AnasaziTsqrOrthoManagerImpl.hpp

// @HEADER
// *****************************************************************************
//                 Anasazi: Block Eigensolvers Package
//
// Copyright 2004 NTESS and the Anasazi contributors.
// SPDX-License-Identifier: BSD-3-Clause
// *****************************************************************************
// @HEADER

#ifndef __AnasaziTsqrOrthoManagerImpl_hpp
#define __AnasaziTsqrOrthoManagerImpl_hpp

#include "AnasaziConfigDefs.hpp"
#include "AnasaziMultiVecTraits.hpp"
#include "AnasaziOrthoManager.hpp" // OrthoError, etc.

#include "Teuchos_as.hpp"
#include "Teuchos_LAPACK.hpp"
#include "Teuchos_ParameterList.hpp"
#include "Teuchos_ParameterListAcceptorDefaultBase.hpp"
#include <algorithm>


namespace Anasazi {

  class TsqrOrthoError : public OrthoError {
  public: 
    TsqrOrthoError (const std::string& what_arg) : 
      OrthoError (what_arg) {}
  };

  class TsqrOrthoFault : public OrthoError {
  public: 
    TsqrOrthoFault (const std::string& what_arg) : 
      OrthoError (what_arg) {}
  };

  template<class Scalar, class MV>
  class TsqrOrthoManagerImpl :
    public Teuchos::ParameterListAcceptorDefaultBase {
  public:
    typedef Scalar scalar_type;
    typedef typename Teuchos::ScalarTraits<Scalar>::magnitudeType magnitude_type;
    typedef MV multivector_type;
    typedef Teuchos::SerialDenseMatrix<int, Scalar> mat_type;
    typedef Teuchos::RCP<mat_type> mat_ptr;

  private:
    typedef Teuchos::ScalarTraits<Scalar> SCT;
    typedef Teuchos::ScalarTraits<magnitude_type> SCTM;
    typedef MultiVecTraits<Scalar, MV> MVT;
    typedef typename MVT::tsqr_adaptor_type tsqr_adaptor_type;

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

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

    Teuchos::RCP<const Teuchos::ParameterList> getFastParameters ();

    TsqrOrthoManagerImpl (const Teuchos::RCP<Teuchos::ParameterList>& params,
                          const std::string& label);

    TsqrOrthoManagerImpl (const std::string& label);

    void setLabel (const std::string& label) { 
      if (label != label_) {
        label_ = label; 
      }
    }

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

    void 
    innerProd (const MV& X, const MV& Y, mat_type& Z) const
    {
      MVT::MvTransMv (SCT::one(), X, Y, Z);
    }

    void
    norm (const MV& X, std::vector<magnitude_type>& normvec) const;

    void 
    project (MV& X, 
             Teuchos::Array<mat_ptr> C, 
             Teuchos::ArrayView<Teuchos::RCP<const MV> > Q);

    int normalize (MV& X, mat_ptr B);

    int 
    normalizeOutOfPlace (MV& X, MV& Q, mat_ptr B);

    int 
    projectAndNormalize (MV &X,
                         Teuchos::Array<mat_ptr> C,
                         mat_ptr B,
                         Teuchos::ArrayView<Teuchos::RCP<const MV> > Q)
    {
      // "false" means we work on X in place.  The second argument is
      // not read or written in that case.
      return projectAndNormalizeImpl (X, X, false, C, B, Q);
    }

    int 
    projectAndNormalizeOutOfPlace (MV& X_in, 
                                   MV& X_out,
                                   Teuchos::Array<mat_ptr> C,
                                   mat_ptr B,
                                   Teuchos::ArrayView<Teuchos::RCP<const MV> > Q)
    {
      // "true" means we work on X_in out of place, writing the
      // results into X_out.
      return projectAndNormalizeImpl (X_in, X_out, true, C, B, Q);
    }

    magnitude_type
    orthonormError (const MV &X) const
    {
      const Scalar ONE = SCT::one();
      const int ncols = MVT::GetNumberVecs(X);
      mat_type XTX (ncols, ncols);
      innerProd (X, X, XTX);
      for (int k = 0; k < ncols; ++k) {
        XTX(k,k) -= ONE;
      }
      return XTX.normFrobenius();
    }

    magnitude_type 
    orthogError (const MV &X1, 
                 const MV &X2) const
    {
      const int ncols_X1 = MVT::GetNumberVecs (X1);
      const int ncols_X2 = MVT::GetNumberVecs (X2);
      mat_type X1_T_X2 (ncols_X1, ncols_X2);
      innerProd (X1, X2, X1_T_X2);
      return X1_T_X2.normFrobenius();
    }

    magnitude_type blockReorthogThreshold() const { return blockReorthogThreshold_; }

    magnitude_type relativeRankTolerance() const { return relativeRankTolerance_; }

  private:
    Teuchos::RCP<Teuchos::ParameterList> params_;

    mutable Teuchos::RCP<const Teuchos::ParameterList> defaultParams_;

    std::string label_;

    tsqr_adaptor_type tsqrAdaptor_;

    Teuchos::RCP<MV> Q_;

    magnitude_type eps_;


    bool randomizeNullSpace_;

    bool reorthogonalizeBlocks_;

    bool throwOnReorthogFault_;

    magnitude_type blockReorthogThreshold_;

    magnitude_type relativeRankTolerance_;

    bool forceNonnegativeDiagonal_;

    void
    raiseReorthogFault (const std::vector<magnitude_type>& normsAfterFirstPass,
                        const std::vector<magnitude_type>& normsAfterSecondPass,
                        const std::vector<int>& faultIndices);

    void
    checkProjectionDims (int& ncols_X, 
                         int& num_Q_blocks,
                         int& ncols_Q_total,
                         const MV& X, 
                         Teuchos::ArrayView<Teuchos::RCP<const MV> > Q) const;

    void
    allocateProjectionCoefficients (Teuchos::Array<mat_ptr>& C, 
                                    Teuchos::ArrayView<Teuchos::RCP<const MV> > Q,
                                    const MV& X,
                                    const bool attemptToRecycle = true) const;

    int 
    projectAndNormalizeImpl (MV& X_in, 
                             MV& X_out,
                             const bool outOfPlace,
                             Teuchos::Array<mat_ptr> C,
                             mat_ptr B,
                             Teuchos::ArrayView<Teuchos::RCP<const MV> > Q);

    void
    rawProject (MV& X, 
                Teuchos::ArrayView<Teuchos::RCP<const MV> > Q,
                Teuchos::ArrayView<mat_ptr> C) const;

    void
    rawProject (MV& X, 
                const Teuchos::RCP<const MV>& Q, 
                const mat_ptr& C) const;

    int rawNormalize (MV& X, MV& Q, mat_type& B);

    int normalizeOne (MV& X, mat_ptr B) const;

    int normalizeImpl (MV& X, MV& Q, mat_ptr B, const bool outOfPlace);
  };

  template<class Scalar, class MV>
  void
  TsqrOrthoManagerImpl<Scalar, MV>::
  setParameterList (const Teuchos::RCP<Teuchos::ParameterList>& params)
  {
    using Teuchos::ParameterList;
    using Teuchos::parameterList;
    using Teuchos::RCP;
    using Teuchos::sublist;
    typedef magnitude_type M; // abbreviation.

    RCP<const ParameterList> defaultParams = getValidParameters ();
    // Sublist of TSQR implementation parameters; to get below.
    RCP<ParameterList> tsqrParams;

    RCP<ParameterList> theParams;
    if (params.is_null()) {
      theParams = parameterList (*defaultParams);
    } else {
      theParams = params;

      // Don't call validateParametersAndSetDefaults(); we prefer to
      // ignore parameters that we don't recognize, at least for now.
      // However, we do fill in missing parameters with defaults.

      randomizeNullSpace_ = 
        theParams->get<bool> ("randomizeNullSpace", 
                              defaultParams->get<bool> ("randomizeNullSpace"));
      reorthogonalizeBlocks_ = 
        theParams->get<bool> ("reorthogonalizeBlocks", 
                              defaultParams->get<bool> ("reorthogonalizeBlocks"));
      throwOnReorthogFault_ = 
        theParams->get<bool> ("throwOnReorthogFault", 
                              defaultParams->get<bool> ("throwOnReorthogFault"));
      blockReorthogThreshold_ = 
        theParams->get<M> ("blockReorthogThreshold",
                           defaultParams->get<M> ("blockReorthogThreshold"));
      relativeRankTolerance_ = 
        theParams->get<M> ("relativeRankTolerance", 
                           defaultParams->get<M> ("relativeRankTolerance"));
      forceNonnegativeDiagonal_ = 
        theParams->get<bool> ("forceNonnegativeDiagonal", 
                              defaultParams->get<bool> ("forceNonnegativeDiagonal"));

      // Get the sublist of TSQR implementation parameters.  Use the
      // default sublist if one isn't provided.
      if (! theParams->isSublist ("TSQR implementation")) {
        theParams->set ("TSQR implementation", 
                        defaultParams->sublist ("TSQR implementation"));
      }
      tsqrParams = sublist (theParams, "TSQR implementation", true);
    }

    // Send the TSQR implementation parameters to the TSQR adaptor.
    tsqrAdaptor_.setParameterList (tsqrParams);

    // Save the input parameter list.
    setMyParamList (theParams);
  }

  template<class Scalar, class MV>
  TsqrOrthoManagerImpl<Scalar, MV>::
  TsqrOrthoManagerImpl (const Teuchos::RCP<Teuchos::ParameterList>& params,
                        const std::string& label) :
    label_ (label),
    Q_ (Teuchos::null),               // Initialized on demand
    eps_ (SCTM::eps()),               // Machine precision
    randomizeNullSpace_ (true),
    reorthogonalizeBlocks_ (true),
    throwOnReorthogFault_ (false),
    blockReorthogThreshold_ (0),
    relativeRankTolerance_ (0),
    forceNonnegativeDiagonal_ (false)
  {
    setParameterList (params); // This also sets tsqrAdaptor_'s parameters.
  }

  template<class Scalar, class MV>
  TsqrOrthoManagerImpl<Scalar, MV>::
  TsqrOrthoManagerImpl (const std::string& label) :
    label_ (label),
    Q_ (Teuchos::null),               // Initialized on demand
    eps_ (SCTM::eps()),               // Machine precision
    randomizeNullSpace_ (true),
    reorthogonalizeBlocks_ (true),
    throwOnReorthogFault_ (false),
    blockReorthogThreshold_ (0),
    relativeRankTolerance_ (0), 
    forceNonnegativeDiagonal_ (false) 
  {
    setParameterList (Teuchos::null); // Set default parameters.
  }

  template<class Scalar, class MV>
  void
  TsqrOrthoManagerImpl<Scalar, MV>::
  norm (const MV& X, std::vector<magnitude_type>& normVec) const
  {
    const int numCols = MVT::GetNumberVecs (X);
    // std::vector<T>::size_type is unsigned; int is signed.  Mixed
    // unsigned/signed comparisons trigger compiler warnings.
    if (normVec.size() < static_cast<size_t>(numCols))
      normVec.resize (numCols); // Resize normvec if necessary.
    MVT::MvNorm (X, normVec);
  }

  template<class Scalar, class MV>
  void
  TsqrOrthoManagerImpl<Scalar, MV>::project (MV& X, 
                                             Teuchos::Array<mat_ptr> C,
                                             Teuchos::ArrayView<Teuchos::RCP<const MV> > Q)
  {
    int ncols_X, num_Q_blocks, ncols_Q_total;
    checkProjectionDims (ncols_X, num_Q_blocks, ncols_Q_total, X, Q);
    // Test for quick exit: any dimension of X is zero, or there are
    // zero Q blocks, or the total number of columns of the Q blocks
    // is zero.
    if (ncols_X == 0 || num_Q_blocks == 0 || ncols_Q_total == 0)
      return;

    // Make space for first-pass projection coefficients
    allocateProjectionCoefficients (C, Q, X, true);

    // We only use columnNormsBefore and compute pre-projection column
    // norms if doing block reorthogonalization.
    std::vector<magnitude_type> columnNormsBefore (ncols_X, magnitude_type(0));
    if (reorthogonalizeBlocks_)
      MVT::MvNorm (X, columnNormsBefore);

    // Project (first block orthogonalization step): 
    // C := Q^* X, X := X - Q C.
    rawProject (X, Q, C); 

    // If we are doing block reorthogonalization, reorthogonalize X if
    // necessary.
    if (reorthogonalizeBlocks_)
      {
        std::vector<magnitude_type> columnNormsAfter (ncols_X, magnitude_type(0));
        MVT::MvNorm (X, columnNormsAfter);
        
        // Relative block reorthogonalization threshold
        const magnitude_type relThres = blockReorthogThreshold();
        // Reorthogonalize X if any of its column norms decreased by a
        // factor more than the block reorthogonalization threshold.
        // Don't bother trying to subset the columns; that will make
        // the columns noncontiguous and thus hinder BLAS 3
        // optimizations.
        bool reorthogonalize = false;
        for (int j = 0; j < ncols_X; ++j)
          if (columnNormsAfter[j] < relThres * columnNormsBefore[j])
            {
              reorthogonalize = true;
              break;
            }
        if (reorthogonalize)
          {
            // Second-pass projection coefficients
            Teuchos::Array<mat_ptr> C2;
            allocateProjectionCoefficients (C2, Q, X, false);

            // Perform the second projection pass:
            // C2 = Q' X, X = X - Q*C2
            rawProject (X, Q, C2);
            // Update the projection coefficients
            for (int k = 0; k < num_Q_blocks; ++k)
              *C[k] += *C2[k];
          }
      }
  }



  template<class Scalar, class MV>
  int 
  TsqrOrthoManagerImpl<Scalar, MV>::normalize (MV& X, mat_ptr B)
  {
    using Teuchos::Range1D;
    using Teuchos::RCP;

    // MVT returns int for this, even though the "local ordinal
    // type" of the MV may be some other type (for example,
    // Tpetra::MultiVector<double, int32_t, int64_t, ...>).
    const int numCols = MVT::GetNumberVecs (X);

    // This special case (for X having only one column) makes
    // TsqrOrthoManagerImpl equivalent to Modified Gram-Schmidt
    // orthogonalization with conditional full reorthogonalization,
    // if all multivector inputs have only one column.  It also
    // avoids allocating Q_ scratch space and copying data when we
    // don't need to invoke TSQR at all.
    if (numCols == 1) {
      return normalizeOne (X, B);
    }

    // We use Q_ as scratch space for the normalization, since TSQR
    // requires a scratch multivector (it can't factor in place).  Q_
    // should come from a vector space compatible with X's vector
    // space, and Q_ should have at least as many columns as X.
    // Otherwise, we have to reallocate.  We also have to allocate
    // (not "re-") Q_ if we haven't allocated it before.  (We can't
    // allocate Q_ until we have some X, so we need a multivector as
    // the "prototype.")
    //
    // NOTE (mfh 11 Jan 2011) We only increase the number of columsn
    // in Q_, never decrease.  This is OK for typical uses of TSQR,
    // but you might prefer different behavior in some cases.
    //
    // NOTE (mfh 10 Mar 2011) We should only reuse the scratch space
    // Q_ if X and Q_ have compatible data distributions.  However,
    // Belos' current MultiVecTraits interface does not let us check
    // for this.  Thus, we can only check whether X and Q_ have the
    // same number of rows.  This will behave correctly for the common
    // case in Belos that all multivectors with the same number of
    // rows have the same data distribution.
    //
    // The specific MV implementation may do more checks than this on
    // its own and throw an exception if X and Q_ are not compatible,
    // but it may not.  If you find that recycling the Q_ space causes
    // troubles, you may consider modifying the code below to
    // reallocate Q_ for every X that comes in.  
    if (Q_.is_null() || 
        MVT::GetGlobalLength(*Q_) != MVT::GetGlobalLength(X) ||
        numCols > MVT::GetNumberVecs (*Q_)) {
      Q_ = MVT::Clone (X, numCols);
    }

    // normalizeImpl() wants the second MV argument to have the same
    // number of columns as X.  To ensure this, we pass it a view of
    // Q_ if Q_ has more columns than X.  (This is possible if we
    // previously called normalize() with a different multivector,
    // since we never reallocate Q_ if it has more columns than
    // necessary.)
    if (MVT::GetNumberVecs(*Q_) == numCols) {
      return normalizeImpl (X, *Q_, B, false);
    } else {
      RCP<MV> Q_view = MVT::CloneViewNonConst (*Q_, Range1D(0, numCols-1));
      return normalizeImpl (X, *Q_view, B, false);
    }
  }

  template<class Scalar, class MV>
  void
  TsqrOrthoManagerImpl<Scalar, MV>::
  allocateProjectionCoefficients (Teuchos::Array<mat_ptr>& C, 
                                  Teuchos::ArrayView<Teuchos::RCP<const MV> > Q,
                                  const MV& X,
                                  const bool attemptToRecycle) const
  {
    const int num_Q_blocks = Q.size();
    const int ncols_X = MVT::GetNumberVecs (X);
    C.resize (num_Q_blocks);
    if (attemptToRecycle)
      {
        for (int i = 0; i < num_Q_blocks; ++i) 
          {
            const int ncols_Qi = MVT::GetNumberVecs (*Q[i]);
            // Create a new C[i] if necessary, otherwise resize if
            // necessary, otherwise fill with zeros.
            if (C[i].is_null())
              C[i] = rcp (new mat_type (ncols_Qi, ncols_X));
            else 
              {
                mat_type& Ci = *C[i];
                if (Ci.numRows() != ncols_Qi || Ci.numCols() != ncols_X)
                  Ci.shape (ncols_Qi, ncols_X);
                else
                  Ci.putScalar (SCT::zero());
              }
          }
      }
    else
      {
        for (int i = 0; i < num_Q_blocks; ++i) 
          {
            const int ncols_Qi = MVT::GetNumberVecs (*Q[i]);
            C[i] = rcp (new mat_type (ncols_Qi, ncols_X));
          }
      }
  }

  template<class Scalar, class MV>
  int 
  TsqrOrthoManagerImpl<Scalar, MV>::
  normalizeOutOfPlace (MV& X, MV& Q, mat_ptr B)
  {
    const int numVecs = MVT::GetNumberVecs(X);
    if (numVecs == 0) {
      return 0; // Nothing to do.
    } else if (numVecs == 1) {
      // Special case for a single column; scale and copy over.
      using Teuchos::Range1D;
      using Teuchos::RCP;
      using Teuchos::rcp;

      // Normalize X in place (faster than TSQR for one column).
      const int rank = normalizeOne (X, B);
      // Copy results to first column of Q.
      RCP<MV> Q_0 = MVT::CloneViewNonConst (Q, Range1D(0,0));
      MVT::Assign (X, *Q_0);
      return rank;
    } else {
      // The "true" argument to normalizeImpl() means the output
      // vectors go into Q, and the contents of X are overwritten with
      // invalid values.
      return normalizeImpl (X, Q, B, true);
    }
  }

  template<class Scalar, class MV>
  int 
  TsqrOrthoManagerImpl<Scalar, MV>::
  projectAndNormalizeImpl (MV& X_in, 
                           MV& X_out, // Only written if outOfPlace==false.
                           const bool outOfPlace,
                           Teuchos::Array<mat_ptr> C,
                           mat_ptr B,
                           Teuchos::ArrayView<Teuchos::RCP<const MV> > Q)
  {
    using Teuchos::Range1D;
    using Teuchos::RCP;
    using Teuchos::rcp;

    if (outOfPlace) {
      // Make sure that X_out has at least as many columns as X_in.
      TEUCHOS_TEST_FOR_EXCEPTION(MVT::GetNumberVecs(X_out) < MVT::GetNumberVecs(X_in),
                         std::invalid_argument, 
                         "Belos::TsqrOrthoManagerImpl::"
                         "projectAndNormalizeOutOfPlace(...):"
                         "X_out has " << MVT::GetNumberVecs(X_out) 
                         << " columns, but X_in has "
                         << MVT::GetNumberVecs(X_in) << " columns.");
    }
    // Fetch dimensions of X_in and Q, and allocate space for first-
    // and second-pass projection coefficients (C resp. C2).
    int ncols_X, num_Q_blocks, ncols_Q_total;
    checkProjectionDims (ncols_X, num_Q_blocks, ncols_Q_total, X_in, Q);

    // Test for quick exit: if any dimension of X is zero.
    if (ncols_X == 0) {
      return 0;
    }
    // If there are zero Q blocks or zero Q columns, just normalize!
    if (num_Q_blocks == 0 || ncols_Q_total == 0) {
      if (outOfPlace) {
        return normalizeOutOfPlace (X_in, X_out, B);
      } else {
        return normalize (X_in, B);
      }
    }

    // The typical case is that the entries of C have been allocated
    // before, so we attempt to recycle the allocations.  The call
    // below will reallocate if it cannot recycle.
    allocateProjectionCoefficients (C, Q, X_in, true);

    // If we are doing block reorthogonalization, then compute the
    // column norms of X before projecting for the first time.  This
    // will help us decide whether we need to reorthogonalize X.
    std::vector<magnitude_type> normsBeforeFirstPass (ncols_X, SCTM::zero());
    if (reorthogonalizeBlocks_) {
      MVT::MvNorm (X_in, normsBeforeFirstPass);
    }

    // First (Modified) Block Gram-Schmidt pass, in place in X_in.
    rawProject (X_in, Q, C);

    // Make space for the normalization coefficients.  This will
    // either be a freshly allocated matrix (if B is null), or a view
    // of the appropriately sized upper left submatrix of *B (if B is
    // not null).
    //
    // Note that if we let the normalize() routine allocate (in the
    // case that B is null), that storage will go away at the end of
    // normalize().  (This is because it passes the RCP by value, not
    // by reference.)
    mat_ptr B_out;
    if (B.is_null()) {
      B_out = rcp (new mat_type (ncols_X, ncols_X));
    } else {
      // Make sure that B is no smaller than numCols x numCols.
      TEUCHOS_TEST_FOR_EXCEPTION(B->numRows() < ncols_X || B->numCols() < ncols_X,
                         std::invalid_argument,
                         "normalizeOne: Input matrix B must be at "
                         "least " << ncols_X << " x " << ncols_X 
                         << ", but is instead " << B->numRows()
                         << " x " << B->numCols() << ".");
      // Create a view of the ncols_X by ncols_X upper left
      // submatrix of *B.  TSQR will write the normalization
      // coefficients there.
      B_out = rcp (new mat_type (Teuchos::View, *B, ncols_X, ncols_X));
    }

    // Rank of X(_in) after first projection pass.  If outOfPlace,
    // this overwrites X_in with invalid values, and the results go in
    // X_out.  Otherwise, it's done in place in X_in.
    const int firstPassRank = outOfPlace ? 
      normalizeOutOfPlace (X_in, X_out, B_out) : 
      normalize (X_in, B_out);
    if (B.is_null()) {
      // The input matrix B is null, so assign B_out to it.  If B was
      // not null on input, then B_out is a view of *B, so we don't
      // have to do anything here.  Note that SerialDenseMatrix uses
      // raw pointers to store data and represent views, so we have to
      // be careful about scope.
      B = B_out;
    }
    int rank = firstPassRank; // Current rank of X.

    // If X was not full rank after projection and randomizeNullSpace_
    // is true, then normalize(OutOfPlace)() replaced the null space
    // basis of X with random vectors, and orthogonalized them against
    // the column space basis of X.  However, we still need to
    // orthogonalize the random vectors against the Q[i], after which
    // we will need to renormalize them.
    //
    // If outOfPlace, then we need to work in X_out (where
    // normalizeOutOfPlace() wrote the normalized vectors).
    // Otherwise, we need to work in X_in.
    //
    // Note: We don't need to keep the new projection coefficients,
    // since they are multiplied by the "small" part of B
    // corresponding to the null space of the original X.
    if (firstPassRank < ncols_X && randomizeNullSpace_) {
      const int numNullSpaceCols = ncols_X - firstPassRank;
      const Range1D nullSpaceIndices (firstPassRank, ncols_X - 1);

      // Space for projection coefficients (will be thrown away)
      Teuchos::Array<mat_ptr> C_null (num_Q_blocks);
      for (int k = 0; k < num_Q_blocks; ++k) {
        const int numColsQk = MVT::GetNumberVecs(*Q[k]);
        C_null[k] = rcp (new mat_type (numColsQk, numNullSpaceCols));
      }
      // Space for normalization coefficients (will be thrown away).
      RCP<mat_type> B_null (new mat_type (numNullSpaceCols, numNullSpaceCols));

      int randomVectorsRank;
      if (outOfPlace) {
        // View of the null space basis columns of X.
        // normalizeOutOfPlace() wrote them into X_out.
        RCP<MV> X_out_null = MVT::CloneViewNonConst (X_out, nullSpaceIndices);
        // Use X_in for scratch space.  Copy X_out_null into the
        // last few columns of X_in (X_in_null) and do projections
        // in there.  (This saves us a copy wen we renormalize
        // (out of place) back into X_out.)
        RCP<MV> X_in_null = MVT::CloneViewNonConst (X_in, nullSpaceIndices);
        MVT::Assign (*X_out_null, *X_in_null);
        // Project the new random vectors against the Q blocks, and
        // renormalize the result into X_out_null.  
        rawProject (*X_in_null, Q, C_null);
        randomVectorsRank = normalizeOutOfPlace (*X_in_null, *X_out_null, B_null);
      } else {
        // View of the null space columns of X.  
        // They live in X_in.
        RCP<MV> X_null = MVT::CloneViewNonConst (X_in, nullSpaceIndices);
        // Project the new random vectors against the Q blocks,
        // and renormalize the result (in place).
        rawProject (*X_null, Q, C_null);
        randomVectorsRank = normalize (*X_null, B_null);
      }
      // While unusual, it is still possible for the random data not
      // to be full rank after projection and normalization.  In that
      // case, we could try another set of random data and recurse as
      // necessary, but instead for now we just raise an exception.
      TEUCHOS_TEST_FOR_EXCEPTION(randomVectorsRank != numNullSpaceCols, 
                         TsqrOrthoError, 
                         "Belos::TsqrOrthoManagerImpl::projectAndNormalize"
                         "OutOfPlace(): After projecting and normalizing the "
                         "random vectors (used to replace the null space "
                         "basis vectors from normalizing X), they have rank "
                         << randomVectorsRank << ", but should have full "
                         "rank " << numNullSpaceCols << ".");
    }

    // Whether or not X_in was full rank after projection, we still
    // might want to reorthogonalize against Q.
    if (reorthogonalizeBlocks_) {
      // We are only interested in the column space basis of X
      // resp. X_out.
      std::vector<magnitude_type> 
        normsAfterFirstPass (firstPassRank, SCTM::zero());
      std::vector<magnitude_type> 
        normsAfterSecondPass (firstPassRank, SCTM::zero());

      // Compute post-first-pass (pre-normalization) norms.  We could
      // have done that using MVT::MvNorm() on X_in after projecting,
      // but before the first normalization.  However, that operation
      // may be expensive.  It is also unnecessary: after calling
      // normalize(), the 2-norm of B(:,j) is the 2-norm of X_in(:,j)
      // before normalization, in exact arithmetic.
      //
      // NOTE (mfh 06 Nov 2010) This is one way that combining
      // projection and normalization into a single kernel --
      // projectAndNormalize() -- pays off.  In project(), we have to
      // compute column norms of X before and after projection.  Here,
      // we get them for free from the normalization coefficients.
      Teuchos::BLAS<int, Scalar> blas;
      for (int j = 0; j < firstPassRank; ++j) {
        const Scalar* const B_j = &(*B_out)(0,j);
        // Teuchos::BLAS::NRM2 returns a magnitude_type result on
        // Scalar inputs.
        normsAfterFirstPass[j] = blas.NRM2 (firstPassRank, B_j, 1);
      }
      // Test whether any of the norms dropped below the
      // reorthogonalization threshold.
      bool reorthogonalize = false;
      for (int j = 0; j < firstPassRank; ++j) {
        // If any column's norm decreased too much, mark this block
        // for reorthogonalization.  Note that this test will _not_
        // activate reorthogonalization if a column's norm before the
        // first project-and-normalize step was zero.  It _will_
        // activate reorthogonalization if the column's norm before
        // was not zero, but is zero now.
        const magnitude_type curThreshold = 
          blockReorthogThreshold() * normsBeforeFirstPass[j];
        if (normsAfterFirstPass[j] < curThreshold) {
          reorthogonalize = true; 
          break;
        }
      }
      // Perform another Block Gram-Schmidt pass if necessary.  "Twice
      // is enough" (Kahan's theorem) for a Krylov method, unless
      // (using Stewart's term) there is an "orthogonalization fault"
      // (indicated by reorthogFault).
      //
      // NOTE (mfh 07 Nov 2010) For now, we include the entire block
      // of X, including possible random data (that was already
      // projected and normalized above).  It might make more sense
      // just to process the first firstPassRank columns of X.
      // However, the resulting reorthogonalization should still be
      // correct regardless.
      bool reorthogFault = false;
      // Indices of X at which there was an orthogonalization fault.
      std::vector<int> faultIndices;
      if (reorthogonalize) {
        using Teuchos::Copy;
        using Teuchos::NO_TRANS;

        // If we're using out-of-place normalization, copy X_out
        // (results of first project and normalize pass) back into
        // X_in, for the second project and normalize pass.
        if (outOfPlace)
          MVT::Assign (X_out, X_in);

        // C2 is only used internally, so we know that we are
        // allocating fresh and not recycling allocations.  Stating
        // this lets us save time checking dimensions.
        Teuchos::Array<mat_ptr> C2;
        allocateProjectionCoefficients (C2, Q, X_in, false);

        // Block Gram-Schmidt (again).  Delay updating the block
        // coefficients until we have the new normalization
        // coefficients, which we need in order to do the update.
        rawProject (X_in, Q, C2);
            
        // Coefficients for (re)normalization of X_in.
        RCP<mat_type> B2 (new mat_type (ncols_X, ncols_X));

        // Normalize X_in (into X_out, if working out of place).
        const int secondPassRank = outOfPlace ? 
          normalizeOutOfPlace (X_in, X_out, B2) : 
          normalize (X_in, B2);
        rank = secondPassRank; // Current rank of X

        // Update normalization coefficients.  We begin with copying
        // B_out, since the BLAS' _GEMM routine doesn't let us alias
        // its input and output arguments.
        mat_type B_copy (Copy, *B_out, B_out->numRows(), B_out->numCols());
        // B_out := B2 * B_out (where input B_out is in B_copy).
        const int err = B_out->multiply (NO_TRANS, NO_TRANS, SCT::one(), 
                                         *B2, B_copy, SCT::zero());
        TEUCHOS_TEST_FOR_EXCEPTION(err != 0, std::logic_error, 
                           "Teuchos::SerialDenseMatrix::multiply "
                           "returned err = " << err << " != 0");
        // Update the block coefficients from the projection step.  We
        // use B_copy for this (a copy of B_out, the first-pass
        // normalization coefficients).
        for (int k = 0; k < num_Q_blocks; ++k) {
          mat_type C_k_copy (Copy, *C[k], C[k]->numRows(), C[k]->numCols());

          // C[k] := C2[k]*B_copy + C[k].
          const int err2 = C[k]->multiply (NO_TRANS, NO_TRANS, SCT::one(), 
                                          *C2[k], B_copy, SCT::one());
          TEUCHOS_TEST_FOR_EXCEPTION(err2 != 0, std::logic_error, 
                             "Teuchos::SerialDenseMatrix::multiply "
                             "returned err = " << err << " != 0");
        }
        // Compute post-second-pass (pre-normalization) norms, using
        // B2 (the coefficients from the second normalization step) in
        // the same way as with B_out before.
        for (int j = 0; j < rank; ++j) {
          const Scalar* const B2_j = &(*B2)(0,j);
          normsAfterSecondPass[j] = blas.NRM2 (rank, B2_j, 1);
        }
        // Test whether any of the norms dropped below the
        // reorthogonalization threshold.  If so, it's an
        // orthogonalization fault, which requires expensive recovery.
        reorthogFault = false;
        for (int j = 0; j < rank; ++j) {
          const magnitude_type relativeLowerBound = 
            blockReorthogThreshold() * normsAfterFirstPass[j];
          if (normsAfterSecondPass[j] < relativeLowerBound) {
            reorthogFault = true; 
            faultIndices.push_back (j);
          }
        }
      } // if (reorthogonalize) // reorthogonalization pass

      if (reorthogFault) {
        if (throwOnReorthogFault_) {
          raiseReorthogFault (normsAfterFirstPass, 
                              normsAfterSecondPass, 
                              faultIndices);
        } else {
          // NOTE (mfh 19 Jan 2011) We could handle the fault here by
          // slowly reorthogonalizing, one vector at a time, the
          // offending vectors of X.  However, we choose not to
          // implement this for now.  If it becomes a problem, let us
          // know and we will prioritize implementing this.
          TEUCHOS_TEST_FOR_EXCEPTION(true, std::logic_error, 
                             "TsqrOrthoManagerImpl has not yet implemented"
                             " recovery from an orthogonalization fault.");
        }
      }
    } // if (reorthogonalizeBlocks_)
    return rank;
  }


  template<class Scalar, class MV>
  void
  TsqrOrthoManagerImpl<Scalar, MV>::
  raiseReorthogFault (const std::vector<magnitude_type>& normsAfterFirstPass,
                      const std::vector<magnitude_type>& normsAfterSecondPass,
                      const std::vector<int>& faultIndices)
  {
    using std::endl;
    typedef std::vector<int>::size_type size_type;
    std::ostringstream os;
                  
    os << "Orthogonalization fault at the following column(s) of X:" << endl;
    os << "Column\tNorm decrease factor" << endl;
    for (size_type k = 0; k < faultIndices.size(); ++k)
      {
        const int index = faultIndices[k];
        const magnitude_type decreaseFactor = 
          normsAfterSecondPass[index] / normsAfterFirstPass[index];
        os << index << "\t" << decreaseFactor << endl;
      }
    throw TsqrOrthoFault (os.str());
  }

  template<class Scalar, class MV>
  Teuchos::RCP<const Teuchos::ParameterList>
  TsqrOrthoManagerImpl<Scalar, MV>::getValidParameters () const
  {
    using Teuchos::ParameterList;
    using Teuchos::parameterList;
    using Teuchos::RCP;

    if (defaultParams_.is_null()) {
      RCP<ParameterList> params = parameterList ("TsqrOrthoManagerImpl");
      //
      // TSQR parameters (set as a sublist).
      //
      params->set ("TSQR implementation", *(tsqrAdaptor_.getValidParameters()),
                   "TSQR implementation parameters.");
      // 
      // Orthogonalization parameters
      //
      const bool defaultRandomizeNullSpace = true;
      params->set ("randomizeNullSpace", defaultRandomizeNullSpace, 
                   "Whether to fill in null space vectors with random data.");

      const bool defaultReorthogonalizeBlocks = true;
      params->set ("reorthogonalizeBlocks", defaultReorthogonalizeBlocks,
                   "Whether to do block reorthogonalization as necessary.");

      // This parameter corresponds to the "blk_tol_" parameter in
      // Belos' DGKSOrthoManager.  We choose the same default value.
      const magnitude_type defaultBlockReorthogThreshold = 
        magnitude_type(10) * SCTM::squareroot (SCTM::eps());
      params->set ("blockReorthogThreshold", defaultBlockReorthogThreshold, 
                   "If reorthogonalizeBlocks==true, and if the norm of "
                   "any column within a block decreases by this much or "
                   "more after orthogonalization, we reorthogonalize.");

      // This parameter corresponds to the "sing_tol_" parameter in
      // Belos' DGKSOrthoManager.  We choose the same default value.
      const magnitude_type defaultRelativeRankTolerance = 
        Teuchos::as<magnitude_type>(10) * SCTM::eps();

      // If the relative rank tolerance is zero, then we will always
      // declare blocks to be numerically full rank, as long as no
      // singular values are zero.
      params->set ("relativeRankTolerance", defaultRelativeRankTolerance,
                   "Relative tolerance to determine the numerical rank of a "
                   "block when normalizing.");

      // See Stewart's 2008 paper on block Gram-Schmidt for a definition
      // of "orthogonalization fault."
      const bool defaultThrowOnReorthogFault = true;
      params->set ("throwOnReorthogFault", defaultThrowOnReorthogFault,
                   "Whether to throw an exception if an orthogonalization "
                   "fault occurs.  This only matters if reorthogonalization "
                   "is enabled (reorthogonalizeBlocks==true).");

      const bool defaultForceNonnegativeDiagonal = false;
      params->set ("forceNonnegativeDiagonal", defaultForceNonnegativeDiagonal,
                   "Whether to force the R factor produced by the normalization "
                   "step to have a nonnegative diagonal.");

      defaultParams_ = params;
    }
    return defaultParams_;
  }

  template<class Scalar, class MV>
  Teuchos::RCP<const Teuchos::ParameterList>
  TsqrOrthoManagerImpl<Scalar, MV>::getFastParameters ()
  {
    using Teuchos::ParameterList;
    using Teuchos::RCP;
    using Teuchos::rcp;

    RCP<const ParameterList> defaultParams = getValidParameters();
    // Start with a clone of the default parameters.
    RCP<ParameterList> params = rcp (new ParameterList (*defaultParams));
        
    // Disable reorthogonalization and randomization of the null
    // space basis.  Reorthogonalization tolerances don't matter,
    // since we aren't reorthogonalizing blocks in the fast
    // settings.  We can leave the default values.  Also,
    // (re)orthogonalization faults may only occur with
    // reorthogonalization, so we don't have to worry about the
    // "throwOnReorthogFault" setting.
    const bool randomizeNullSpace = false;
    params->set ("randomizeNullSpace", randomizeNullSpace);      
    const bool reorthogonalizeBlocks = false;
    params->set ("reorthogonalizeBlocks", reorthogonalizeBlocks);

    return params;
  }

  template<class Scalar, class MV>
  int
  TsqrOrthoManagerImpl<Scalar, MV>::
  rawNormalize (MV& X, 
                MV& Q, 
                Teuchos::SerialDenseMatrix<int, Scalar>& B)
  {
    int rank;
    try {
      // This call only computes the QR factorization X = Q B.
      // It doesn't compute the rank of X.  That comes from
      // revealRank() below.
      tsqrAdaptor_.factorExplicit (X, Q, B, forceNonnegativeDiagonal_);
      // This call will only modify *B if *B on input is not of full
      // numerical rank.
      rank = tsqrAdaptor_.revealRank (Q, B, relativeRankTolerance_);
    } catch (std::exception& e) {
      throw TsqrOrthoError (e.what()); // Toss the exception up the chain.
    }
    return rank;
  }

  template<class Scalar, class MV>
  int
  TsqrOrthoManagerImpl<Scalar, MV>::
  normalizeOne (MV& X, 
                Teuchos::RCP<Teuchos::SerialDenseMatrix<int, Scalar> > B) const
  {
    // Make space for the normalization coefficient.  This will either
    // be a freshly allocated matrix (if B is null), or a view of the
    // 1x1 upper left submatrix of *B (if B is not null).
    mat_ptr B_out;
    if (B.is_null()) {
      B_out = rcp (new mat_type (1, 1));
    } else {
      const int numRows = B->numRows();
      const int numCols = B->numCols();
      TEUCHOS_TEST_FOR_EXCEPTION(numRows < 1 || numCols < 1, 
                         std::invalid_argument,
                         "normalizeOne: Input matrix B must be at "
                         "least 1 x 1, but is instead " << numRows
                         << " x " << numCols << ".");
      // Create a view of the 1x1 upper left submatrix of *B.
      B_out = rcp (new mat_type (Teuchos::View, *B, 1, 1));
    }

    // Compute the norm of X, and write the result to B_out.
    std::vector<magnitude_type> theNorm (1, SCTM::zero());
    MVT::MvNorm (X, theNorm);
    (*B_out)(0,0) = theNorm[0];

    if (B.is_null()) {
      // The input matrix B is null, so assign B_out to it.  If B was
      // not null on input, then B_out is a view of *B, so we don't
      // have to do anything here.  Note that SerialDenseMatrix uses
      // raw pointers to store data and represent views, so we have to
      // be careful about scope.
      B = B_out;
    }

    // Scale X by its norm, if its norm is zero.  Otherwise, do the
    // right thing based on whether the user wants us to fill the null
    // space with random vectors.
    if (theNorm[0] == SCTM::zero()) {
      // Make a view of the first column of Q, fill it with random
      // data, and normalize it.  Throw away the resulting norm.
      if (randomizeNullSpace_) {
        MVT::MvRandom(X);
        MVT::MvNorm (X, theNorm);
        if (theNorm[0] == SCTM::zero()) {
          // It is possible that a random vector could have all zero
          // entries, but unlikely.  We could try again, but it's also
          // possible that multiple tries could result in zero
          // vectors.  We choose instead to give up.
          throw TsqrOrthoError("normalizeOne: a supposedly random "
                               "vector has norm zero!");
        } else {
          // NOTE (mfh 09 Nov 2010) I'm assuming that dividing a
          // Scalar by a magnitude_type is defined and that it results
          // in a Scalar.
          const Scalar alpha = SCT::one() / theNorm[0];
          MVT::MvScale (X, alpha);
        }
      }
      return 0; // The rank of the matrix (actually just one vector) X.
    } else {
      // NOTE (mfh 09 Nov 2010) I'm assuming that dividing a Scalar by
      // a magnitude_type is defined and that it results in a Scalar.
      const Scalar alpha = SCT::one() / theNorm[0];
      MVT::MvScale (X, alpha);
      return 1; // The rank of the matrix (actually just one vector) X.
    }
  }


  template<class Scalar, class MV>
  void
  TsqrOrthoManagerImpl<Scalar, MV>::
  rawProject (MV& X, 
              Teuchos::ArrayView<Teuchos::RCP<const MV> > Q,
              Teuchos::ArrayView<Teuchos::RCP<Teuchos::SerialDenseMatrix<int, Scalar> > > C) const
  {
    // "Modified Gram-Schmidt" version of Block Gram-Schmidt.
    const int num_Q_blocks = Q.size();
    for (int i = 0; i < num_Q_blocks; ++i)
      {
        // TEUCHOS_TEST_FOR_EXCEPTION(C[i].is_null(), std::logic_error,
        //                    "TsqrOrthoManagerImpl::rawProject(): C[" 
        //                    << i << "] is null");
        // TEUCHOS_TEST_FOR_EXCEPTION(Q[i].is_null(), std::logic_error,
        //                    "TsqrOrthoManagerImpl::rawProject(): Q[" 
        //                    << i << "] is null");
        mat_type& Ci = *C[i];
        const MV& Qi = *Q[i];
        innerProd (Qi, X, Ci);
        MVT::MvTimesMatAddMv (-SCT::one(), Qi, Ci, SCT::one(), X);
      }
  }


  template<class Scalar, class MV>
  void
  TsqrOrthoManagerImpl<Scalar, MV>::
  rawProject (MV& X, 
              const Teuchos::RCP<const MV>& Q, 
              const Teuchos::RCP<Teuchos::SerialDenseMatrix<int, Scalar> >& C) const
  {
    // Block Gram-Schmidt
    innerProd (*Q, X, *C);
    MVT::MvTimesMatAddMv (-SCT::one(), *Q, *C, SCT::one(), X);
  }

  template<class Scalar, class MV>
  int
  TsqrOrthoManagerImpl<Scalar, MV>::
  normalizeImpl (MV& X, 
                 MV& Q, 
                 Teuchos::RCP<Teuchos::SerialDenseMatrix<int, Scalar> > B, 
                 const bool outOfPlace)
  {
    using Teuchos::Range1D;
    using Teuchos::RCP;
    using Teuchos::rcp;
    using Teuchos::ScalarTraits;
    using Teuchos::tuple;
    using std::cerr;
    using std::endl;
    // Don't set this to true unless you want lots of debugging
    // messages written to stderr on every MPI process.
    const bool extraDebug = false;

    const int numCols = MVT::GetNumberVecs (X);
    if (numCols == 0) {
      return 0; // Fast exit for an empty input matrix.
    }

    // We allow Q to have more columns than X.  In that case, we only
    // touch the first numCols columns of Q.
    TEUCHOS_TEST_FOR_EXCEPTION(MVT::GetNumberVecs(Q) < numCols, 
                       std::invalid_argument, 
                       "TsqrOrthoManagerImpl::normalizeImpl(X,Q,B): Q has "
                       << MVT::GetNumberVecs(Q) << " columns.  This is too "
                       "few, since X has " << numCols << " columns.");
    // TSQR wants a Q with the same number of columns as X, so have it
    // work on a nonconstant view of Q with the same number of columns
    // as X.
    RCP<MV> Q_view = MVT::CloneViewNonConst (Q, Range1D(0, numCols-1));

    // Make space for the normalization coefficients.  This will
    // either be a freshly allocated matrix (if B is null), or a view
    // of the appropriately sized upper left submatrix of *B (if B is
    // not null).
    mat_ptr B_out;
    if (B.is_null()) {
      B_out = rcp (new mat_type (numCols, numCols));
    } else {
      // Make sure that B is no smaller than numCols x numCols.
      TEUCHOS_TEST_FOR_EXCEPTION(B->numRows() < numCols || B->numCols() < numCols,
                         std::invalid_argument,
                         "normalizeOne: Input matrix B must be at "
                         "least " << numCols << " x " << numCols 
                         << ", but is instead " << B->numRows()
                         << " x " << B->numCols() << ".");
      // Create a view of the numCols x numCols upper left submatrix
      // of *B.  TSQR will write the normalization coefficients there.
      B_out = rcp (new mat_type (Teuchos::View, *B, numCols, numCols));
    }

    if (extraDebug) {
      std::vector<magnitude_type> norms (numCols);
      MVT::MvNorm (X, norms);
      cerr << "Column norms of X before orthogonalization: ";
      typedef typename std::vector<magnitude_type>::const_iterator iter_type;
      for (iter_type iter = norms.begin(); iter != norms.end(); ++iter) {
        cerr << *iter;
        if (iter+1 != norms.end())
          cerr << ", ";
      }
    }

    // Compute rank-revealing decomposition (in this case, TSQR of X
    // followed by SVD of the R factor and appropriate updating of the
    // resulting Q factor) of X.  X is modified in place and filled
    // with garbage, and Q_view contains the resulting explicit Q
    // factor.  Later, we will copy this back into X.
    //
    // The matrix *B_out will only be upper triangular if X is of full
    // numerical rank.  Otherwise, the entries below the diagonal may
    // be filled in as well.
    const int rank = rawNormalize (X, *Q_view, *B_out);
    if (B.is_null()) {
      // The input matrix B is null, so assign B_out to it.  If B was
      // not null on input, then B_out is a view of *B, so we don't
      // have to do anything here.  Note that SerialDenseMatrix uses
      // raw pointers to store data and represent views, so we have to
      // be careful about scope.
      B = B_out;
    }

    if (extraDebug) {
      std::vector<magnitude_type> norms (numCols);
      MVT::MvNorm (*Q_view, norms);
      cerr << "Column norms of Q_view after orthogonalization: ";
      for (typename std::vector<magnitude_type>::const_iterator iter = norms.begin(); 
           iter != norms.end(); ++iter) {
        cerr << *iter;
        if (iter+1 != norms.end())
          cerr << ", ";
      }
    }
    TEUCHOS_TEST_FOR_EXCEPTION(rank < 0 || rank > numCols, std::logic_error,
                       "Belos::TsqrOrthoManagerImpl::normalizeImpl: "
                       "rawNormalize() returned rank = " << rank << " for a "
                       "matrix X with " << numCols << " columns.  Please report"
                       " this bug to the Belos developers.");
    if (extraDebug && rank == 0) {
      // Sanity check: ensure that the columns of X are sufficiently
      // small for X to be reported as rank zero.
      const mat_type& B_ref = *B;
      std::vector<magnitude_type> norms (B_ref.numCols());
      for (typename mat_type::ordinalType j = 0; j < B_ref.numCols(); ++j) {
        typedef typename mat_type::scalarType mat_scalar_type;
        mat_scalar_type sumOfSquares = ScalarTraits<mat_scalar_type>::zero();
        for (typename mat_type::ordinalType i = 0; i <= j; ++i) {
          const mat_scalar_type B_ij = 
            ScalarTraits<mat_scalar_type>::magnitude (B_ref(i,j));
          sumOfSquares += B_ij*B_ij;
        }
        norms[j] = ScalarTraits<mat_scalar_type>::squareroot (sumOfSquares);
      }
      using std::cerr;
      using std::endl;
      cerr << "Norms of columns of B after orthogonalization: ";
      for (typename mat_type::ordinalType j = 0; j < B_ref.numCols(); ++j) {
        cerr << norms[j];
        if (j != B_ref.numCols() - 1)
          cerr << ", ";
      }
      cerr << endl;
    }

    // If X is full rank or we don't want to replace its null space
    // basis with random vectors, then we're done.
    if (rank == numCols || ! randomizeNullSpace_) {
      // If we're supposed to be working in place in X, copy the
      // results back from Q_view into X.
      if (! outOfPlace) {
        MVT::Assign (*Q_view, X);
      }
      return rank;
    }

    if (randomizeNullSpace_ && rank < numCols) {
      // X wasn't full rank.  Replace the null space basis of X (in
      // the last numCols-rank columns of Q_view) with random data,
      // project it against the first rank columns of Q_view, and
      // normalize.
      //
      // Number of columns to fill with random data.
      const int nullSpaceNumCols = numCols - rank;
      // Inclusive range of indices of columns of X to fill with
      // random data.
      Range1D nullSpaceIndices (rank, numCols-1);

      // rawNormalize() wrote the null space basis vectors into
      // Q_view.  We continue to work in place in Q_view by writing
      // the random data there and (if there is a nontrival column
      // space) projecting in place against the column space basis
      // vectors (also in Q_view).
      RCP<MV> Q_null = MVT::CloneViewNonConst (*Q_view, nullSpaceIndices);
      // Replace the null space basis with random data.
      MVT::MvRandom (*Q_null); 

      // Make sure that the "random" data isn't all zeros.  This is
      // statistically nearly impossible, but we test for debugging
      // purposes.
      {
        std::vector<magnitude_type> norms (MVT::GetNumberVecs(*Q_null));
        MVT::MvNorm (*Q_null, norms);

        bool anyZero = false;
        typedef typename std::vector<magnitude_type>::const_iterator iter_type;
        for (iter_type it = norms.begin(); it != norms.end(); ++it) {
          if (*it == SCTM::zero()) {
            anyZero = true;
          }
        }
        if (anyZero) {
          std::ostringstream os;
          os << "TsqrOrthoManagerImpl::normalizeImpl: "
            "We are being asked to randomize the null space, for a matrix "
            "with " << numCols << " columns and reported column rank "
             << rank << ".  The inclusive range of columns to fill with "
            "random data is [" << nullSpaceIndices.lbound() << "," 
             << nullSpaceIndices.ubound() << "].  After filling the null "
            "space vectors with random numbers, at least one of the vectors"
            " has norm zero.  Here are the norms of all the null space "
            "vectors: [";
          for (iter_type it = norms.begin(); it != norms.end(); ++it) {
            os << *it;
            if (it+1 != norms.end())
              os << ", ";
          }
          os << "].)  There is a tiny probability that this could happen "
            "randomly, but it is likely a bug.  Please report it to the "
            "Belos developers, especially if you are able to reproduce the "
            "behavior.";
          TEUCHOS_TEST_FOR_EXCEPTION(anyZero, TsqrOrthoError, os.str());
        }
      }

      if (rank > 0) {
        // Project the random data against the column space basis of
        // X, using a simple block projection ("Block Classical
        // Gram-Schmidt").  This is accurate because we've already
        // orthogonalized the column space basis of X nearly to
        // machine precision via a QR factorization (TSQR) with
        // accuracy comparable to Householder QR.
        RCP<const MV> Q_col = MVT::CloneView (*Q_view, Range1D(0, rank-1));

        // Temporary storage for projection coefficients.  We don't
        // need to keep them, since they represent the null space
        // basis (for which the coefficients are logically zero).
        mat_ptr C_null (new mat_type (rank, nullSpaceNumCols));
        rawProject (*Q_null, Q_col, C_null);
      }
      // Normalize the projected random vectors, so that they are
      // mutually orthogonal (as well as orthogonal to the column
      // space basis of X).  We use X for the output of the
      // normalization: for out-of-place normalization (outOfPlace ==
      // true), X is overwritten with "invalid values" anyway, and for
      // in-place normalization (outOfPlace == false), we want the
      // result to be in X anyway.
      RCP<MV> X_null = MVT::CloneViewNonConst (X, nullSpaceIndices);
      // Normalization coefficients for projected random vectors.
      // Will be thrown away.
      mat_type B_null (nullSpaceNumCols, nullSpaceNumCols);
      // Write the normalized vectors to X_null (in X).
      const int nullSpaceBasisRank = rawNormalize (*Q_null, *X_null, B_null);
          
      // It's possible, but unlikely, that X_null doesn't have full
      // rank (after the projection step).  We could recursively fill
      // in more random vectors until we finally get a full rank
      // matrix, but instead we just throw an exception.
      //
      // NOTE (mfh 08 Nov 2010) Perhaps we should deal with this case
      // more elegantly.  Recursion might be one way to solve it, but
      // be sure to check that the recursion will terminate.  We could
      // do this by supplying an additional argument to rawNormalize,
      // which is the null space basis rank from the previous
      // iteration.  The rank has to decrease each time, or the
      // recursion may go on forever.
      if (nullSpaceBasisRank < nullSpaceNumCols) {
        std::vector<magnitude_type> norms (MVT::GetNumberVecs(*X_null));
        MVT::MvNorm (*X_null, norms);
        std::ostringstream os;
        os << "TsqrOrthoManagerImpl::normalizeImpl: "
           << "We are being asked to randomize the null space, "
           << "for a matrix with " << numCols << " columns and "
           << "column rank " << rank << ".  After projecting and "
           << "normalizing the generated random vectors, they "
           << "only have rank " << nullSpaceBasisRank << ".  They"
           << " should have full rank " << nullSpaceNumCols 
           << ".  (The inclusive range of columns to fill with "
           << "random data is [" << nullSpaceIndices.lbound() 
           << "," << nullSpaceIndices.ubound() << "].  The "
           << "column norms of the resulting Q factor are: [";
        for (typename std::vector<magnitude_type>::size_type k = 0; 
             k < norms.size(); ++k) {
          os << norms[k];
          if (k != norms.size()-1) {
            os << ", ";
          }
        }
        os << "].)  There is a tiny probability that this could "
           << "happen randomly, but it is likely a bug.  Please "
           << "report it to the Belos developers, especially if "
           << "you are able to reproduce the behavior.";

        TEUCHOS_TEST_FOR_EXCEPTION(nullSpaceBasisRank < nullSpaceNumCols, 
                           TsqrOrthoError, os.str());
      }
      // If we're normalizing out of place, copy the X_null
      // vectors back into Q_null; the Q_col vectors are already
      // where they are supposed to be in that case.
      //
      // If we're normalizing in place, leave X_null alone (it's
      // already where it needs to be, in X), but copy Q_col back
      // into the first rank columns of X.
      if (outOfPlace) {
        MVT::Assign (*X_null, *Q_null);
      } else if (rank > 0) {
        // MVT::Assign() doesn't accept empty ranges of columns.
        RCP<const MV> Q_col = MVT::CloneView (*Q_view, Range1D(0, rank-1));
        RCP<MV> X_col = MVT::CloneViewNonConst (X, Range1D(0, rank-1));
        MVT::Assign (*Q_col, *X_col);
      }
    }
    return rank;
  }


  template<class Scalar, class MV>
  void
  TsqrOrthoManagerImpl<Scalar, MV>::
  checkProjectionDims (int& ncols_X, 
                       int& num_Q_blocks,
                       int& ncols_Q_total,
                       const MV& X, 
                       Teuchos::ArrayView<Teuchos::RCP<const MV> > Q) const
  {
    // First assign to temporary values, so the function won't
    // commit any externally visible side effects unless it will
    // return normally (without throwing an exception).  (I'm being
    // cautious; MVT::GetNumberVecs() probably shouldn't have any
    // externally visible side effects, unless it is logging to a
    // file or something.)
    int the_ncols_X, the_num_Q_blocks, the_ncols_Q_total;
    the_num_Q_blocks = Q.size();
    the_ncols_X = MVT::GetNumberVecs (X);

    // Compute the total number of columns of all the Q[i] blocks.
    the_ncols_Q_total = 0;
    // You should be angry if your compiler doesn't support type
    // inference ("auto").  That's why I need this awful typedef.
    using Teuchos::ArrayView;
    using Teuchos::RCP;
    typedef typename ArrayView<RCP<const MV> >::const_iterator iter_type;
    for (iter_type it = Q.begin(); it != Q.end(); ++it) {
      const MV& Qi = **it;
      the_ncols_Q_total += MVT::GetNumberVecs (Qi);
    }

    // Commit temporary values to the output arguments.
    ncols_X = the_ncols_X;
    num_Q_blocks = the_num_Q_blocks;
    ncols_Q_total = the_ncols_Q_total;
  }

} // namespace Anasazi

#endif // __AnasaziTsqrOrthoManagerImpl_hpp