state rank.");

      curDim_ = newstate.curDim;
      // pick an integral amount
      curDim_ = (int)(curDim_ / blockSize_)*blockSize_;

      TEUCHOS_TEST_FOR_EXCEPTION( curDim_ != newstate.curDim, std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Rank of new state must be a multiple of getBlockSize().");

      // put data in V
      std::vector<int> nevind(curDim_);
      for (int i=0; i<curDim_; ++i) nevind[i] = i;
      if (newstate.V != V_) {
        if (curDim_ < MVT::GetNumberVecs(*newstate.V)) {
          newstate.V = MVT::CloneView(*newstate.V,nevind);
        }
        MVT::SetBlock(*newstate.V,nevind,*V_);
      }
      lclV = MVT::CloneViewNonConst(*V_,nevind);
    } // end if user specified V
    else 
    {
      // get vectors from problem or generate something, projectAndNormalize
      RCP<const MV> ivec = problem_->getInitVec();
      TEUCHOS_TEST_FOR_EXCEPTION(ivec == Teuchos::null,std::invalid_argument,
             "Anasazi::TraceMinBase::initialize(newstate): Eigenproblem did not specify initial vectors to clone from.");
      // clear newstate so we won't use any data from it below
      newstate.X       = Teuchos::null;
      newstate.MX      = Teuchos::null;
      newstate.KX      = Teuchos::null;
      newstate.R       = Teuchos::null;
      newstate.T       = Teuchos::null;
      newstate.RV      = Teuchos::null;
      newstate.KK      = Teuchos::null;
      newstate.KV      = Teuchos::null;
      newstate.MopV    = Teuchos::null;
      newstate.isOrtho = false;

      // If the user did not specify a current dimension, use that of the initial vectors they provided
      if(newstate.curDim > 0)
        curDim_ = newstate.curDim;
      else
        curDim_ = MVT::GetNumberVecs(*ivec);

      // pick the largest multiple of blockSize_
      curDim_ = (int)(curDim_ / blockSize_)*blockSize_;
      if (curDim_ > blockSize_*numBlocks_) {
        // user specified too many vectors... truncate
        // this produces a full subspace, but that is okay
        curDim_ = blockSize_*numBlocks_;
      }
      bool userand = false;
      if (curDim_ == 0) {
        // we need at least blockSize_ vectors
        // use a random multivec: ignore everything from InitVec
        userand = true;
        curDim_ = blockSize_;
      }

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

      // get a pointer into V
      // lclV has curDim vectors
      //
      // get pointer to first curDim vectors in V_
      lclV = MVT::CloneViewNonConst(*V_,nevind);
      if (userand) 
      {
        // generate random vector data
        MVT::MvRandom(*lclV);
      }
      else 
      {
        if(newstate.curDim > 0)
        {
          if(MVT::GetNumberVecs(*newstate.V) > curDim_) {
            RCP<const MV> helperMV = MVT::CloneView(*newstate.V,nevind);
            MVT::SetBlock(*helperMV,nevind,*lclV);
          }
          // assign ivec to first part of lclV
          MVT::SetBlock(*newstate.V,nevind,*lclV);
        }
        else
        {
          if(MVT::GetNumberVecs(*ivec) > curDim_) {
            ivec = MVT::CloneView(*ivec,nevind);
          }
          // assign ivec to first part of lclV
          MVT::SetBlock(*ivec,nevind,*lclV);
        }
      }
    } // end if user did not specify V

    // A vector of relevant indices
    std::vector<int> dimind(curDim_);
    for (int i=0; i<curDim_; ++i) dimind[i] = i;

    // Compute MV if necessary
    if(hasM_ && newstate.MopV == Teuchos::null)
    {
      om_->stream(Debug) << "Computing MV\n";

#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
      Teuchos::TimeMonitor lcltimer( *timerMOp_ );
#endif
      count_ApplyM_+= curDim_;

      newstate.isOrtho = false;
      // Get a pointer to the relevant parts of MV
      lclMV = MVT::CloneViewNonConst(*MV_,dimind);
      OPT::Apply(*MOp_,*lclV,*lclMV);
    }
    // Copy MV if necessary
    else if(hasM_)
    {
      om_->stream(Debug) << "Copying MV\n";

      TEUCHOS_TEST_FOR_EXCEPTION( MVText::GetGlobalLength(*newstate.MopV) != MVText::GetGlobalLength(*MV_), std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Vector length of MV not correct." );

      TEUCHOS_TEST_FOR_EXCEPTION( MVT::GetNumberVecs(*newstate.MopV) < curDim_, std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Number of vectors in MV not correct.");

      if(newstate.MopV != MV_) {
        if(curDim_ < MVT::GetNumberVecs(*newstate.MopV)) {
          newstate.MopV = MVT::CloneView(*newstate.MopV,dimind);
        }
        MVT::SetBlock(*newstate.MopV,dimind,*MV_);
      }
      lclMV = MVT::CloneViewNonConst(*MV_,dimind);
    }
    // There is no M, so there is no MV
    else
    {
      om_->stream(Debug) << "There is no MV\n";

      lclMV = lclV;
      MV_ = V_;
    }

    // Project and normalize so that V'V = I and Q'V=0
    if(newstate.isOrtho == false)
    {
      om_->stream(Debug) << "Project and normalize\n";

#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
      Teuchos::TimeMonitor lcltimer( *timerOrtho_ );
#endif

      // These things are now invalid
      newstate.X       = Teuchos::null;
      newstate.MX      = Teuchos::null;
      newstate.KX      = Teuchos::null;
      newstate.R       = Teuchos::null;
      newstate.T       = Teuchos::null;
      newstate.RV      = Teuchos::null;
      newstate.KK      = Teuchos::null;
      newstate.KV      = Teuchos::null;

      int rank;
      if(auxVecs_.size() > 0)
      {
        rank = orthman_->projectAndNormalizeMat(*lclV, auxVecs_, 
               Teuchos::tuple(RCP< Teuchos::SerialDenseMatrix< int, ScalarType > >(Teuchos::null)), 
               Teuchos::null, lclMV, MauxVecs_);
      }
      else
      {
        rank = orthman_->normalizeMat(*lclV,Teuchos::null,lclMV);
      }

      TEUCHOS_TEST_FOR_EXCEPTION(rank != curDim_,TraceMinBaseInitFailure,
             "Anasazi::TraceMinBase::initialize(): Couldn't generate initial basis of full rank.");
    }

    // Compute KV
    if(Op_ != Teuchos::null && newstate.KV == Teuchos::null)
    {
      om_->stream(Debug) << "Computing KV\n";

#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
      Teuchos::TimeMonitor lcltimer( *timerOp_ );
#endif
      count_ApplyOp_+= curDim_;

      // These things are now invalid
      newstate.X       = Teuchos::null;
      newstate.MX      = Teuchos::null;
      newstate.KX      = Teuchos::null;
      newstate.R       = Teuchos::null;
      newstate.T       = Teuchos::null;
      newstate.RV      = Teuchos::null;
      newstate.KK      = Teuchos::null;
      newstate.KV      = Teuchos::null;

      lclKV = MVT::CloneViewNonConst(*KV_,dimind);
      OPT::Apply(*Op_,*lclV,*lclKV);
    }
    // Copy KV
    else if(Op_ != Teuchos::null)
    {
      om_->stream(Debug) << "Copying MV\n";

      TEUCHOS_TEST_FOR_EXCEPTION( MVText::GetGlobalLength(*newstate.KV) != MVText::GetGlobalLength(*KV_), std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Vector length of MV not correct." );

      TEUCHOS_TEST_FOR_EXCEPTION( MVT::GetNumberVecs(*newstate.KV) < curDim_, std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Number of vectors in KV not correct.");

      if (newstate.KV != KV_) {
        if (curDim_ < MVT::GetNumberVecs(*newstate.KV)) {
          newstate.KV = MVT::CloneView(*newstate.KV,dimind);
        }
        MVT::SetBlock(*newstate.KV,dimind,*KV_);
      }
      lclKV = MVT::CloneViewNonConst(*KV_,dimind);
    }
    else
    {
      om_->stream(Debug) << "There is no KV\n";

      lclKV = lclV;
      KV_ = V_;
    }      

    // Compute KK
    if(newstate.KK == Teuchos::null)
    {
      om_->stream(Debug) << "Computing KK\n";

      // These things are now invalid
      newstate.X       = Teuchos::null;
      newstate.MX      = Teuchos::null;
      newstate.KX      = Teuchos::null;
      newstate.R       = Teuchos::null;
      newstate.T       = Teuchos::null;
      newstate.RV      = Teuchos::null;

      // Note: there used to be a bug here.
      // We can't just call computeKK because it only computes the new parts of KK

      // Get a pointer to the part of KK we're interested in
      lclKK = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*KK_,curDim_,curDim_) );

      // KK := V'KV
      MVT::MvTransMv(ONE,*lclV,*lclKV,*lclKK);
    }
    // Copy KK
    else
    {
      om_->stream(Debug) << "Copying KK\n";

      // check size of KK
      TEUCHOS_TEST_FOR_EXCEPTION( newstate.KK->numRows() < curDim_ || newstate.KK->numCols() < curDim_, std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Projected matrix in new state must be as large as specified state rank.");

      // put data into KK_
      lclKK = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*KK_,curDim_,curDim_) );
      if (newstate.KK != KK_) {
        if (newstate.KK->numRows() > curDim_ || newstate.KK->numCols() > curDim_) {
          newstate.KK = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*newstate.KK,curDim_,curDim_) );
        }
        lclKK->assign(*newstate.KK);
      }
    }

    // Compute Ritz pairs
    if(newstate.T == Teuchos::null || newstate.RV == Teuchos::null)
    {
      om_->stream(Debug) << "Computing Ritz pairs\n";

      // These things are now invalid
      newstate.X       = Teuchos::null;
      newstate.MX      = Teuchos::null;
      newstate.KX      = Teuchos::null;
      newstate.R       = Teuchos::null;
      newstate.T       = Teuchos::null;
      newstate.RV      = Teuchos::null;

      computeRitzPairs();
    }
    // Copy Ritz pairs
    else
    {
      om_->stream(Debug) << "Copying Ritz pairs\n";

      TEUCHOS_TEST_FOR_EXCEPTION((signed int)(newstate.T->size()) != curDim_,
             std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): Size of T must be consistent with dimension of V.");

      TEUCHOS_TEST_FOR_EXCEPTION( newstate.RV->numRows() < curDim_ || newstate.RV->numCols() < curDim_, std::invalid_argument, 
             "Anasazi::TraceMinBase::initialize(newstate): Ritz vectors in new state must be as large as specified state rank.");

      std::copy(newstate.T->begin(),newstate.T->end(),theta_.begin());

      lclRV = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*ritzVecs_,curDim_,curDim_) );
      if (newstate.RV != ritzVecs_) {
        if (newstate.RV->numRows() > curDim_ || newstate.RV->numCols() > curDim_) {
          newstate.RV = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*newstate.RV,curDim_,curDim_) );
        }
        lclRV->assign(*newstate.RV);
      }
    }

    // Compute X
    if(newstate.X == Teuchos::null)
    {
      om_->stream(Debug) << "Computing X\n";

      // These things are now invalid
      newstate.MX      = Teuchos::null;
      newstate.KX      = Teuchos::null;
      newstate.R       = Teuchos::null;

      computeX();
    }
    // Copy X
    else
    {
      om_->stream(Debug) << "Copying X\n";

      if(computeAllRes_ == false)
      {
        TEUCHOS_TEST_FOR_EXCEPTION(MVT::GetNumberVecs(*newstate.X) != blockSize_ || MVText::GetGlobalLength(*newstate.X) != MVText::GetGlobalLength(*X_),
               std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): Size of X must be consistent with block size and length of V.");

        if(newstate.X != X_) {
          MVT::SetBlock(*newstate.X,bsind,*X_);
        }
      }
      else
      {
        TEUCHOS_TEST_FOR_EXCEPTION(MVT::GetNumberVecs(*newstate.X) != curDim_ || MVText::GetGlobalLength(*newstate.X) != MVText::GetGlobalLength(*X_),
               std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): Size of X must be consistent with current dimension and length of V.");

        if(newstate.X != X_) {
          MVT::SetBlock(*newstate.X,dimind,*X_);
        }
      }
    }

    // Compute KX and MX if necessary
    // TODO: These technically should be separate; it won't matter much in terms of running time though
    if((Op_ != Teuchos::null && newstate.KX == Teuchos::null) || (hasM_ && newstate.MX == Teuchos::null))
    {
      om_->stream(Debug) << "Computing KX and MX\n";

      // These things are now invalid
      newstate.R = Teuchos::null;

      updateKXMX();
    }
    // Copy KX and MX if necessary
    else
    {
      om_->stream(Debug) << "Copying KX and MX\n";
      if(Op_ != Teuchos::null)
      {
        if(computeAllRes_ == false)
        {
          TEUCHOS_TEST_FOR_EXCEPTION(MVT::GetNumberVecs(*newstate.KX) != blockSize_ || MVText::GetGlobalLength(*newstate.KX) != MVText::GetGlobalLength(*KX_),
                 std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): Size of KX must be consistent with block size and length of V.");

          if(newstate.KX != KX_) {
            MVT::SetBlock(*newstate.KX,bsind,*KX_);
          }
        }
        else
        {
          TEUCHOS_TEST_FOR_EXCEPTION(MVT::GetNumberVecs(*newstate.KX) != curDim_ || MVText::GetGlobalLength(*newstate.KX) != MVText::GetGlobalLength(*KX_),
                 std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): Size of KX must be consistent with current dimension and length of V.");

          if (newstate.KX != KX_) {
            MVT::SetBlock(*newstate.KX,dimind,*KX_);
          }
        }
      }

      if(hasM_)
      {
        if(computeAllRes_ == false)
        {
          TEUCHOS_TEST_FOR_EXCEPTION(MVT::GetNumberVecs(*newstate.MX) != blockSize_ || MVText::GetGlobalLength(*newstate.MX) != MVText::GetGlobalLength(*MX_),
                 std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): Size of MX must be consistent with block size and length of V.");

          if (newstate.MX != MX_) {
            MVT::SetBlock(*newstate.MX,bsind,*MX_);
          }
        }
        else
        {
          TEUCHOS_TEST_FOR_EXCEPTION(MVT::GetNumberVecs(*newstate.MX) != curDim_ || MVText::GetGlobalLength(*newstate.MX) != MVText::GetGlobalLength(*MX_),
                 std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): Size of MX must be consistent with current dimension and length of V.");

          if (newstate.MX != MX_) {
            MVT::SetBlock(*newstate.MX,dimind,*MX_);
          }
        }
      }
    }

    // Compute R
    if(newstate.R == Teuchos::null)
    {
      om_->stream(Debug) << "Computing R\n";

      updateResidual();
    }
    // Copy R
    else
    {
      om_->stream(Debug) << "Copying R\n";

      if(computeAllRes_ == false)
      {
        TEUCHOS_TEST_FOR_EXCEPTION(MVText::GetGlobalLength(*newstate.R) != MVText::GetGlobalLength(*X_),
               std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): vector length of newstate.R not correct." );
        TEUCHOS_TEST_FOR_EXCEPTION(MVT::GetNumberVecs(*newstate.R) != blockSize_,
               std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): newstate.R must have at least block size vectors." );
        if (newstate.R != R_) {
          MVT::SetBlock(*newstate.R,bsind,*R_);
        }
      }
      else
      {
        TEUCHOS_TEST_FOR_EXCEPTION(MVText::GetGlobalLength(*newstate.R) != MVText::GetGlobalLength(*X_),
               std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): vector length of newstate.R not correct." );
        TEUCHOS_TEST_FOR_EXCEPTION(MVT::GetNumberVecs(*newstate.R) != curDim_,
               std::invalid_argument, "Anasazi::TraceMinBase::initialize(newstate): newstate.R must have at least curDim vectors." );
        if (newstate.R != R_) {
          MVT::SetBlock(*newstate.R,dimind,*R_);
        }
      }
    }

    // R has been updated; mark the norms as out-of-date
    Rnorms_current_ = false;
    R2norms_current_ = false;

    // Set the largest safe shift
    largestSafeShift_ = newstate.largestSafeShift;

    // Copy over the Ritz shifts
    if(newstate.ritzShifts != Teuchos::null)
    {
      om_->stream(Debug) << "Copying Ritz shifts\n";
      std::copy(newstate.ritzShifts->begin(),newstate.ritzShifts->end(),ritzShifts_.begin());
    }
    else
    {
      om_->stream(Debug) << "Setting Ritz shifts to 0\n";
      for(size_t i=0; i<ritzShifts_.size(); i++)
        ritzShifts_[i] = ZERO;
    }

    for(size_t i=0; i<ritzShifts_.size(); i++)
      om_->stream(Debug) << "Ritz shifts[" << i << "] = " << ritzShifts_[i] << std::endl;

    // finally, we are initialized
    initialized_ = true;

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

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


  // Return initialized state
  template <class ScalarType, class MV, class OP>
  bool TraceMinBase<ScalarType,MV,OP>::isInitialized() const { return initialized_; }


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

    TEUCHOS_TEST_FOR_EXCEPTION(blockSize < 1, std::invalid_argument, "Anasazi::TraceMinBase::setSize(blocksize,numblocks): blocksize must be strictly positive.");

    if (blockSize == blockSize_ && numBlocks == numBlocks_) {
      // do nothing
      return;
    }

    blockSize_ = blockSize;
    numBlocks_ = numBlocks;

    RCP<const MV> tmp;
    // grab some Multivector to Clone
    // in practice, getInitVec() should always provide this, but it is possible to use a 
    // Eigenproblem with nothing in getInitVec() by manually initializing with initialize(); 
    // in case of that strange scenario, we will try to Clone from X_ first, then resort to getInitVec()
    if (X_ != Teuchos::null) { // this is equivalent to blockSize_ > 0
      tmp = X_;
    }
    else {
      tmp = problem_->getInitVec();
      TEUCHOS_TEST_FOR_EXCEPTION(tmp == Teuchos::null,std::invalid_argument,
             "Anasazi::TraceMinBase::setSize(): eigenproblem did not specify initial vectors to clone from.");
    }

    TEUCHOS_TEST_FOR_EXCEPTION(numAuxVecs_+blockSize*static_cast<ptrdiff_t>(numBlocks) > MVText::GetGlobalLength(*tmp),std::invalid_argument,
           "Anasazi::TraceMinBase::setSize(): max subspace dimension and auxilliary subspace too large.  Potentially impossible orthogonality constraints.");

    // New subspace dimension
    int newsd = blockSize_*numBlocks_;

    // blockSize dependent
    //
    ritzShifts_.resize(blockSize_,ZERO);
    if(computeAllRes_ == false)
    {
      // grow/allocate vectors
      Rnorms_.resize(blockSize_,NANVAL);
      R2norms_.resize(blockSize_,NANVAL);
      //
      // clone multivectors off of tmp
      //
      // free current allocation first, to make room for new allocation
      X_ = Teuchos::null;
      KX_ = Teuchos::null;
      MX_ = Teuchos::null;
      R_ = Teuchos::null;
      V_ = Teuchos::null;
      KV_ = Teuchos::null;
      MV_ = Teuchos::null;

      om_->print(Debug," >> Allocating X_\n");
      X_ = MVT::Clone(*tmp,blockSize_);
      if(Op_ != Teuchos::null) {
        om_->print(Debug," >> Allocating KX_\n");
        KX_ = MVT::Clone(*tmp,blockSize_);
      }
      else {
        KX_ = X_;
      }
      if (hasM_) {
        om_->print(Debug," >> Allocating MX_\n");
        MX_ = MVT::Clone(*tmp,blockSize_);
      }
      else {
        MX_ = X_;
      }
      om_->print(Debug," >> Allocating R_\n");
      R_ = MVT::Clone(*tmp,blockSize_);
    }
    else
    {
      // grow/allocate vectors
      Rnorms_.resize(newsd,NANVAL);
      R2norms_.resize(newsd,NANVAL);
      //
      // clone multivectors off of tmp
      //
      // free current allocation first, to make room for new allocation
      X_ = Teuchos::null;
      KX_ = Teuchos::null;
      MX_ = Teuchos::null;
      R_ = Teuchos::null;
      V_ = Teuchos::null;
      KV_ = Teuchos::null;
      MV_ = Teuchos::null;

      om_->print(Debug," >> Allocating X_\n");
      X_ = MVT::Clone(*tmp,newsd);
      if(Op_ != Teuchos::null) {
        om_->print(Debug," >> Allocating KX_\n");
        KX_ = MVT::Clone(*tmp,newsd);
      }
      else {
        KX_ = X_;
      }
      if (hasM_) {
        om_->print(Debug," >> Allocating MX_\n");
        MX_ = MVT::Clone(*tmp,newsd);
      }
      else {
        MX_ = X_;
      }
      om_->print(Debug," >> Allocating R_\n");
      R_ = MVT::Clone(*tmp,newsd);
    }

    // blockSize*numBlocks dependent
    //
    theta_.resize(newsd,NANVAL);
    om_->print(Debug," >> Allocating V_\n");
    V_ = MVT::Clone(*tmp,newsd);
    KK_ = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(newsd,newsd) );
    ritzVecs_ = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(newsd,newsd) );

    if(Op_ != Teuchos::null) {
      om_->print(Debug," >> Allocating KV_\n");
      KV_ = MVT::Clone(*tmp,newsd);
    }
    else {
      KV_ = V_;
    }
    if (hasM_) {
      om_->print(Debug," >> Allocating MV_\n");
      MV_ = MVT::Clone(*tmp,newsd);
    }
    else {
      MV_ = V_;
    }

    om_->print(Debug," >> done allocating.\n");

    initialized_ = false;
    curDim_ = 0;
  }


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

    // set new auxiliary vectors
    auxVecs_ = auxvecs;

    if(hasM_)
      MauxVecs_.resize(0);
    numAuxVecs_ = 0;

    for (tarcpmv i=auxVecs_.begin(); i != auxVecs_.end(); ++i) {
      numAuxVecs_ += MVT::GetNumberVecs(**i);

      if(hasM_)
      {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
      Teuchos::TimeMonitor lcltimer( *timerMOp_ );
#endif
        count_ApplyM_+= MVT::GetNumberVecs(**i);

        RCP<MV> helperMV = MVT::Clone(**i,MVT::GetNumberVecs(**i));
        OPT::Apply(*MOp_,**i,*helperMV);
        MauxVecs_.push_back(helperMV);
      }
    }

    // If the solver has been initialized, V is not necessarily orthogonal to new auxiliary vectors
    if (numAuxVecs_ > 0 && initialized_) {
      initialized_ = false;
    }

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


  // compute/return residual M-norms
  template <class ScalarType, class MV, class OP>
  std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> 
  TraceMinBase<ScalarType,MV,OP>::getResNorms() {
    if (Rnorms_current_ == false) {
      // Update the residual norms
      if(computeAllRes_)
      {
        std::vector<int> curind(curDim_);
        for(int i=0; i<curDim_; i++)
          curind[i] = i;

        RCP<const MV> locR = MVT::CloneView(*R_,curind);
        std::vector<ScalarType> locNorms(curDim_);
        orthman_->norm(*locR,locNorms);

        for(int i=0; i<curDim_; i++)
          Rnorms_[i] = locNorms[i];
        for(int i=curDim_+1; i<blockSize_*numBlocks_; i++)
          Rnorms_[i] = NANVAL;

        Rnorms_current_ = true;
        locNorms.resize(blockSize_);
        return locNorms;
      }
      else
        orthman_->norm(*R_,Rnorms_);
      Rnorms_current_ = true;
    }
    else if(computeAllRes_)
    {
      std::vector<ScalarType> locNorms(blockSize_);
      for(int i=0; i<blockSize_; i++)
        locNorms[i] = Rnorms_[i];
      return locNorms;
    }

    return Rnorms_;
  }

  
  // compute/return residual 2-norms
  template <class ScalarType, class MV, class OP>
  std::vector<typename Teuchos::ScalarTraits<ScalarType>::magnitudeType> 
  TraceMinBase<ScalarType,MV,OP>::getRes2Norms() {
    if (R2norms_current_ == false) {
      // Update the residual 2-norms 
      if(computeAllRes_)
      {
        std::vector<int> curind(curDim_);
        for(int i=0; i<curDim_; i++)
          curind[i] = i;

        RCP<const MV> locR = MVT::CloneView(*R_,curind);
        std::vector<ScalarType> locNorms(curDim_);
        MVT::MvNorm(*locR,locNorms);

        for(int i=0; i<curDim_; i++)
        {
          R2norms_[i] = locNorms[i];
        }
        for(int i=curDim_+1; i<blockSize_*numBlocks_; i++)
          R2norms_[i] = NANVAL;

        R2norms_current_ = true;
        locNorms.resize(blockSize_);
        return locNorms;
      }
      else
        MVT::MvNorm(*R_,R2norms_);
      R2norms_current_ = true;
    }
    else if(computeAllRes_)
    {
      std::vector<ScalarType> locNorms(blockSize_);
      for(int i=0; i<blockSize_; i++)
        locNorms[i] = R2norms_[i];
      return locNorms;
    }

    return R2norms_;
  }

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

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

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

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

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

    if (initialized_) {
      os << endl;
      os <<"CURRENT EIGENVALUE ESTIMATES             "<<endl;
      os << std::setw(20) << "Eigenvalue" 
         << std::setw(20) << "Residual(M)"
         << std::setw(20) << "Residual(2)"
         << endl;
      os <<"--------------------------------------------------------------------------------"<<endl;
      for (int i=0; i<blockSize_; ++i) {
        os << std::setw(20) << theta_[i];
        if (Rnorms_current_) os << std::setw(20) << Rnorms_[i];
        else os << std::setw(20) << "not current";
        if (R2norms_current_) os << std::setw(20) << R2norms_[i];
        else os << std::setw(20) << "not current";
        os << endl;
      }
    }
    os <<"================================================================================" << endl;
    os << endl;
  }

  template <class ScalarType, class MV, class OP>
  ScalarType TraceMinBase<ScalarType,MV,OP>::getTrace() const
  {
    ScalarType currentTrace = ZERO;

    int neigs = std::min(NEV_,curDim_);
    for(int i=0; i < neigs; i++)
      currentTrace += theta_[i];

    return currentTrace;
  }

  template <class ScalarType, class MV, class OP>
  bool TraceMinBase<ScalarType,MV,OP>::traceLeveled()
  {
    ScalarType ratioOfChange = traceThresh_;

    if(previouslyLeveled_)
    {
      om_->stream(Debug) << "The trace already leveled, so we're not going to check it again\n";
      return true;
    }

    ScalarType currentTrace = getTrace();

    om_->stream(Debug) << "The current trace is " << currentTrace << std::endl;

    // Compute the ratio of the change
    // We seek the point where the trace has leveled off
    // It should be reasonably safe to shift at this point
    if(previousTrace_ != ZERO)
    {
      om_->stream(Debug) << "The previous trace was " << previousTrace_ << std::endl;

      ratioOfChange = std::abs(previousTrace_-currentTrace)/std::abs(previousTrace_);
      om_->stream(Debug) << "The ratio of change is " << ratioOfChange << std::endl;
    }

    previousTrace_ = currentTrace;

    if(ratioOfChange < traceThresh_)
    {
      previouslyLeveled_ = true;
      return true;
    }
    return false;
  }

  // Compute the residual of each CLUSTER of eigenvalues
  // This is important for selecting the Ritz shifts
  template <class ScalarType, class MV, class OP>
  std::vector<ScalarType> TraceMinBase<ScalarType,MV,OP>::getClusterResids()
  {
    int nvecs;

    if(computeAllRes_)
      nvecs = curDim_;
    else
      nvecs = blockSize_;

    std::vector<ScalarType> clusterResids(nvecs);
    std::vector<int> clusterIndices;
    for(int i=0; i < nvecs; i++)
    {
      // test for cluster
      if(clusterIndices.empty() || (theta_[i-1] + R2norms_[i-1] >= theta_[i] - R2norms_[i]))
      {
        // Add to cluster
        if(!clusterIndices.empty())  om_->stream(Debug) << theta_[i-1] << " is in a cluster with " << theta_[i] << " because " << theta_[i-1] + R2norms_[i-1] << " >= " << theta_[i] - R2norms_[i] << std::endl;
        clusterIndices.push_back(i);
      }
      // Cluster completed
      else
      {
        om_->stream(Debug) << theta_[i-1] << " is NOT in a cluster with " << theta_[i] << " because " << theta_[i-1] + R2norms_[i-1] << " < " << theta_[i] - R2norms_[i] << std::endl;
        ScalarType totalRes = ZERO;
        for(size_t j=0; j < clusterIndices.size(); j++)
          totalRes += (R2norms_[clusterIndices[j]]*R2norms_[clusterIndices[j]]);

        // If the smallest magnitude value of this sign is in a cluster with the 
        // largest magnitude cluster of this sign, it is not safe for the smallest
        // eigenvalue to use a shift
        if(theta_[clusterIndices[0]] < 0 && theta_[i] < 0)
          negSafeToShift_ = true;
        else if(theta_[clusterIndices[0]] > 0 && theta_[i] > 0)
          posSafeToShift_ = true;

        for(size_t j=0; j < clusterIndices.size(); j++)
          clusterResids[clusterIndices[j]] = sqrt(totalRes);

        clusterIndices.clear();
        clusterIndices.push_back(i);
      }
    }

    // Handle last cluster
    ScalarType totalRes = ZERO;
    for(size_t j=0; j < clusterIndices.size(); j++)
      totalRes += R2norms_[clusterIndices[j]];
    for(size_t j=0; j < clusterIndices.size(); j++)
      clusterResids[clusterIndices[j]] = totalRes;

    return clusterResids;
  }

  // Compute the Ritz shifts based on the Ritz values and residuals
  // A note on shifting: if the matrix is indefinite, you NEED to use a large block size
  // TODO: resids[i] on its own is unsafe for the generalized EVP
  //       See "A Parallel Implementation of the Trace Minimization Eigensolver"
  //       by Eloy Romero and Jose E. Roman
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::computeRitzShifts(const std::vector<ScalarType>& clusterResids)
  {
    std::vector<ScalarType> thetaMag(theta_);
    bool traceHasLeveled = traceLeveled();

    // Compute the magnitude of the eigenvalues
    for(int i=0; i<blockSize_; i++)
      thetaMag[i] = std::abs(thetaMag[i]);

    // Test whether it is safe to shift
    // TODO: Add residual shift option
    // Note: If you choose single shift with an indefinite matrix, you're gonna have a bad time...
    // Note: This is not safe for indefinite matrices, and I don't even know that it CAN be made safe.
    //       There just isn't any theoretical reason it should work.
    // TODO: It feels like this should be different for TraceMin-Base; we should be able to use all eigenvalues in the current subspace to determine whether we have a cluster
    if(whenToShift_ == ALWAYS_SHIFT || (whenToShift_ == SHIFT_WHEN_TRACE_LEVELS && traceHasLeveled))
    {
      // Set the shift to the largest safe shift
      if(howToShift_ == LARGEST_CONVERGED_SHIFT)
      {
        for(int i=0; i<blockSize_; i++)
          ritzShifts_[i] = largestSafeShift_;
      }
      // Set the shifts to the Ritz values
      else if(howToShift_ == RITZ_VALUES_SHIFT)
      {
        ritzShifts_[0] = theta_[0];

        // If we're using mulitple shifts, set them to EACH Ritz value.
        // Otherwise, only use the smallest
        if(useMultipleShifts_)
        {
          for(int i=1; i<blockSize_; i++)
            ritzShifts_[i] = theta_[i];
        }
        else
        {
          for(int i=1; i<blockSize_; i++)
            ritzShifts_[i] = ritzShifts_[0];
        }
      }
      // Use Dr. Sameh's original shifting strategy
      else if(howToShift_ == ADJUSTED_RITZ_SHIFT)
      {
        om_->stream(Debug) << "\nSeeking a shift for theta[0]=" << thetaMag[0] << std::endl;

        // This is my adjustment.  If all eigenvalues are in a single cluster, it's probably a bad idea to shift the smallest one.
        // If all of your eigenvalues are in one cluster, it's either way to early to shift or your subspace is too small
        if((theta_[0] > 0 && posSafeToShift_) || (theta_[0] < 0 && negSafeToShift_) || considerClusters_ == false)
        {
          // Initialize with a conservative shift, either the biggest safe shift or the eigenvalue adjusted by its cluster's residual
          ritzShifts_[0] = std::max(largestSafeShift_,thetaMag[0]-clusterResids[0]);

          om_->stream(Debug) << "Initializing with a conservative shift, either the most positive converged eigenvalue (" 
                             << largestSafeShift_ << ") or the eigenvalue adjusted by the residual (" << thetaMag[0] << "-" 
                             << clusterResids[0] << ").\n";

          // If this eigenvalue is NOT in a cluster, do an aggressive shift
          if(R2norms_[0] == clusterResids[0])
          {
            ritzShifts_[0] = thetaMag[0];
            om_->stream(Debug) << "Since this eigenvalue is NOT in a cluster, we can use the eigenvalue itself as a shift: ritzShifts[0]=" << ritzShifts_[0] << std::endl;
          }
          else
            om_->stream(Debug) << "This eigenvalue is in a cluster, so it would not be safe to use the eigenvalue itself as a shift\n";
        }
        else
        {
          if(largestSafeShift_ > std::abs(ritzShifts_[0]))
          {
            om_->stream(Debug) << "Initializing with a conservative shift...the most positive converged eigenvalue: " << largestSafeShift_ << std::endl;
            ritzShifts_[0] = largestSafeShift_;
          }
          else
            om_->stream(Debug) << "Using the previous value of ritzShifts[0]=" << ritzShifts_[0];
            
        }

        om_->stream(Debug) << "ritzShifts[0]=" << ritzShifts_[0] << std::endl;

        if(useMultipleShifts_)
        {
          // Compute shifts for other eigenvalues
          for(int i=1; i < blockSize_; i++)
          {
            om_->stream(Debug) << "\nSeeking a shift for theta[" << i << "]=" << thetaMag[i] << std::endl;

            // If the previous shift was aggressive and we are not in a cluster, do an aggressive shift
            if(ritzShifts_[i-1] == thetaMag[i-1] && i < blockSize_-1 && thetaMag[i] < thetaMag[i+1] - clusterResids[i+1])
            {
              ritzShifts_[i] = thetaMag[i];
              om_->stream(Debug) << "Using an aggressive shift: ritzShifts_[" << i << "]=" << ritzShifts_[i] << std::endl;
            }
            else
            {
              if(ritzShifts_[0] > std::abs(ritzShifts_[i]))
              {
                om_->stream(Debug) << "It was unsafe to use the aggressive shift.  Choose the shift used by theta[0]=" 
                                   << thetaMag[0] << ": ritzShifts[0]=" << ritzShifts_[0] << std::endl;

                // Choose a conservative shift, that of the smallest positive eigenvalue
                ritzShifts_[i] = ritzShifts_[0];
              }
              else
                om_->stream(Debug) << "It was unsafe to use the aggressive shift.  We will use the shift from the previous iteration: " << ritzShifts_[i] << std::endl;                

              om_->stream(Debug) << "Check whether any less conservative shifts would work (such as the biggest eigenvalue outside of the cluster, namely theta[ell] < " 
                                 << thetaMag[i] << "-" << clusterResids[i] << " (" << thetaMag[i] - clusterResids[i] << ")\n";

              // If possible, choose a less conservative shift, that of the biggest eigenvalue outside of the cluster
              for(int ell=0; ell < i; ell++)
              {
                if(thetaMag[ell] < thetaMag[i] - clusterResids[i])
                {
                  ritzShifts_[i] = thetaMag[ell];
                  om_->stream(Debug) << "ritzShifts_[" << i << "]=" << ritzShifts_[i] << " is valid\n";
                }
                else
                  break;
              }
            } // end else

            om_->stream(Debug) << "ritzShifts[" << i << "]=" << ritzShifts_[i] << std::endl;
          } // end for
        } // end if(useMultipleShifts_)
        else
        {
          for(int i=1; i<blockSize_; i++)
            ritzShifts_[i] = ritzShifts_[0];          
        }
      } // end if(howToShift_ == "Adjusted Ritz Values")
    } // end if(whenToShift_ == "Always" || (whenToShift_ == "After Trace Levels" && traceHasLeveled))

    // Set the correct sign
    for(int i=0; i<blockSize_; i++)
    {
      if(theta_[i] < 0)
        ritzShifts_[i] = -ritzShifts_[i];
    }
  }

  template <class ScalarType, class MV, class OP>
  std::vector<ScalarType> TraceMinBase<ScalarType,MV,OP>::computeTol()
  {
    ScalarType temp1, temp2;
    int nvecs = ritzShifts_.size();
    std::vector<ScalarType> tolerances;
  
    for(int i=0; i < nvecs; i++)
    {
      if(std::abs(theta_[0]) != std::abs(ritzShifts_[i]))
        temp1 = std::abs(theta_[i]-ritzShifts_[i])/std::abs(std::abs(theta_[0])-std::abs(ritzShifts_[i]));
      else
        temp1 = ONE;

      temp2 = pow(2.0,-iter_);

      // TODO: The min and max tolerances should not be hard coded
      //       Neither should the maximum number of iterations
      tolerances.push_back(std::max(std::min(temp1,temp2),1e-8));
    }
  
    if(nvecs > 1)
      tolerances[nvecs-1] = tolerances[nvecs-2];

    return tolerances;
  }

  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::solveSaddlePointProblem(RCP<MV> Delta)
  {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
  Teuchos::TimeMonitor lcltimer( *timerSaddle_ );
#endif

    // This case can arise when looking for the largest eigenpairs
    if(Op_ == Teuchos::null)
    {
      // dense solver
      Teuchos::SerialDenseSolver<int,ScalarType> My_Solver;  

      // Schur complement
      RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > lclS = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(blockSize_,blockSize_) );

      if(computeAllRes_)
      {
        // Get the valid indices of X
        std::vector<int> curind(blockSize_);
        for(int i=0; i<blockSize_; i++)
          curind[i] = i;

        // Get a view of MX
        RCP<const MV> lclMX = MVT::CloneView(*MX_, curind);

        // form S = X' M^2 X
        MVT::MvTransMv(ONE,*lclMX,*lclMX,*lclS);

        // compute the inverse of S
        My_Solver.setMatrix(lclS);
        My_Solver.invert();

        // Delta = X - MX/S
        RCP<const MV> lclX = MVT::CloneView(*X_, curind);
        MVT::Assign(*lclX,*Delta);
        MVT::MvTimesMatAddMv( -ONE, *lclMX, *lclS, ONE, *Delta);
      }
      else
      {
        // form S = X' M^2 X
        MVT::MvTransMv(ONE,*MX_,*MX_,*lclS);

        // compute the inverse of S
        My_Solver.setMatrix(lclS);
        My_Solver.invert();

        // Delta = X - MX/S
        MVT::Assign(*X_,*Delta);
        MVT::MvTimesMatAddMv( -ONE, *MX_, *lclS, ONE, *Delta);
      }
    }
    else
    {
      std::vector<int> order(curDim_);
      std::vector<ScalarType> tempvec(blockSize_);
      RCP<BasicSort<MagnitudeType> > sorter = rcp( new BasicSort<MagnitudeType>("SR") );

      // Stores the residual of each CLUSTER of eigenvalues
      std::vector<ScalarType> clusterResids;

      // Sort the eigenvalues in ascending order for the Ritz shift selection
      sorter->sort(theta_, Teuchos::rcpFromRef(order), curDim_);   // don't catch exception

      // Apply the same ordering to the residual norms
      getRes2Norms();
      for (int i=0; i<blockSize_; i++)
        tempvec[i] = R2norms_[order[i]];
      R2norms_ = tempvec;

      // Compute the residual of each CLUSTER of eigenvalues
      // This is important for selecting the Ritz shifts
      clusterResids = getClusterResids();

      // Sort the eigenvalues based on what the user wanted
      sm_->sort(theta_, Teuchos::rcpFromRef(order), blockSize_);

      // Apply the same ordering to the residual norms and cluster residuals
      for (int i=0; i<blockSize_; i++)
        tempvec[i] = R2norms_[order[i]];
      R2norms_ = tempvec;

      for (int i=0; i<blockSize_; i++)
        tempvec[i] = clusterResids[order[i]];
      clusterResids = tempvec;      

      // Compute the Ritz shifts
      computeRitzShifts(clusterResids);

      // Compute the tolerances for the inner solves
      std::vector<ScalarType> tolerances = computeTol();

      for(int i=0; i<blockSize_; i++)
      {
        om_->stream(IterationDetails) << "Choosing Ritz shifts...theta[" << i << "]=" 
            << theta_[i] << ", resids[" << i << "]=" << R2norms_[i] << ", clusterResids[" << i << "]=" << clusterResids[i]
            << ", ritzShifts[" << i << "]=" << ritzShifts_[i] << ", and tol[" << i << "]=" << tolerances[i] << std::endl;
      }

      // Set the Ritz shifts for the solver
      ritzOp_->setRitzShifts(ritzShifts_);

      // Set the inner stopping tolerance
      // This uses the Ritz values to determine when to stop
      ritzOp_->setInnerTol(tolerances);

      // Solve the saddle point problem
      if(saddleSolType_ == PROJECTED_KRYLOV_SOLVER)
      {
        if(Prec_ != Teuchos::null)
          solveSaddleProjPrec(Delta);
        else
          solveSaddleProj(Delta);
      }
      else if(saddleSolType_ == SCHUR_COMPLEMENT_SOLVER)
      {
        if(Z_ == Teuchos::null || MVT::GetNumberVecs(*Z_) != blockSize_)
        {
          // We do NOT want Z to be 0, because that could result in stagnation
          Z_ = MVT::Clone(*X_,blockSize_);
          MVT::MvRandom(*Z_);
        }
        solveSaddleSchur(Delta);
      }
      else if(saddleSolType_ == BD_PREC_MINRES)
      {
        solveSaddleBDPrec(Delta);
//        Delta->describe(*(Teuchos::VerboseObjectBase::getDefaultOStream()),Teuchos::VERB_EXTREME);
      }
      else
        std::cout << "Invalid saddle solver type\n";
    }
  }

  // Solve the saddle point problem using projected minres
  // TODO: We should be able to choose KX or -R as RHS.
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::solveSaddleProj (RCP<MV> Delta) const
  {
    RCP<TraceMinProjRitzOp<ScalarType,MV,OP> > projOp;

    if(computeAllRes_)
    {
      // Get the valid indices of X
      std::vector<int> curind(blockSize_);
      for(int i=0; i<blockSize_; i++)
        curind[i] = i;

      RCP<const MV> locMX = MVT::CloneView(*MX_, curind);

      // We should really project out the converged eigenvectors too
      if(projectAllVecs_)
      {
        if(projectLockedVecs_ && numAuxVecs_ > 0)
          projOp = rcp( new TraceMinProjRitzOp<ScalarType,MV,OP>(ritzOp_,locMX,orthman_,auxVecs_) );
        else
          projOp = rcp( new TraceMinProjRitzOp<ScalarType,MV,OP>(ritzOp_,locMX,orthman_) );
      }
      else
        projOp = rcp( new TraceMinProjRitzOp<ScalarType,MV,OP>(ritzOp_,locMX) );

      // Remember, Delta0 must equal 0
      // This ensures B-orthogonality between Delta and X
      MVT::MvInit(*Delta);

      if(useRHSR_)
      {
        RCP<const MV> locR = MVT::CloneView(*R_, curind);
        projOp->ApplyInverse(*locR, *Delta);
      }
      else
      {
        RCP<const MV> locKX = MVT::CloneView(*KX_, curind);
        projOp->ApplyInverse(*locKX, *Delta);
      }
    }
    else
    {
      // We should really project out the converged eigenvectors too
      if(projectAllVecs_)
      {
        if(projectLockedVecs_ && numAuxVecs_ > 0)
          projOp = rcp( new TraceMinProjRitzOp<ScalarType,MV,OP>(ritzOp_,MX_,orthman_,auxVecs_) );
        else
          projOp = rcp( new TraceMinProjRitzOp<ScalarType,MV,OP>(ritzOp_,MX_,orthman_) );
      }
      else
        projOp = rcp( new TraceMinProjRitzOp<ScalarType,MV,OP>(ritzOp_,MX_) );

      // Remember, Delta0 must equal 0
      // This ensures B-orthogonality between Delta and X
      MVT::MvInit(*Delta);

      if(useRHSR_) {
        projOp->ApplyInverse(*R_, *Delta);
      }
      else {
        projOp->ApplyInverse(*KX_, *Delta);
      }
    }
  }

  // TODO: Fix preconditioning
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::solveSaddleProjPrec (RCP<MV> Delta) const
  {
    // If we don't have Belos installed, we can't use TraceMinProjRitzOpWithPrec
    // Of course, this problem will be detected in the constructor and an exception will be thrown
    // This is only here to make sure the code compiles properly
#ifdef HAVE_ANASAZI_BELOS
    RCP<TraceMinProjRitzOpWithPrec<ScalarType,MV,OP> > projOp;

    if(computeAllRes_)
    {
      // Get the valid indices of X
      std::vector<int> curind(blockSize_);
      for(int i=0; i<blockSize_; i++)
        curind[i] = i;

      RCP<const MV> locMX = MVT::CloneView(*MX_, curind);

      // We should really project out the converged eigenvectors too
      if(projectAllVecs_)
      {
        if(projectLockedVecs_ && numAuxVecs_ > 0)
          projOp = rcp( new TraceMinProjRitzOpWithPrec<ScalarType,MV,OP>(ritzOp_,locMX,orthman_,auxVecs_) );
        else
          projOp = rcp( new TraceMinProjRitzOpWithPrec<ScalarType,MV,OP>(ritzOp_,locMX,orthman_) );
      }
      else
        projOp = rcp( new TraceMinProjRitzOpWithPrec<ScalarType,MV,OP>(ritzOp_,locMX) );

      // Remember, Delta0 must equal 0
      // This ensures B-orthogonality between Delta and X
      MVT::MvInit(*Delta);

      if(useRHSR_)
      {
        RCP<const MV> locR = MVT::CloneView(*R_, curind);
        projOp->ApplyInverse(*locR, *Delta);
        MVT::MvScale(*Delta, -ONE);
      }
      else
      {
        RCP<const MV> locKX = MVT::CloneView(*KX_, curind);
        projOp->ApplyInverse(*locKX, *Delta);
      }
    }
    else
    {
      // We should really project out the converged eigenvectors too
      if(projectAllVecs_)
      {
        if(projectLockedVecs_ && numAuxVecs_ > 0)
          projOp = rcp( new TraceMinProjRitzOpWithPrec<ScalarType,MV,OP>(ritzOp_,MX_,orthman_,auxVecs_) );
        else
          projOp = rcp( new TraceMinProjRitzOpWithPrec<ScalarType,MV,OP>(ritzOp_,MX_,orthman_) );
      }
      else
        projOp = rcp( new TraceMinProjRitzOpWithPrec<ScalarType,MV,OP>(ritzOp_,MX_) );

      // Remember, Delta0 must equal 0
      // This ensures B-orthogonality between Delta and X
      MVT::MvInit(*Delta);

      if(useRHSR_)
      {
        projOp->ApplyInverse(*R_, *Delta);
        MVT::MvScale(*Delta,-ONE);
      }
      else
        projOp->ApplyInverse(*KX_, *Delta);
    }
#endif
  }

  // TODO: We can hold the Schur complement constant in later iterations
  // TODO: Make sure we're using the preconditioner correctly
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::solveSaddleSchur (RCP<MV> Delta) const 
  {
    // dense solver
    Teuchos::SerialDenseSolver<int,ScalarType> My_Solver;  

    RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > lclL;
    RCP<Teuchos::SerialDenseMatrix<int,ScalarType> > lclS = rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(blockSize_,blockSize_) );

    if(computeAllRes_)
    {
      // Get the valid indices of X
      std::vector<int> curind(blockSize_);
      for(int i=0; i<blockSize_; i++)
        curind[i] = i;

      // Z = K \ MX
      MVT::SetBlock(*X_,curind,*Z_);
      RCP<const MV> lclMX = MVT::CloneView(*MX_, curind);
      ritzOp_->ApplyInverse(*lclMX,*Z_);

      // form S = X' M Z
      MVT::MvTransMv(ONE,*Z_,*lclMX,*lclS);

      // solve S L = I
      My_Solver.setMatrix(lclS);
      My_Solver.invert();
      lclL = lclS;

      // Delta = X - Z L
      RCP<const MV> lclX = MVT::CloneView(*X_, curind);
      MVT::Assign(*lclX,*Delta);
      MVT::MvTimesMatAddMv( -ONE, *Z_, *lclL, ONE, *Delta);
    }
    else
    {
      // Z = K \ MX
      ritzOp_->ApplyInverse(*MX_,*Z_);

      // form S = X' M Z
      MVT::MvTransMv(ONE,*Z_,*MX_,*lclS);

      // solve S L = I
      My_Solver.setMatrix(lclS);
      My_Solver.invert();
      lclL = lclS;

      // Delta = X - Z L
      MVT::Assign(*X_,*Delta);
      MVT::MvTimesMatAddMv( -ONE, *Z_, *lclL, ONE, *Delta);
    }
  }


  // TODO: We can hold the Schur complement constant in later iterations
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::solveSaddleBDPrec (RCP<MV> Delta) const 
  {
    RCP<MV> locKX, locMX;
    if(computeAllRes_)
    {
      std::vector<int> curind(blockSize_);
      for(int i=0; i<blockSize_; i++)
        curind[i] = i;
      locKX = MVT::CloneViewNonConst(*KX_, curind);
      locMX = MVT::CloneViewNonConst(*MX_, curind);
    }
    else
    {
      locKX = KX_;
      locMX = MX_;
    }

    // Create the operator [A BX; X'B 0]
    RCP< SaddleOperator<ScalarType, MV, TraceMinRitzOp<ScalarType,MV,OP> > > sadOp = rcp(new SaddleOperator<ScalarType, MV, TraceMinRitzOp<ScalarType,MV,OP> >(ritzOp_,locMX));

    // Create the RHS [AX; 0]
    RCP< SaddleContainer<ScalarType, MV> > sadRHS = rcp(new SaddleContainer<ScalarType,MV>(locKX));

    // Create the solution vector [Delta; L]
    MVT::MvInit(*Delta);
    RCP< SaddleContainer<ScalarType, MV> > sadSol = rcp(new SaddleContainer<ScalarType,MV>(Delta));

    // Create a minres solver
    RCP<PseudoBlockMinres<ScalarType,SaddleContainer<ScalarType, MV>,SaddleOperator<ScalarType, MV, TraceMinRitzOp<ScalarType,MV,OP> > > > sadSolver;
    if(Prec_ != Teuchos::null)
    {
      RCP< SaddleOperator<ScalarType, MV, TraceMinRitzOp<ScalarType,MV,OP> > > sadPrec = rcp(new SaddleOperator<ScalarType, MV, TraceMinRitzOp<ScalarType,MV,OP> >(ritzOp_,locMX,BD_PREC));
      sadSolver = rcp(new PseudoBlockMinres<ScalarType,SaddleContainer<ScalarType, MV>,SaddleOperator<ScalarType, MV, TraceMinRitzOp<ScalarType,MV,OP> > >(sadOp, sadPrec));
    }
    else {
      sadSolver = rcp(new PseudoBlockMinres<ScalarType,SaddleContainer<ScalarType, MV>,SaddleOperator<ScalarType, MV, TraceMinRitzOp<ScalarType,MV,OP> > >(sadOp));
    }

    // Set the tolerance for the minres solver
    std::vector<ScalarType> tol;
    ritzOp_->getInnerTol(tol);
    sadSolver->setTol(tol);

    // Set the maximum number of iterations
    sadSolver->setMaxIter(ritzOp_->getMaxIts());

    // Set the solution vector
    sadSolver->setSol(sadSol);

    // Set the RHS
    sadSolver->setRHS(sadRHS);

    // Solve the saddle point problem
    sadSolver->solve();
  }


  // Compute KK := V'KV
  // We only compute the NEW elements
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::computeKK()
  {
    // Get the valid indices of V
    std::vector<int> curind(curDim_);
    for(int i=0; i<curDim_; i++)
      curind[i] = i;

    // Get a pointer to the valid parts of V
    RCP<const MV> lclV = MVT::CloneView(*V_,curind);

    // Get the valid indices of KV
    curind.resize(blockSize_);
    for(int i=0; i<blockSize_; i++)
      curind[i] = curDim_-blockSize_+i;
    RCP<const MV> lclKV = MVT::CloneView(*KV_,curind);

    // Get a pointer to the valid part of KK
    RCP< Teuchos::SerialDenseMatrix<int,ScalarType> > lclKK = 
        rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*KK_,curDim_,blockSize_,0,curDim_-blockSize_) );

    // KK := V'KV
    MVT::MvTransMv(ONE,*lclV,*lclKV,*lclKK);

    // We only constructed the upper triangular part of the matrix, but that's okay because KK is symmetric!
    for(int r=0; r<curDim_; r++)
    {
      for(int c=0; c<r; c++)
      {
        (*KK_)(r,c) = (*KK_)(c,r);
      }
    }
  }

  // Compute the eigenpairs of KK, i.e. the Ritz pairs
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::computeRitzPairs()
  {
    // Get a pointer to the valid part of KK
    RCP< Teuchos::SerialDenseMatrix<int,ScalarType> > lclKK = 
        rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*KK_,curDim_,curDim_) );

    // Get a pointer to the valid part of ritzVecs
    RCP< Teuchos::SerialDenseMatrix<int,ScalarType> > lclRV = 
        rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*ritzVecs_,curDim_,curDim_) );

    // Compute Ritz pairs from KK
    {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
      Teuchos::TimeMonitor lcltimer( *timerDS_ );
#endif
      int rank = curDim_;
      Utils::directSolver(curDim_, *lclKK, Teuchos::null, *lclRV, theta_, rank, 10);
      // we want all ritz values back
      // TODO: This probably should not be an ortho failure
      TEUCHOS_TEST_FOR_EXCEPTION(rank != curDim_,TraceMinBaseOrthoFailure,
             "Anasazi::TraceMinBase::computeRitzPairs(): Failed to compute all eigenpairs of KK.");
    }

    // Sort ritz pairs
    {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
      Teuchos::TimeMonitor lcltimer( *timerSortEval_ );
#endif

      std::vector<int> order(curDim_);
      //
      // sort the first curDim_ values in theta_
      sm_->sort(theta_, Teuchos::rcpFromRef(order), curDim_);   // don't catch exception
      //
      // apply the same ordering to the primitive ritz vectors
      Utils::permuteVectors(order,*lclRV);
    }
  }

  // Compute X := V evecs
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::computeX()
  {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor lcltimer( *timerLocal_ );
#endif

    // Get the valid indices of V
    std::vector<int> curind(curDim_);
    for(int i=0; i<curDim_; i++)
      curind[i] = i;

    // Get a pointer to the valid parts of V
    RCP<const MV> lclV = MVT::CloneView(*V_,curind);

    if(computeAllRes_)
    {
      // Capture the relevant eigenvectors
      RCP< Teuchos::SerialDenseMatrix<int,ScalarType> > relevantEvecs = 
          rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*ritzVecs_,curDim_,curDim_) ); 

      // X <- lclV*S
      RCP<MV> lclX = MVT::CloneViewNonConst(*X_,curind);
      MVT::MvTimesMatAddMv( ONE, *lclV, *relevantEvecs, ZERO, *lclX );
    }
    else
    {
      // Capture the relevant eigenvectors
      RCP< Teuchos::SerialDenseMatrix<int,ScalarType> > relevantEvecs = 
          rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*ritzVecs_,curDim_,blockSize_) ); 

      // X <- lclV*S
      MVT::MvTimesMatAddMv( ONE, *lclV, *relevantEvecs, ZERO, *X_ );
    }
  }

  // Compute KX := KV evecs and (if necessary) MX := MV evecs
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::updateKXMX()
  {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor lcltimer( *timerLocal_ );
#endif

    // Get the valid indices of V
    std::vector<int> curind(curDim_);
    for(int i=0; i<curDim_; i++)
      curind[i] = i;

    // Get pointers to the valid parts of V, KV, and MV (if necessary)
    RCP<const MV> lclV = MVT::CloneView(*V_,curind);
    RCP<const MV> lclKV = MVT::CloneView(*KV_,curind);

    if(computeAllRes_)
    {
      // Capture the relevant eigenvectors
      RCP< Teuchos::SerialDenseMatrix<int,ScalarType> > relevantEvecs = 
          rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*ritzVecs_,curDim_,curDim_) ); 

      // Update KX and MX
      RCP<MV> lclKX = MVT::CloneViewNonConst(*KX_,curind);
      MVT::MvTimesMatAddMv( ONE, *lclKV, *relevantEvecs, ZERO, *lclKX );
      if(hasM_)
      {
        RCP<const MV> lclMV = MVT::CloneView(*MV_,curind);
        RCP<MV> lclMX = MVT::CloneViewNonConst(*MX_,curind);
        MVT::MvTimesMatAddMv( ONE, *lclMV, *relevantEvecs, ZERO, *lclMX );
      }
    }
    else
    {
      // Capture the relevant eigenvectors
      RCP< Teuchos::SerialDenseMatrix<int,ScalarType> > relevantEvecs = 
          rcp( new Teuchos::SerialDenseMatrix<int,ScalarType>(Teuchos::View,*ritzVecs_,curDim_,blockSize_) ); 

      // Update KX and MX
      MVT::MvTimesMatAddMv( ONE, *lclKV, *relevantEvecs, ZERO, *KX_ );
      if(hasM_)
      {
        RCP<const MV> lclMV = MVT::CloneView(*MV_,curind);
        MVT::MvTimesMatAddMv( ONE, *lclMV, *relevantEvecs, ZERO, *MX_ );
      }
    }
  }

  // Update the residual R := KX - MX*T
  template <class ScalarType, class MV, class OP>
  void TraceMinBase<ScalarType,MV,OP>::updateResidual () {
#ifdef ANASAZI_TEUCHOS_TIME_MONITOR
    Teuchos::TimeMonitor lcltimer( *timerCompRes_ );
#endif

    if(computeAllRes_)
    {
      // Get the valid indices of X
      std::vector<int> curind(curDim_);
      for(int i=0; i<curDim_; i++)
        curind[i] = i;

      // Holds MX*T
      RCP<MV> MXT = MVT::CloneCopy(*MX_,curind);

      // Holds the relevant part of theta
      std::vector<ScalarType> locTheta(curDim_);
      for(int i=0; i<curDim_; i++)
        locTheta[i] = theta_[i];

      // Compute MX*T
      MVT::MvScale(*MXT,locTheta);

      // form R <- KX - MX*T
      RCP<const MV> locKX = MVT::CloneView(*KX_,curind);
      RCP<MV> locR = MVT::CloneViewNonConst(*R_,curind);
      MVT::MvAddMv(ONE,*locKX,-ONE,*MXT,*locR);
    }
    else
    {
      // Holds MX*T
      RCP<MV> MXT = MVT::CloneCopy(*MX_);

      // Holds the relevant part of theta
      std::vector<ScalarType> locTheta(blockSize_);
      for(int i=0; i<blockSize_; i++)
        locTheta[i] = theta_[i];

      // Compute MX*T
      MVT::MvScale(*MXT,locTheta);

      // form R <- KX - MX*T
      MVT::MvAddMv(ONE,*KX_,-ONE,*MXT,*R_);
    }

    // R has been updated; mark the norms as out-of-date
    Rnorms_current_ = false;
    R2norms_current_ = false;
  }


  // Check accuracy, orthogonality, and other debugging stuff
  // 
  // bools specify which tests we want to run (instead of running more than we actually care about)
  //
  // we don't bother checking the following because they are computed explicitly:
  //    H == Prec*R
  //   KH == K*H
  //
  // 
  // checkV : V orthonormal
  //          orthogonal to auxvecs
  // checkX : X orthonormal
  //          orthogonal to auxvecs
  // checkMX: check MX == M*X
  // checkKX: check KX == K*X
  // checkH : H orthonormal 
  //          orthogonal to V and H and auxvecs
  // checkMH: check MH == M*H
  // checkR : check R orthogonal to X
  // checkQ : check that auxiliary vectors are actually orthonormal
  // checkKK: check that KK is symmetric in memory 
  //
  // TODO: 
  //  add checkTheta 
  //
  template <class ScalarType, class MV, class OP>
  std::string TraceMinBase<ScalarType,MV,OP>::accuracyCheck( const CheckList &chk, const std::string &where ) const 
  {
    using std::endl;

    std::stringstream os;
    os.precision(2);
    os.setf(std::ios::scientific, std::ios::floatfield);

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

    // V and friends
    std::vector<int> lclind(curDim_);
    for (int i=0; i<curDim_; ++i) lclind[i] = i;
    RCP<const MV> lclV;
    if (initialized_) {
      lclV = MVT::CloneView(*V_,lclind);
    }
    if (chk.checkV && initialized_) {
      MagnitudeType err = orthman_->orthonormError(*lclV);
      os << " >> Error in V^H M V == I  : " << err << endl;
      for (Array_size_type i=0; i<auxVecs_.size(); ++i) {
        err = orthman_->orthogError(*lclV,*auxVecs_[i]);
        os << " >> Error in V^H M Q[" << i << "] == 0 : " << err << endl;
      }
    }

    // X and friends
    RCP<const MV> lclX;
    if(initialized_)
    {
      if(computeAllRes_)
        lclX = MVT::CloneView(*X_,lclind);
      else
        lclX = X_;
    }

    if (chk.checkX && initialized_) {
      MagnitudeType err = orthman_->orthonormError(*lclX);
      os << " >> Error in X^H M X == I  : " << err << endl;
      for (Array_size_type i=0; i<auxVecs_.size(); ++i) {
        err = orthman_->orthogError(*lclX,*auxVecs_[i]);
        os << " >> Error in X^H M Q[" << i << "] == 0 : " << err << endl;
      }
    }
    if (chk.checkMX && hasM_ && initialized_) {
      RCP<const MV> lclMX;
      if(computeAllRes_)
        lclMX = MVT::CloneView(*MX_,lclind);
      else
        lclMX = MX_;

      MagnitudeType err = Utils::errorEquality(*lclX, *lclMX, MOp_);
      os << " >> Error in MX == M*X     : " << err << endl;
    }
    if (Op_ != Teuchos::null && chk.checkKX && initialized_) {
      RCP<const MV> lclKX;
      if(computeAllRes_)
        lclKX = MVT::CloneView(*KX_,lclind);
      else
        lclKX = KX_;

      MagnitudeType err = Utils::errorEquality(*lclX, *lclKX, Op_);
      os << " >> Error in KX == K*X     : " << err << endl;
    }

    // KK
    if (chk.checkKK && initialized_) {
      Teuchos::SerialDenseMatrix<int,ScalarType> curKK(curDim_,curDim_);
      if(Op_ != Teuchos::null) {
        RCP<MV> lclKV = MVT::Clone(*V_,curDim_);
        OPT::Apply(*Op_,*lclV,*lclKV);
        MVT::MvTransMv(ONE,*lclV,*lclKV,curKK);
      }
      else {
        MVT::MvTransMv(ONE,*lclV,*lclV,curKK);
      }
      Teuchos::SerialDenseMatrix<int,ScalarType> subKK(Teuchos::View,*KK_,curDim_,curDim_);
      curKK -= subKK;
      os << " >> Error in V^H K V == KK : " << curKK.normFrobenius() << endl;

      Teuchos::SerialDenseMatrix<int,ScalarType> SDMerr(curDim_,curDim_);
      for (int j=0; j<curDim_; ++j) {
        for (int i=0; i<curDim_; ++i) {
          SDMerr(i,j) = subKK(i,j) - SCT::conjugate(subKK(j,i));
        }
      }
      os << " >> Error in KK - KK^H == 0 : " << SDMerr.normFrobenius() << endl;
    }

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

    os << endl;

    return os.str();
  }
  
}} // End of namespace Anasazi

#endif

// End of file AnasaziTraceMinBase.hpp