BelosOrthoManagerTest.hpp

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// @HEADER
// *****************************************************************************
//                 Belos: Block Linear Solvers Package
//
// Copyright 2004-2016 NTESS and the Belos contributors.
// SPDX-License-Identifier: BSD-3-Clause
// *****************************************************************************
// @HEADER
 
 
#include <BelosConfigDefs.hpp>
#include <BelosMultiVecTraits.hpp>
#include <BelosOutputManager.hpp>
#include <BelosOrthoManagerFactory.hpp>
#include <Teuchos_StandardCatchMacros.hpp>
#include <Teuchos_TimeMonitor.hpp>
#include <Teuchos_SerialDenseHelpers.hpp>
#include <iostream>
#include <stdexcept>
 
using std::endl;
 
namespace Belos {
  namespace Test {
 
    template<class Scalar, class MV>
    class OrthoManagerBenchmarker {
    private:
      typedef Scalar scalar_type;
      typedef typename Teuchos::ScalarTraits<Scalar>::magnitudeType magnitude_type;
      typedef MultiVecTraits<Scalar, MV> MVT;
      typedef Teuchos::SerialDenseMatrix<int, Scalar> mat_type;
 
    public:
      static void
      baseline (const Teuchos::RCP<const MV>& X,
                const int numCols,
                const int numBlocks,
                const int numTrials)
      {
        using Teuchos::Array;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::Time;
        using Teuchos::TimeMonitor;
 
        // Make some blocks to "orthogonalize."  Fill with random
        // data.  We only need X so that we can make clones (it knows
        // its data distribution).
        Array<RCP<MV> > V (numBlocks);
        for (int k = 0; k < numBlocks; ++k) {
          V[k] = MVT::Clone (*X, numCols);
          MVT::MvRandom (*V[k]);
        }
 
        // Make timers with informative labels
        RCP<Time> timer = TimeMonitor::getNewCounter ("Baseline for OrthoManager benchmark");
 
        // Baseline benchmark just copies data.  It's sort of a lower
        // bound proxy for the volume of data movement done by a real
        // OrthoManager.
        {
          TimeMonitor monitor (*timer);
          for (int trial = 0; trial < numTrials; ++trial) {
            for (int k = 0; k < numBlocks; ++k) {
              for (int j = 0; j < k; ++j)
                MVT::Assign (*V[j], *V[k]);
              MVT::Assign (*X, *V[k]);
            }
          }
        }
      }
 
      static void
      benchmark (const Teuchos::RCP<OrthoManager<Scalar, MV> >& orthoMan,
                 const std::string& orthoManName,
                 const std::string& normalization,
                 const Teuchos::RCP<const MV>& X,
                 const int numCols,
                 const int numBlocks,
                 const int numTrials,
                 const Teuchos::RCP<OutputManager<Scalar> >& outMan,
                 std::ostream& resultStream,
                 const bool displayResultsCompactly=false)
      {
        using Teuchos::Array;
        using Teuchos::ArrayView;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::Time;
        using Teuchos::TimeMonitor;
        using std::endl;
 
        TEUCHOS_TEST_FOR_EXCEPTION(orthoMan.is_null(), std::invalid_argument,
                           "orthoMan is null");
        TEUCHOS_TEST_FOR_EXCEPTION(X.is_null(), std::invalid_argument,
                           "X is null");
        TEUCHOS_TEST_FOR_EXCEPTION(numCols < 1, std::invalid_argument,
                           "numCols = " << numCols << " < 1");
        TEUCHOS_TEST_FOR_EXCEPTION(numBlocks < 1, std::invalid_argument,
                           "numBlocks = " << numBlocks << " < 1");
        TEUCHOS_TEST_FOR_EXCEPTION(numTrials < 1, std::invalid_argument,
                           "numTrials = " << numTrials << " < 1");
        // Debug output stream
        std::ostream& debugOut = outMan->stream(Debug);
 
        // If you like, you can add the "baseline" as an approximate
        // lower bound for orthogonalization performance.  It may be
        // useful as a sanity check to make sure that your
        // orthogonalizations are really computing something, though
        // testing accuracy can help with that too.
        //
        //baseline (X, numCols, numBlocks, numTrials);
 
        // Make space to put the projection and normalization
        // coefficients.
        Array<RCP<mat_type> > C (numBlocks);
        for (int k = 0; k < numBlocks; ++k) {
          C[k] = rcp (new mat_type (numCols, numCols));
        }
        RCP<mat_type> B (new mat_type (numCols, numCols));
 
        // Make some blocks to orthogonalize.  Fill with random data.
        // We won't be orthogonalizing X, or even modifying X.  We
        // only need X so that we can make clones (since X knows its
        // data distribution).
        Array<RCP<MV> > V (numBlocks);
        for (int k = 0; k < numBlocks; ++k) {
          V[k] = MVT::Clone (*X, numCols);
          MVT::MvRandom (*V[k]);
        }
 
        // Make timers with informative labels.  We time an additional
        // first run to measure the startup costs, if any, of the
        // OrthoManager instance.
        RCP<Time> firstRunTimer;
        {
          std::ostringstream os;
          os << "OrthoManager: " << orthoManName << " first run";
          firstRunTimer = TimeMonitor::getNewCounter (os.str());
        }
        RCP<Time> timer;
        {
          std::ostringstream os;
          os << "OrthoManager: " << orthoManName << " total over "
             << numTrials << " trials (excluding first run above)";
          timer = TimeMonitor::getNewCounter (os.str());
        }
        // The first run lets us measure the startup costs, if any, of
        // the OrthoManager instance, without these costs influencing
        // the following timing runs.
        {
          TimeMonitor monitor (*firstRunTimer);
          {
            (void) orthoMan->normalize (*V[0], B);
            for (int k = 1; k < numBlocks; ++k) {
              // k is the number of elements in the ArrayView.  We
              // have to assign first to an ArrayView-of-RCP-of-MV,
              // rather than to an ArrayView-of-RCP-of-const-MV, since
              // the latter requires a reinterpret cast.  Don't you
              // love C++ type inference?
              ArrayView<RCP<MV> > V_0k_nonconst = V.view (0, k);
              ArrayView<RCP<const MV> > V_0k =
                Teuchos::av_reinterpret_cast<RCP<const MV> > (V_0k_nonconst);
              (void) orthoMan->projectAndNormalize (*V[k], C, B, V_0k);
            }
          }
          // "Test" that the trial run actually orthogonalized
          // correctly.  Results are printed to the OutputManager's
          // Belos::Debug output stream, so depending on the
          // OutputManager's chosen verbosity level, you may or may
          // not see the results of the test.
          //
          // NOTE (mfh 22 Jan 2011) For now, these results have to be
          // inspected visually.  We should add a simple automatic
          // test.
          debugOut << "Orthogonality of V[0:" << (numBlocks-1)
                   << "]:" << endl;
          for (int k = 0; k < numBlocks; ++k) {
            // Orthogonality of each block
            debugOut << "For block V[" << k << "]:" << endl;
            debugOut << "  ||<V[" << k << "], V[" << k << "]> - I|| = "
                     << orthoMan->orthonormError(*V[k]) << endl;
            // Relative orthogonality with the previous blocks
            for (int j = 0; j < k; ++j) {
              debugOut << "  ||< V[" << j << "], V[" << k << "] >|| = "
                       << orthoMan->orthogError(*V[j], *V[k]) << endl;
            }
          }
        }
 
        // Run the benchmark for numTrials trials.  Time all trials as
        // a single run.
        {
          TimeMonitor monitor (*timer);
 
          for (int trial = 0; trial < numTrials; ++trial) {
            (void) orthoMan->normalize (*V[0], B);
            for (int k = 1; k < numBlocks; ++k) {
              ArrayView<RCP<MV> > V_0k_nonconst = V.view (0, k);
              ArrayView<RCP<const MV> > V_0k =
                Teuchos::av_reinterpret_cast<RCP<const MV> > (V_0k_nonconst);
              (void) orthoMan->projectAndNormalize (*V[k], C, B, V_0k);
            }
          }
        }
 
        // Report timing results.
        if (displayResultsCompactly)
          {
            // The "compact" format is suitable for automatic parsing,
            // using a CSV (Comma-Delimited Values) parser.  The first
            // "comment" line may be parsed to extract column
            // ("field") labels; the second line contains the actual
            // data, in ASCII comma-delimited format.
            using std::endl;
            resultStream << "#orthoManName"
                         << ",normalization"
                         << ",numRows"
                         << ",numCols"
                         << ",numBlocks"
                         << ",firstRunTimeInSeconds"
                         << ",timeInSeconds"
                         << ",numTrials"
                         << endl;
            resultStream << orthoManName
                         << "," << (orthoManName=="Simple" ? normalization : "N/A")
                         << "," << MVT::GetGlobalLength(*X)
                         << "," << numCols
                         << "," << numBlocks
                         << "," << firstRunTimer->totalElapsedTime()
                         << "," << timer->totalElapsedTime()
                         << "," << numTrials
                         << endl;
          }
        else {
          TimeMonitor::summarize (resultStream);
        }
      }
    };
 
    template< class Scalar, class MV >
    class OrthoManagerTester {
    private:
      typedef typename Teuchos::Array<Teuchos::RCP<MV> >::size_type size_type;
 
    public:
      typedef Scalar scalar_type;
      typedef Teuchos::ScalarTraits<scalar_type> SCT;
      typedef typename SCT::magnitudeType magnitude_type;
      typedef Teuchos::ScalarTraits<magnitude_type> SMT;
      typedef MultiVecTraits<scalar_type, MV> MVT;
      typedef Teuchos::SerialDenseMatrix<int, scalar_type> mat_type;
 
      static int
      runTests (const Teuchos::RCP<OrthoManager<Scalar, MV> >& OM,
                const bool isRankRevealing,
                const Teuchos::RCP<MV>& S,
                const int sizeX1,
                const int sizeX2,
                const Teuchos::RCP<OutputManager<Scalar> >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::null;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::rcp_dynamic_cast;
        using Teuchos::tuple;
 
        // Number of tests that have failed thus far.
        int numFailed = 0;
 
        // Relative tolerance against which all tests are performed.
        const magnitude_type TOL = 1.0e-12;
        // Absolute tolerance constant
        //const magnitude_type ATOL = 10;
 
        const scalar_type ZERO = SCT::zero();
        const scalar_type ONE = SCT::one();
 
        // Debug output stream
        std::ostream& debugOut = MyOM->stream(Debug);
 
        // Number of columns in the input "prototype" multivector S.
        const int sizeS = MVT::GetNumberVecs (*S);
 
        // Create multivectors X1 and X2, using the same map as multivector
        // S.  Then, test orthogonalizing X2 against X1.  After doing so, X1
        // and X2 should each be M-orthonormal, and should be mutually
        // M-orthogonal.
        debugOut << "Generating X1,X2 for testing... ";
        RCP< MV > X1 = MVT::Clone (*S, sizeX1);
        RCP< MV > X2 = MVT::Clone (*S, sizeX2);
        debugOut << "done." << endl;
        {
          magnitude_type err;
 
          //
          // Fill X1 with random values, and test the normalization error.
          //
          debugOut << "Filling X1 with random values... ";
          MVT::MvRandom(*X1);
          debugOut << "done." << endl
                   << "Calling normalize() on X1... ";
          // The Anasazi and Belos OrthoManager interfaces differ.
          // For example, Anasazi's normalize() method accepts either
          // one or two arguments, whereas Belos' normalize() requires
          // two arguments.
          const int initialX1Rank = OM->normalize(*X1, Teuchos::null);
          TEUCHOS_TEST_FOR_EXCEPTION(initialX1Rank != sizeX1,
                             std::runtime_error,
                             "normalize(X1) returned rank "
                             << initialX1Rank << " from " << sizeX1
                             << " vectors. Cannot continue.");
          debugOut << "done." << endl
                   << "Calling orthonormError() on X1... ";
          err = OM->orthonormError(*X1);
          TEUCHOS_TEST_FOR_EXCEPTION(err > TOL, std::runtime_error,
                             "After normalize(X1), orthonormError(X1) = "
                             << err << " > TOL = " << TOL);
          debugOut << "done: ||<X1,X1> - I|| = " << err << endl;
 
          //
          // Fill X2 with random values, project against X1 and normalize,
          // and test the orthogonalization error.
          //
          debugOut << "Filling X2 with random values... ";
          MVT::MvRandom(*X2);
          debugOut << "done." << endl
                   << "Calling projectAndNormalize(X2, C, B, tuple(X1))... "
                   << std::flush;
          // The projectAndNormalize() interface also differs between
          // Anasazi and Belos.  Anasazi's projectAndNormalize() puts
          // the multivector and the array of multivectors first, and
          // the (array of) SerialDenseMatrix arguments (which are
          // optional) afterwards.  Belos puts the (array of)
          // SerialDenseMatrix arguments in the middle, and they are
          // not optional.
          int initialX2Rank;
          {
            Array<RCP<mat_type> > C (1);
            RCP<mat_type> B = Teuchos::null;
            initialX2Rank =
              OM->projectAndNormalize (*X2, C, B, tuple<RCP<const MV> >(X1));
          }
          TEUCHOS_TEST_FOR_EXCEPTION(initialX2Rank != sizeX2,
                             std::runtime_error,
                             "projectAndNormalize(X2,X1) returned rank "
                             << initialX2Rank << " from " << sizeX2
                             << " vectors. Cannot continue.");
          debugOut << "done." << endl
                   << "Calling orthonormError() on X2... ";
          err = OM->orthonormError (*X2);
          TEUCHOS_TEST_FOR_EXCEPTION(err > TOL,
                             std::runtime_error,
                             "projectAndNormalize(X2,X1) did not meet tolerance: "
                             "orthonormError(X2) = " << err << " > TOL = " << TOL);
          debugOut << "done: || <X2,X2> - I || = " << err << endl
                   << "Calling orthogError(X2, X1)... ";
          err = OM->orthogError (*X2, *X1);
          TEUCHOS_TEST_FOR_EXCEPTION(err > TOL,
                             std::runtime_error,
                             "projectAndNormalize(X2,X1) did not meet tolerance: "
                             "orthogError(X2,X1) = " << err << " > TOL = " << TOL);
          debugOut << "done: || <X2,X1> || = " << err << endl;
        }
 
#ifdef HAVE_BELOS_TSQR
        //
        // If OM is an OutOfPlaceNormalizerMixin, exercise the
        // out-of-place normalization routines.
        //
        typedef Belos::OutOfPlaceNormalizerMixin<Scalar, MV> mixin_type;
        RCP<mixin_type> tsqr = rcp_dynamic_cast<mixin_type>(OM);
        if (! tsqr.is_null())
          {
            magnitude_type err;
            debugOut << endl
                     << "=== OutOfPlaceNormalizerMixin tests ==="
                     << endl << endl;
            //
            // Fill X1_in with random values, and test the normalization
            // error with normalizeOutOfPlace().
            //
            // Don't overwrite X1, else you'll mess up the tests that
            // follow!
            //
            RCP<MV> X1_in = MVT::CloneCopy (*X1);
            debugOut << "Filling X1_in with random values... ";
            MVT::MvRandom(*X1_in);
            debugOut << "done." << endl;
            debugOut << "Filling X1_out with different random values...";
            RCP<MV> X1_out = MVT::Clone(*X1_in, MVT::GetNumberVecs(*X1_in));
            MVT::MvRandom(*X1_out);
            debugOut << "done." << endl
                     << "Calling normalizeOutOfPlace(*X1_in, *X1_out, null)... ";
            const int initialX1Rank =
              tsqr->normalizeOutOfPlace(*X1_in, *X1_out, Teuchos::null);
            TEUCHOS_TEST_FOR_EXCEPTION(initialX1Rank != sizeX1, std::runtime_error,
                               "normalizeOutOfPlace(*X1_in, *X1_out, null) "
                               "returned rank " << initialX1Rank << " from "
                               << sizeX1 << " vectors. Cannot continue.");
            debugOut << "done." << endl
                     << "Calling orthonormError() on X1_out... ";
            err = OM->orthonormError(*X1_out);
            TEUCHOS_TEST_FOR_EXCEPTION(err > TOL, std::runtime_error,
                               "After calling normalizeOutOfPlace(*X1_in, "
                               "*X1_out, null), orthonormError(X1) = "
                               << err << " > TOL = " << TOL);
            debugOut << "done: ||<X1_out,X1_out> - I|| = " << err << endl;
 
            //
            // Fill X2_in with random values, project against X1_out
            // and normalize via projectAndNormalizeOutOfPlace(), and
            // test the orthogonalization error.
            //
            // Don't overwrite X2, else you'll mess up the tests that
            // follow!
            //
            RCP<MV> X2_in = MVT::CloneCopy (*X2);
            debugOut << "Filling X2_in with random values... ";
            MVT::MvRandom(*X2_in);
            debugOut << "done." << endl
                     << "Filling X2_out with different random values...";
            RCP<MV> X2_out = MVT::Clone(*X2_in, MVT::GetNumberVecs(*X2_in));
            MVT::MvRandom(*X2_out);
            debugOut << "done." << endl
                     << "Calling projectAndNormalizeOutOfPlace(X2_in, X2_out, "
                     << "C, B, X1_out)...";
            int initialX2Rank;
            {
              Array<RCP<mat_type> > C (1);
              RCP<mat_type> B = Teuchos::null;
              initialX2Rank =
                tsqr->projectAndNormalizeOutOfPlace (*X2_in, *X2_out, C, B,
                                                     tuple<RCP<const MV> >(X1_out));
            }
            TEUCHOS_TEST_FOR_EXCEPTION(initialX2Rank != sizeX2,
                               std::runtime_error,
                               "projectAndNormalizeOutOfPlace(*X2_in, "
                               "*X2_out, C, B, tuple(X1_out)) returned rank "
                               << initialX2Rank << " from " << sizeX2
                               << " vectors. Cannot continue.");
            debugOut << "done." << endl
                     << "Calling orthonormError() on X2_out... ";
            err = OM->orthonormError (*X2_out);
            TEUCHOS_TEST_FOR_EXCEPTION(err > TOL, std::runtime_error,
                               "projectAndNormalizeOutOfPlace(*X2_in, *X2_out, "
                               "C, B, tuple(X1_out)) did not meet tolerance: "
                               "orthonormError(X2_out) = "
                               << err << " > TOL = " << TOL);
            debugOut << "done: || <X2_out,X2_out> - I || = " << err << endl
                     << "Calling orthogError(X2_out, X1_out)... ";
            err = OM->orthogError (*X2_out, *X1_out);
            TEUCHOS_TEST_FOR_EXCEPTION(err > TOL, std::runtime_error,
                               "projectAndNormalizeOutOfPlace(*X2_in, *X2_out, "
                               "C, B, tuple(X1_out)) did not meet tolerance: "
                               "orthogError(X2_out, X1_out) = "
                               << err << " > TOL = " << TOL);
            debugOut << "done: || <X2_out,X1_out> || = " << err << endl;
            debugOut << endl
                     << "=== Done with OutOfPlaceNormalizerMixin tests ==="
                     << endl << endl;
          }
#endif // HAVE_BELOS_TSQR
 
        {
          //
          // Test project() on a random multivector S, by projecting S
          // against various combinations of X1 and X2.
          //
          MVT::MvRandom(*S);
 
          debugOut << "Testing project() by projecting a random multivector S "
            "against various combinations of X1 and X2 " << endl;
          const int thisNumFailed = testProject(OM,S,X1,X2,MyOM);
          numFailed += thisNumFailed;
          if (thisNumFailed > 0)
            debugOut << "  *** " << thisNumFailed
                     << (thisNumFailed > 1 ? " tests" : " test")
                     << " failed." << endl;
        }
 
        {
          //
          // Test normalize() for various deficient cases 
          //
          debugOut << "Testing normalize() on bad multivectors " << endl;
          const int thisNumFailed = testNormalize(OM,S,MyOM);
          numFailed += thisNumFailed;
        }
 
        if (isRankRevealing)
          {
            // run a X1,Y2 range multivector against P_{X1,X1} P_{Y2,Y2}
            // note, this is allowed under the restrictions on project(),
            // because <X1,Y2> = 0
            // also, <Y2,Y2> = I, but <X1,X1> != I, so biOrtho must be set to false
            // it should require randomization, as
            // P_{X1,X1} P_{Y2,Y2} (X1*C1 + Y2*C2) = P_{X1,X1} X1*C1 = 0
            mat_type C1(sizeX1,sizeS), C2(sizeX2,sizeS);
            Teuchos::randomSyncedMatrix(C1);
            Teuchos::randomSyncedMatrix(C2);
            // S := X1*C1
            MVT::MvTimesMatAddMv(ONE,*X1,C1,ZERO,*S);
            // S := S + X2*C2
            MVT::MvTimesMatAddMv(ONE,*X2,C2,ONE,*S);
 
            debugOut << "Testing project() by projecting [X1 X2]-range multivector "
              "against P_X1 P_X2 " << endl;
            const int thisNumFailed = testProject(OM,S,X1,X2,MyOM);
            numFailed += thisNumFailed;
            if (thisNumFailed > 0)
              debugOut << "  *** " << thisNumFailed
                       << (thisNumFailed > 1 ? " tests" : " test")
                       << " failed." << endl;
          }
 
        // This test is only distinct from the rank-1 multivector test
        // (below) if S has at least 3 columns.
        if (isRankRevealing && sizeS > 2)
          {
            MVT::MvRandom(*S);
            RCP<MV> mid = MVT::Clone(*S,1);
            mat_type c(sizeS,1);
            MVT::MvTimesMatAddMv(ONE,*S,c,ZERO,*mid);
            std::vector<int> ind(1);
            ind[0] = sizeS-1;
            MVT::SetBlock(*mid,ind,*S);
 
            debugOut << "Testing normalize() on a rank-deficient multivector " << endl;
            const int thisNumFailed = testNormalizeRankReveal(OM,S,MyOM);
            numFailed += thisNumFailed;
            if (thisNumFailed > 0)
              debugOut << "  *** " << thisNumFailed
                       << (thisNumFailed > 1 ? " tests" : " test")
                       << " failed." << endl;
          }
 
        // This test will only exercise rank deficiency if S has at least 2
        // columns.
        if (isRankRevealing && sizeS > 1)
          {
            // rank-1
            RCP<MV> one = MVT::Clone(*S,1);
            MVT::MvRandom(*one);
            mat_type scaleS(sizeS,1);
            Teuchos::randomSyncedMatrix(scaleS);
            // put multiple of column 0 in columns 0:sizeS-1
            for (int i=0; i<sizeS; i++)
              {
                std::vector<int> ind(1);
                ind[0] = i;
                RCP<MV> Si = MVT::CloneViewNonConst(*S,ind);
                MVT::MvAddMv(scaleS(i,0),*one,ZERO,*one,*Si);
              }
            debugOut << "Testing normalize() on a rank-1 multivector " << endl;
            const int thisNumFailed = testNormalizeRankReveal(OM,S,MyOM);
            numFailed += thisNumFailed;
            if (thisNumFailed > 0)
              debugOut << "  *** " << thisNumFailed
                       << (thisNumFailed > 1 ? " tests" : " test")
                       << " failed." << endl;
          }
 
        {
          std::vector<int> ind(1);
          MVT::MvRandom(*S);
 
          debugOut << "Testing projectAndNormalize() on a random multivector " << endl;
          const int thisNumFailed = testProjectAndNormalize(OM,S,X1,X2,MyOM);
          numFailed += thisNumFailed;
          if (thisNumFailed > 0)
            debugOut << "  *** " << thisNumFailed
                     << (thisNumFailed > 1 ? " tests" : " test")
                     << " failed." << endl;
        }
 
        if (isRankRevealing)
          {
            // run a X1,X2 range multivector against P_X1 P_X2
            // this is allowed as <X1,X2> == 0
            // it should require randomization, as
            // P_X1 P_X2 (X1*C1 + X2*C2) = P_X1 X1*C1 = 0
            // and
            // P_X2 P_X1 (X2*C2 + X1*C1) = P_X2 X2*C2 = 0
            mat_type C1(sizeX1,sizeS), C2(sizeX2,sizeS);
            Teuchos::randomSyncedMatrix(C1);
            Teuchos::randomSyncedMatrix(C2);
            MVT::MvTimesMatAddMv(ONE,*X1,C1,ZERO,*S);
            MVT::MvTimesMatAddMv(ONE,*X2,C2,ONE,*S);
 
            debugOut << "Testing projectAndNormalize() by projecting [X1 X2]-range "
              "multivector against P_X1 P_X2 " << endl;
            const int thisNumFailed = testProjectAndNormalize(OM,S,X1,X2,MyOM);
            numFailed += thisNumFailed;
            if (thisNumFailed > 0)
              debugOut << "  *** " << thisNumFailed
                       << (thisNumFailed > 1 ? " tests" : " test")
                       << " failed." << endl;
          }
 
        // This test is only distinct from the rank-1 multivector test
        // (below) if S has at least 3 columns.
        if (isRankRevealing && sizeS > 2)
          {
            MVT::MvRandom(*S);
            RCP<MV> mid = MVT::Clone(*S,1);
            mat_type c(sizeS,1);
            MVT::MvTimesMatAddMv(ONE,*S,c,ZERO,*mid);
            std::vector<int> ind(1);
            ind[0] = sizeS-1;
            MVT::SetBlock(*mid,ind,*S);
 
            debugOut << "Testing projectAndNormalize() on a rank-deficient "
              "multivector " << endl;
            const int thisNumFailed = testProjectAndNormalize(OM,S,X1,X2,MyOM);
            numFailed += thisNumFailed;
            if (thisNumFailed > 0)
              debugOut << "  *** " << thisNumFailed
                       << (thisNumFailed > 1 ? " tests" : " test")
                       << " failed." << endl;
          }
 
        // This test will only exercise rank deficiency if S has at least 2
        // columns.
        if (isRankRevealing && sizeS > 1)
          {
            // rank-1
            RCP<MV> one = MVT::Clone(*S,1);
            MVT::MvRandom(*one);
            mat_type scaleS(sizeS,1);
            Teuchos::randomSyncedMatrix(scaleS);
            // Put a multiple of column 0 in columns 0:sizeS-1.
            for (int i=0; i<sizeS; i++)
              {
                std::vector<int> ind(1);
                ind[0] = i;
                RCP<MV> Si = MVT::CloneViewNonConst(*S,ind);
                MVT::MvAddMv(scaleS(i,0),*one,ZERO,*one,*Si);
              }
            debugOut << "Testing projectAndNormalize() on a rank-1 multivector " << endl;
            bool constantStride = true;
            if (! MVT::HasConstantStride(*S)) {
              debugOut << "-- S does not have constant stride" << endl;
              constantStride = false;
            }
            if (! MVT::HasConstantStride(*X1)) {
              debugOut << "-- X1 does not have constant stride" << endl;
              constantStride = false;
            }
            if (! MVT::HasConstantStride(*X2)) {
              debugOut << "-- X2 does not have constant stride" << endl;
              constantStride = false;
            }
            if (! constantStride) {
              debugOut << "-- Skipping this test, since TSQR does not work on "
                "multivectors with nonconstant stride" << endl;
            }
            else {
              const int thisNumFailed = testProjectAndNormalize(OM,S,X1,X2,MyOM);
              numFailed += thisNumFailed;
              if (thisNumFailed > 0) {
                debugOut << "  *** " << thisNumFailed
                         << (thisNumFailed > 1 ? " tests" : " test")
                         << " failed." << endl;
              }
            }
          }
 
        if (numFailed != 0) {
          MyOM->stream(Errors) << numFailed << " total test failures." << endl;
        }
        return numFailed;
      }
 
    private:
 
      static magnitude_type
      MVDiff (const MV& X, const MV& Y)
      {
        using Teuchos::RCP;
 
        const scalar_type ONE = SCT::one();
        const int numCols = MVT::GetNumberVecs(X);
        TEUCHOS_TEST_FOR_EXCEPTION( (MVT::GetNumberVecs(Y) != numCols),
                            std::logic_error,
                            "MVDiff: X and Y should have the same number of columns."
                            "  X has " << numCols << " column(s) and Y has "
                            << MVT::GetNumberVecs(Y) << " columns." );
        // Resid := X
        RCP< MV > Resid = MVT::CloneCopy(X);
        // Resid := Resid - Y
        MVT::MvAddMv (-ONE, Y, ONE, *Resid, *Resid);
 
        return frobeniusNorm (*Resid);
      }
 
 
      static magnitude_type
      frobeniusNorm (const MV& X)
      {
        const scalar_type ONE = SCT::one();
        const int numCols = MVT::GetNumberVecs(X);
        mat_type C (numCols, numCols);
 
        // $C := X^* X$
        MVT::MvTransMv (ONE, X, X, C);
 
        magnitude_type err (0);
        for (int i = 0; i < numCols; ++i)
          err += SCT::magnitude (C(i,i));
 
        return SCT::magnitude (SCT::squareroot (err));
      }
 
 
      static int
      testProjectAndNormalize (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > > OM,
                               const Teuchos::RCP< const MV >& S,
                               const Teuchos::RCP< const MV >& X1,
                               const Teuchos::RCP< const MV >& X2,
                               const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        return testProjectAndNormalizeNew (OM, S, X1, X2, MyOM);
      }
 
      static int
      testProjectAndNormalizeOld (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > >& OM,
                                  const Teuchos::RCP< const MV >& S,
                                  const Teuchos::RCP< const MV >& X1,
                                  const Teuchos::RCP< const MV >& X2,
                                  const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::null;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::tuple;
 
        const scalar_type ONE = SCT::one();
        const magnitude_type ZERO = SCT::magnitude(SCT::zero());
 
        // Relative tolerance against which all tests are performed.
        const magnitude_type TOL = 1.0e-12;
        // Absolute tolerance constant
        const magnitude_type ATOL = 10;
 
        const int sizeS = MVT::GetNumberVecs(*S);
        const int sizeX1 = MVT::GetNumberVecs(*X1);
        const int sizeX2 = MVT::GetNumberVecs(*X2);
        int numerr = 0;
        std::ostringstream sout;
 
        //
        // output tests:
        //   <S_out,S_out> = I
        //   <S_out,X1> = 0
        //   <S_out,X2> = 0
        //   S_in = S_out B + X1 C1 + X2 C2
        //
        // we will loop over an integer specifying the test combinations
        // the bit pattern for the different tests is listed in parenthesis
        //
        // for the projectors, test the following combinations:
        // none              (00)
        // P_X1              (01)
        // P_X2              (10)
        // P_X1 P_X2         (11)
        // P_X2 P_X1         (11)
        // the latter two should be tested to give the same answer
        //
        // for each of these, we should test with C1, C2 and B
        //
        // if hasM:
        // with and without MX1   (1--)
        // with and without MX2  (1---)
        // with and without MS  (1----)
        //
        // as hasM controls the upper level bits, we need only run test cases 0-3 if hasM==false
        // otherwise, we run test cases 0-31
        //
 
        int numtests = 4;
 
        // test ortho error before orthonormalizing
        if (X1 != null) {
          magnitude_type err = OM->orthogError(*S,*X1);
          sout << "   || <S,X1> || before     : " << err << endl;
        }
        if (X2 != null) {
          magnitude_type err = OM->orthogError(*S,*X2);
          sout << "   || <S,X2> || before     : " << err << endl;
        }
 
        for (int t=0; t<numtests; t++) {
 
          Array< RCP< const MV > > theX;
          RCP<mat_type > B = rcp( new mat_type(sizeS,sizeS) );
          Array<RCP<mat_type > > C;
          if ( (t % 3) == 0 ) {
            // neither <X1,Y1> nor <X2,Y2>
            // C, theX and theY are already empty
          }
          else if ( (t % 3) == 1 ) {
            // X1
            theX = tuple(X1);
            C = tuple( rcp(new mat_type(sizeX1,sizeS)) );
          }
          else if ( (t % 3) == 2 ) {
            // X2
            theX = tuple(X2);
            C = tuple( rcp(new mat_type(sizeX2,sizeS)) );
          }
          else {
            // X1 and X2, and the reverse.
            theX = tuple(X1,X2);
            C = tuple( rcp(new mat_type(sizeX1,sizeS)),
                       rcp(new mat_type(sizeX2,sizeS)) );
          }
 
          // We wrap up all the OrthoManager calls in a try-catch
          // block, in order to check whether any of the methods throw
          // an exception.  For the tests we perform, every thrown
          // exception is a failure.
          try {
            // call routine
            // if (t && 3) == 3, {
            //    call with reversed input: X2 X1
            // }
            // test all outputs for correctness
            // test all outputs for equivalence
 
            // here is where the outputs go
            Array<RCP<MV> > S_outs;
            Array<Array<RCP<mat_type > > > C_outs;
            Array<RCP<mat_type > > B_outs;
            RCP<MV> Scopy;
            Array<int> ret_out;
 
            // copies of S,MS
            Scopy = MVT::CloneCopy(*S);
            // randomize this data, it should be overwritten
            Teuchos::randomSyncedMatrix(*B);
            for (size_type i=0; i<C.size(); i++) {
              Teuchos::randomSyncedMatrix(*C[i]);
            }
            // Run test.  Since S was specified by the caller and
            // Scopy is a copy of S, we don't know what rank to expect
            // here -- though we do require that S have rank at least
            // one.
            //
            // Note that Anasazi and Belos differ, among other places,
            // in the order of arguments to projectAndNormalize().
            int ret = OM->projectAndNormalize(*Scopy,C,B,theX);
            sout << "projectAndNormalize() returned rank " << ret << endl;
            if (ret == 0) {
              sout << "  *** Error: returned rank is zero, cannot continue tests" << endl;
              numerr++;
              break;
            }
            ret_out.push_back(ret);
            // projectAndNormalize() is only required to return a
            // basis of rank "ret"
            // this is what we will test:
            //   the first "ret" columns in Scopy
            //   the first "ret" rows in B
            // save just the parts that we want
            // we allocate S and MS for each test, so we can save these as views
            // however, save copies of the C and B
            if (ret < sizeS) {
              std::vector<int> ind(ret);
              for (int i=0; i<ret; i++) {
                ind[i] = i;
              }
              S_outs.push_back( MVT::CloneViewNonConst(*Scopy,ind) );
              B_outs.push_back( rcp( new mat_type(Teuchos::Copy,*B,ret,sizeS) ) );
            }
            else {
              S_outs.push_back( Scopy );
              B_outs.push_back( rcp( new mat_type(*B) ) );
            }
            C_outs.push_back( Array<RCP<mat_type > >(0) );
            if (C.size() > 0) {
              C_outs.back().push_back( rcp( new mat_type(*C[0]) ) );
            }
            if (C.size() > 1) {
              C_outs.back().push_back( rcp( new mat_type(*C[1]) ) );
            }
 
            // do we run the reversed input?
            if ( (t % 3) == 3 ) {
              // copies of S,MS
              Scopy = MVT::CloneCopy(*S);
 
              // Fill the B and C[i] matrices with random data.  The
              // data will be overwritten by projectAndNormalize().
              // Filling these matrices here is only to catch some
              // bugs in projectAndNormalize().
              Teuchos::randomSyncedMatrix(*B);
              for (size_type i=0; i<C.size(); i++) {
                Teuchos::randomSyncedMatrix(*C[i]);
              }
              // flip the inputs
              theX = tuple( theX[1], theX[0] );
              // Run test.
              // Note that Anasazi and Belos differ, among other places,
              // in the order of arguments to projectAndNormalize().
              ret = OM->projectAndNormalize(*Scopy,C,B,theX);
              sout << "projectAndNormalize() returned rank " << ret << endl;
              if (ret == 0) {
                sout << "  *** Error: returned rank is zero, cannot continue tests" << endl;
                numerr++;
                break;
              }
              ret_out.push_back(ret);
              // projectAndNormalize() is only required to return a
              // basis of rank "ret"
              // this is what we will test:
              //   the first "ret" columns in Scopy
              //   the first "ret" rows in B
              // save just the parts that we want
              // we allocate S and MS for each test, so we can save these as views
              // however, save copies of the C and B
              if (ret < sizeS) {
                std::vector<int> ind(ret);
                for (int i=0; i<ret; i++) {
                  ind[i] = i;
                }
                S_outs.push_back( MVT::CloneViewNonConst(*Scopy,ind) );
                B_outs.push_back( rcp( new mat_type(Teuchos::Copy,*B,ret,sizeS) ) );
              }
              else {
                S_outs.push_back( Scopy );
                B_outs.push_back( rcp( new mat_type(*B) ) );
              }
              C_outs.push_back( Array<RCP<mat_type > >() );
              // reverse the Cs to compensate for the reverse projectors
              C_outs.back().push_back( rcp( new mat_type(*C[1]) ) );
              C_outs.back().push_back( rcp( new mat_type(*C[0]) ) );
              // flip the inputs back
              theX = tuple( theX[1], theX[0] );
            }
 
 
            // test all outputs for correctness
            for (size_type o=0; o<S_outs.size(); o++) {
              // S^T M S == I
              {
                magnitude_type err = OM->orthonormError(*S_outs[o]);
                if (err > TOL) {
                  sout << endl
                       << "  *** Test (number " << (t+1) << " of " << numtests
                       << " total tests) failed: Tolerance exceeded!  Error = "
                       << err << " > TOL = " << TOL << "."
                       << endl << endl;
                  numerr++;
                }
                sout << "   || <S,S> - I || after  : " << err << endl;
              }
              // S_in = X1*C1 + C2*C2 + S_out*B
              {
                RCP<MV> tmp = MVT::Clone(*S,sizeS);
                MVT::MvTimesMatAddMv(ONE,*S_outs[o],*B_outs[o],ZERO,*tmp);
                if (C_outs[o].size() > 0) {
                  MVT::MvTimesMatAddMv(ONE,*X1,*C_outs[o][0],ONE,*tmp);
                  if (C_outs[o].size() > 1) {
                    MVT::MvTimesMatAddMv(ONE,*X2,*C_outs[o][1],ONE,*tmp);
                  }
                }
                magnitude_type err = MVDiff(*tmp,*S);
                if (err > ATOL*TOL) {
                  sout << endl
                       << "  *** Test (number " << (t+1) << " of " << numtests
                       << " total tests) failed: Tolerance exceeded!  Error = "
                       << err << " > ATOL*TOL = " << (ATOL*TOL) << "."
                       << endl << endl;
                  numerr++;
                }
                sout << "  " << t << "|| S_in - X1*C1 - X2*C2 - S_out*B || : " << err << endl;
              }
              // <X1,S> == 0
              if (theX.size() > 0 && theX[0] != null) {
                magnitude_type err = OM->orthogError(*theX[0],*S_outs[o]);
                if (err > TOL) {
                  sout << endl
                       << "  *** Test (number " << (t+1) << " of " << numtests
                       << " total tests) failed: Tolerance exceeded!  Error = "
                       << err << " > TOL = " << TOL << "."
                       << endl << endl;
                  numerr++;
                }
                sout << "  " << t << "|| <X[0],S> || after      : " << err << endl;
              }
              // <X2,S> == 0
              if (theX.size() > 1 && theX[1] != null) {
                magnitude_type err = OM->orthogError(*theX[1],*S_outs[o]);
                if (err > TOL) {
                  sout << endl
                       << "  *** Test (number " << (t+1) << " of " << numtests
                       << " total tests) failed: Tolerance exceeded!  Error = "
                       << err << " > TOL = " << TOL << "."
                       << endl << endl;
                  numerr++;
                }
                sout << "  " << t << "|| <X[1],S> || after      : " << err << endl;
              }
            }
          }
          catch (Belos::OrthoError& e) {
            sout << "  *** Error: OrthoManager threw exception: " << e.what() << endl;
            numerr++;
          }
 
        } // test for
 
        // NOTE (mfh 05 Nov 2010) Since Belos::MsgType is an enum,
        // doing bitwise logical computations on Belos::MsgType values
        // (such as "Debug | Errors") and passing the result into
        // MyOM->stream() confuses the compiler.  As a result, we have
        // to do some type casts to make it work.
        const int msgType = (numerr > 0) ?
          (static_cast<int>(Debug) | static_cast<int>(Errors)) :
          static_cast<int>(Debug);
 
        // We report debug-level messages always.  We also report
        // errors if at least one test failed.
        MyOM->stream(static_cast< MsgType >(msgType)) << sout.str() << endl;
        return numerr;
      }
 
      static int
      testNormalize (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > >& OM,
                     const Teuchos::RCP< const MV >& S,
                     const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::RCP;
 
        int numFailures = 0;
        const scalar_type ZERO = SCT::zero();
 
        const int msgType = (static_cast<int>(Debug) | static_cast<int>(Errors));
 
        // Check that the orthogonalization gracefully handles zero vectors.
        RCP<MV> zeroVec = MVT::Clone(*S,1);
        RCP< mat_type > bZero (new mat_type (1, 1));
        std::vector< magnitude_type > zeroNorm( 1 );
 
        MVT::MvInit( *zeroVec, ZERO );
        OM->normalize( *zeroVec, bZero );
        MVT::MvNorm( *zeroVec, zeroNorm );
        // Check if the number is a NaN, this orthogonalization fails if it is.
        if ( zeroNorm[0] != ZERO )
        {
          MyOM->stream(static_cast< MsgType >(msgType)) << " --> Normalization of zero vector FAILED!" << std::endl;
          numFailures++;
        }
 
        return numFailures;
      }
 
      static int
      testNormalizeRankReveal (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > >& OM,
                               const Teuchos::RCP< const MV >& S,
                               const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::tuple;
 
        const scalar_type ONE = SCT::one();
        std::ostringstream sout;
        // Total number of failed tests in this call of this routine.
        int numerr = 0;
 
        // Relative tolerance against which all tests are performed.
        // We are measuring things in the Frobenius norm $\| \cdot \|_F$.
        // The following bounds hold for all $m \times n$ matrices $A$:
        // \[
        // \|A\|_2 \leq \|A\|_F \leq \sqrt{r} \|A\|_2,
        // \]
        // where $r$ is the (column) rank of $A$.  We bound this above
        // by the number of columns in $A$.
        //
        // An accurate normalization in the Euclidean norm of a matrix
        // $A$ with at least as many rows m as columns n, should
        // produce orthogonality $\|Q^* Q - I\|_2$ less than a factor
        // of machine precision times a low-order polynomial in m and
        // n, and residual $\|A - Q B\|_2$ (where $A = Q B$ is the
        // computed normalization) less than that bound times the norm
        // of $A$.
        //
        // Since we are measuring both of these quantitites in the
        // Frobenius norm instead, we should scale this bound by
        // $\sqrt{n}$.
 
        const int numRows = MVT::GetGlobalLength(*S);
        const int numCols = MVT::GetNumberVecs(*S);
        const int sizeS = MVT::GetNumberVecs(*S);
 
        // A good heuristic is to scale the bound by the square root
        // of the number of floating-point operations.  One could
        // perhaps support this theoretically, since we are using
        // uniform random test problems.
        const magnitude_type fudgeFactor =
          SMT::squareroot(magnitude_type(numRows) *
                          magnitude_type(numCols) *
                          magnitude_type(numCols));
        const magnitude_type TOL = SMT::eps() * fudgeFactor *
          SMT::squareroot(magnitude_type(numCols));
 
        // Absolute tolerance scaling: the Frobenius norm of the test
        // matrix S.  TOL*ATOL is the absolute tolerance for the
        // residual $\|A - Q*B\|_F$.
        const magnitude_type ATOL = frobeniusNorm (*S);
 
        sout << "The test matrix S has Frobenius norm " << ATOL
             << ", and the relative error tolerance is TOL = "
             << TOL << "." << endl;
 
        const int numtests = 1;
        for (int t = 0; t < numtests; ++t) {
 
          try {
            // call routine
            // test all outputs for correctness
 
            // S_copy gets a copy of S; we normalize in place, so we
            // need a copy to check whether the normalization
            // succeeded.
            RCP< MV > S_copy = MVT::CloneCopy (*S);
 
            // Matrix of coefficients from the normalization.
            RCP< mat_type > B (new mat_type (sizeS, sizeS));
            // The contents of B will be overwritten, but fill with
            // random data just to make sure that the normalization
            // operated on all the elements of B on which it should
            // operate.
            Teuchos::randomSyncedMatrix(*B);
 
            const int reportedRank = OM->normalize (*S_copy, B);
            sout << "normalize() returned rank " << reportedRank << endl;
            if (reportedRank == 0) {
              sout << "  *** Error: Cannot continue, since normalize() "
                "reports that S has rank 0" << endl;
              numerr++;
              break;
            }
            //
            // We don't know in this routine whether the input
            // multivector S has full rank; it is only required to
            // have nonzero rank.  Thus, we extract the first
            // reportedRank columns of S_copy and the first
            // reportedRank rows of B, and perform tests on them.
            //
 
            // Construct S_view, a view of the first reportedRank
            // columns of S_copy.
            std::vector<int> indices (reportedRank);
            for (int j = 0; j < reportedRank; ++j)
              indices[j] = j;
            RCP< MV > S_view = MVT::CloneViewNonConst (*S_copy, indices);
            // Construct B_top, a copy of the first reportedRank rows
            // of B.
            //
            // NOTE: We create this as a copy and not a view, because
            // otherwise it would not be safe with respect to RCPs.
            // This is because mat_type uses raw pointers
            // inside, so that a view would become invalid when B
            // would fall out of scope.
            RCP< mat_type > B_top (new mat_type (Teuchos::Copy, *B, reportedRank, sizeS));
 
            // Check ||<S_view,S_view> - I||
            {
              const magnitude_type err = OM->orthonormError(*S_view);
              if (err > TOL) {
                sout << "  *** Error: Tolerance exceeded: err = "
                     << err << " > TOL = " << TOL << endl;
                numerr++;
              }
              sout << "   || <S,S> - I || after  : " << err << endl;
            }
            // Check the residual ||Residual|| = ||S_view * B_top -
            // S_orig||, where S_orig is a view of the first
            // reportedRank columns of S.
            {
              // Residual is allocated with reportedRank columns.  It
              // will contain the result of testing the residual error
              // of the normalization (i.e., $\|S - S_in*B\|$).  It
              // should have the dimensions of S.  Its initial value
              // is a copy of the first reportedRank columns of S.
              RCP< MV > Residual = MVT::CloneCopy (*S);
 
              // Residual := Residual - S_view * B_view
              MVT::MvTimesMatAddMv (-ONE, *S_view, *B_top, ONE, *Residual);
 
              // Compute ||Residual||
              const magnitude_type err = frobeniusNorm (*Residual);
              if (err > ATOL*TOL) {
                sout << "  *** Error: Tolerance exceeded: err = "
                     << err << " > ATOL*TOL = " << (ATOL*TOL) << endl;
                numerr++;
              }
              sout << "  " << t << "|| S - Q*B || : " << err << endl;
            }
          }
          catch (Belos::OrthoError& e) {
            sout << "  *** Error: the OrthoManager's normalize() method "
              "threw an exception: " << e.what() << endl;
            numerr++;
          }
 
        } // test for
 
        const MsgType type = (numerr == 0) ? Debug : static_cast<MsgType> (static_cast<int>(Errors) | static_cast<int>(Debug));
        MyOM->stream(type) << sout.str();
        MyOM->stream(type) << endl;
 
        return numerr;
      }
 
      static int
      testProjectAndNormalizeNew (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > > OM,
                                  const Teuchos::RCP< const MV >& S,
                                  const Teuchos::RCP< const MV >& X1,
                                  const Teuchos::RCP< const MV >& X2,
                                  const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::null;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::tuple;
 
        // We collect all the output in this string wrapper, and print
        // it at the end.
        std::ostringstream sout;
        // Total number of failed tests in this call of this routine.
        int numerr = 0;
 
        const int numRows = MVT::GetGlobalLength(*S);
        const int numCols = MVT::GetNumberVecs(*S);
        const int sizeS = MVT::GetNumberVecs(*S);
 
        // Relative tolerance against which all tests are performed.
        // We are measuring things in the Frobenius norm $\| \cdot \|_F$.
        // The following bounds hold for all $m \times n$ matrices $A$:
        // \[
        // \|A\|_2 \leq \|A\|_F \leq \sqrt{r} \|A\|_2,
        // \]
        // where $r$ is the (column) rank of $A$.  We bound this above
        // by the number of columns in $A$.
        //
        // Since we are measuring both of these quantitites in the
        // Frobenius norm instead, we scale all error tests by
        // $\sqrt{n}$.
        //
        // A good heuristic is to scale the bound by the square root
        // of the number of floating-point operations.  One could
        // perhaps support this theoretically, since we are using
        // uniform random test problems.
        const magnitude_type fudgeFactor =
          SMT::squareroot(magnitude_type(numRows) *
                          magnitude_type(numCols) *
                          magnitude_type(numCols));
        const magnitude_type TOL = SMT::eps() * fudgeFactor *
          SMT::squareroot(magnitude_type(numCols));
 
        // Absolute tolerance scaling: the Frobenius norm of the test
        // matrix S.  TOL*ATOL is the absolute tolerance for the
        // residual $\|A - Q*B\|_F$.
        const magnitude_type ATOL = frobeniusNorm (*S);
 
        sout << "-- The test matrix S has Frobenius norm " << ATOL
             << ", and the relative error tolerance is TOL = "
             << TOL << "." << endl;
 
        // Q will contain the result of projectAndNormalize() on S.
        RCP< MV > Q = MVT::CloneCopy(*S);
        // We use this for collecting the residual error components
        RCP< MV > Residual = MVT::CloneCopy(*S);
        // Number of elements in the X array of blocks against which
        // to project S.
        const int num_X = 2;
        Array< RCP< const MV > > X (num_X);
        X[0] = MVT::CloneCopy(*X1);
        X[1] = MVT::CloneCopy(*X2);
 
        // Coefficients for the normalization
        RCP< mat_type > B (new mat_type (sizeS, sizeS));
 
        // Array of coefficients matrices from the projection.
        // For our first test, we allocate each of these matrices
        // with the proper dimensions.
        Array< RCP< mat_type > > C (num_X);
        for (int k = 0; k < num_X; ++k)
          {
            C[k] = rcp (new mat_type (MVT::GetNumberVecs(*X[k]), sizeS));
            Teuchos::randomSyncedMatrix(*C[k]); // will be overwritten
          }
        try {
          // Q*B := (I - X X^*) S
          const int reportedRank = OM->projectAndNormalize (*Q, C, B, X);
 
          // Pick out the first reportedRank columns of Q.
          std::vector<int> indices (reportedRank);
          for (int j = 0; j < reportedRank; ++j)
            indices[j] = j;
          RCP< const MV > Q_left = MVT::CloneView (*Q, indices);
 
          // Test whether the first reportedRank columns of Q are
          // orthogonal.
          {
            const magnitude_type orthoError = OM->orthonormError (*Q_left);
            sout << "-- ||Q(1:" << reportedRank << ")^* Q(1:" << reportedRank
                 << ") - I||_F = " << orthoError << endl;
            if (orthoError > TOL)
              {
                sout << "   *** Error: ||Q(1:" << reportedRank << ")^* Q(1:"
                     << reportedRank << ") - I||_F = " << orthoError
                     << " > TOL = " << TOL << "." << endl;
                numerr++;
              }
          }
 
          // Compute the residual: if successful, S = Q*B +
          // X (X^* S =: C) in exact arithmetic.  So, the residual is
          // S - Q*B - X1 C1 - X2 C2.
          //
          // Residual := S
          MVT::MvAddMv (SCT::one(), *S, SCT::zero(), *Residual, *Residual);
          {
            // Pick out the first reportedRank rows of B.  Make a deep
            // copy, since mat_type is not safe with respect
            // to RCP-based memory management (it uses raw pointers
            // inside).
            RCP< const mat_type > B_top (new mat_type (Teuchos::Copy, *B, reportedRank, B->numCols()));
            // Residual := Residual - Q(:, 1:reportedRank) * B(1:reportedRank, :)
            MVT::MvTimesMatAddMv (-SCT::one(), *Q_left, *B_top, SCT::one(), *Residual);
          }
          // Residual := Residual - X[k]*C[k]
          for (int k = 0; k < num_X; ++k)
            MVT::MvTimesMatAddMv (-SCT::one(), *X[k], *C[k], SCT::one(), *Residual);
          const magnitude_type residErr = frobeniusNorm (*Residual);
          sout << "-- ||S - Q(:, 1:" << reportedRank << ")*B(1:"
               << reportedRank << ", :) - X1*C1 - X2*C2||_F = "
               << residErr << endl;
          if (residErr > ATOL * TOL)
            {
              sout << "   *** Error: ||S - Q(:, 1:" << reportedRank
                   << ")*B(1:" << reportedRank << ", :) "
                   << "- X1*C1 - X2*C2||_F = " << residErr
                   << " > ATOL*TOL = " << (ATOL*TOL) << "." << endl;
              numerr++;
            }
          // Verify that Q(1:reportedRank) is orthogonal to X[k], for
          // all k.  This test only makes sense if reportedRank > 0.
          if (reportedRank == 0)
            {
              sout << "-- Reported rank of Q is zero: skipping Q, X[k] "
                "orthogonality test." << endl;
            }
          else
            {
              for (int k = 0; k < num_X; ++k)
                {
                  // Q should be orthogonal to X[k], for all k.
                  const magnitude_type projErr = OM->orthogError(*X[k], *Q_left);
                  sout << "-- ||<Q(1:" << reportedRank << "), X[" << k
                       << "]>||_F = " << projErr << endl;
                  if (projErr > ATOL*TOL)
                    {
                      sout << "   *** Error: ||<Q(1:" << reportedRank << "), X["
                           << k << "]>||_F = " << projErr << " > ATOL*TOL = "
                           << (ATOL*TOL) << "." << endl;
                      numerr++;
                    }
                }
            }
        } catch (Belos::OrthoError& e) {
          sout << "  *** Error: The OrthoManager subclass instance threw "
            "an exception: " << e.what() << endl;
          numerr++;
        }
 
        // Print out the collected diagnostic messages, which possibly
        // include error messages.
        const MsgType type = (numerr == 0) ? Debug : static_cast<MsgType> (static_cast<int>(Errors) | static_cast<int>(Debug));
        MyOM->stream(type) << sout.str();
        MyOM->stream(type) << endl;
 
        return numerr;
      }
 
 
      static int
      testProjectNew (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > > OM,
                      const Teuchos::RCP< const MV >& S,
                      const Teuchos::RCP< const MV >& X1,
                      const Teuchos::RCP< const MV >& X2,
                      const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::null;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::tuple;
 
        // We collect all the output in this string wrapper, and print
        // it at the end.
        std::ostringstream sout;
        // Total number of failed tests in this call of this routine.
        int numerr = 0;
 
        const int numRows = MVT::GetGlobalLength(*S);
        const int numCols = MVT::GetNumberVecs(*S);
        const int sizeS = MVT::GetNumberVecs(*S);
 
        // Relative tolerance against which all tests are performed.
        // We are measuring things in the Frobenius norm $\| \cdot \|_F$.
        // The following bounds hold for all $m \times n$ matrices $A$:
        // \[
        // \|A\|_2 \leq \|A\|_F \leq \sqrt{r} \|A\|_2,
        // \]
        // where $r$ is the (column) rank of $A$.  We bound this above
        // by the number of columns in $A$.
        //
        // Since we are measuring both of these quantitites in the
        // Frobenius norm instead, we scale all error tests by
        // $\sqrt{n}$.
        //
        // A good heuristic is to scale the bound by the square root
        // of the number of floating-point operations.  One could
        // perhaps support this theoretically, since we are using
        // uniform random test problems.
        const magnitude_type fudgeFactor =
          SMT::squareroot(magnitude_type(numRows) *
                          magnitude_type(numCols) *
                          magnitude_type(numCols));
        const magnitude_type TOL = SMT::eps() * fudgeFactor *
          SMT::squareroot(magnitude_type(numCols));
 
        // Absolute tolerance scaling: the Frobenius norm of the test
        // matrix S.  TOL*ATOL is the absolute tolerance for the
        // residual $\|A - Q*B\|_F$.
        const magnitude_type ATOL = frobeniusNorm (*S);
 
        sout << "The test matrix S has Frobenius norm " << ATOL
             << ", and the relative error tolerance is TOL = "
             << TOL << "." << endl;
 
        // Make some copies of S, X1, and X2.  The OrthoManager's
        // project() method shouldn't modify X1 or X2, but this is a a
        // test and we don't know that it doesn't!
        RCP< MV > S_copy = MVT::CloneCopy(*S);
        RCP< MV > Residual = MVT::CloneCopy(*S);
        const int num_X = 2;
        Array< RCP< const MV > > X (num_X);
        X[0] = MVT::CloneCopy(*X1);
        X[1] = MVT::CloneCopy(*X2);
 
        // Array of coefficients matrices from the projection.
        // For our first test, we allocate each of these matrices
        // with the proper dimensions.
        Array< RCP< mat_type > > C (num_X);
        for (int k = 0; k < num_X; ++k)
          {
            C[k] = rcp (new mat_type (MVT::GetNumberVecs(*X[k]), sizeS));
            Teuchos::randomSyncedMatrix(*C[k]); // will be overwritten
          }
        try {
          // Compute the projection: S_copy := (I - X X^*) S
          OM->project(*S_copy, C, X);
 
          // Compute the residual: if successful, S = S_copy + X (X^*
          // S =: C) in exact arithmetic.  So, the residual is
          // S - S_copy - X1 C1 - X2 C2.
          //
          // Residual := S - S_copy
          MVT::MvAddMv (SCT::one(), *S, -SCT::one(), *S_copy, *Residual);
          // Residual := Residual - X[k]*C[k]
          for (int k = 0; k < num_X; ++k)
            MVT::MvTimesMatAddMv (-SCT::one(), *X[k], *C[k], SCT::one(), *Residual);
          magnitude_type residErr = frobeniusNorm (*Residual);
          sout << "  ||S - S_copy - X1*C1 - X2*C2||_F = " << residErr;
          if (residErr > ATOL * TOL)
            {
              sout << "  *** Error: ||S - S_copy - X1*C1 - X2*C2||_F = " << residErr
                   << " > ATOL*TOL = " << (ATOL*TOL) << ".";
              numerr++;
            }
          for (int k = 0; k < num_X; ++k)
            {
              // S_copy should be orthogonal to X[k] now.
              const magnitude_type projErr = OM->orthogError(*X[k], *S_copy);
              if (projErr > TOL)
                {
                  sout << "  *** Error: S is not orthogonal to X[" << k
                       << "] by a factor of " << projErr << " > TOL = "
                       << TOL << ".";
                  numerr++;
                }
            }
        } catch (Belos::OrthoError& e) {
          sout << "  *** Error: The OrthoManager subclass instance threw "
            "an exception: " << e.what() << endl;
          numerr++;
        }
 
        // Print out the collected diagnostic messages, which possibly
        // include error messages.
        const MsgType type = (numerr == 0) ? Debug : static_cast<MsgType> (static_cast<int>(Errors) | static_cast<int>(Debug));
        MyOM->stream(type) << sout.str();
        MyOM->stream(type) << endl;
 
        return numerr;
      }
 
      static int
      testProject (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > > OM,
                   const Teuchos::RCP< const MV >& S,
                   const Teuchos::RCP< const MV >& X1,
                   const Teuchos::RCP< const MV >& X2,
                   const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        return testProjectNew (OM, S, X1, X2, MyOM);
      }
 
      static int
      testProjectOld (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > > OM,
                      const Teuchos::RCP< const MV >& S,
                      const Teuchos::RCP< const MV >& X1,
                      const Teuchos::RCP< const MV >& X2,
                      const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::null;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::tuple;
 
        const scalar_type ONE = SCT::one();
        // We collect all the output in this string wrapper, and print
        // it at the end.
        std::ostringstream sout;
        // Total number of failed tests in this call of this routine.
        int numerr = 0;
 
        const int numRows = MVT::GetGlobalLength(*S);
        const int numCols = MVT::GetNumberVecs(*S);
        const int sizeS = MVT::GetNumberVecs(*S);
        const int sizeX1 = MVT::GetNumberVecs(*X1);
        const int sizeX2 = MVT::GetNumberVecs(*X2);
 
        // Relative tolerance against which all tests are performed.
        // We are measuring things in the Frobenius norm $\| \cdot \|_F$.
        // The following bounds hold for all $m \times n$ matrices $A$:
        // \[
        // \|A\|_2 \leq \|A\|_F \leq \sqrt{r} \|A\|_2,
        // \]
        // where $r$ is the (column) rank of $A$.  We bound this above
        // by the number of columns in $A$.
        //
        // Since we are measuring both of these quantitites in the
        // Frobenius norm instead, we scale all error tests by
        // $\sqrt{n}$.
        //
        // A good heuristic is to scale the bound by the square root
        // of the number of floating-point operations.  One could
        // perhaps support this theoretically, since we are using
        // uniform random test problems.
        const magnitude_type fudgeFactor =
          SMT::squareroot(magnitude_type(numRows) *
                          magnitude_type(numCols) *
                          magnitude_type(numCols));
        const magnitude_type TOL = SMT::eps() * fudgeFactor *
          SMT::squareroot(magnitude_type(numCols));
 
        // Absolute tolerance scaling: the Frobenius norm of the test
        // matrix S.  TOL*ATOL is the absolute tolerance for the
        // residual $\|A - Q*B\|_F$.
        const magnitude_type ATOL = frobeniusNorm (*S);
 
        sout << "The test matrix S has Frobenius norm " << ATOL
             << ", and the relative error tolerance is TOL = "
             << TOL << "." << endl;
 
 
        //
        // Output tests:
        //   <S_out,X1> = 0
        //   <S_out,X2> = 0
        //   S_in = S_out + X1 C1 + X2 C2
        //
        // We will loop over an integer specifying the test combinations.
        // The bit pattern for the different tests is listed in parentheses.
        //
        // For the projectors, test the following combinations:
        // none              (00)
        // P_X1              (01)
        // P_X2              (10)
        // P_X1 P_X2         (11)
        // P_X2 P_X1         (11)
        // The latter two should be tested to give the same result.
        //
        // For each of these, we should test with C1 and C2:
        //
        // if hasM:
        // with and without MX1   (1--)
        // with and without MX2  (1---)
        // with and without MS  (1----)
        //
        // As hasM controls the upper level bits, we need only run test
        // cases 0-3 if hasM==false.  Otherwise, we run test cases 0-31.
        //
 
        int numtests = 8;
 
        // test ortho error before orthonormalizing
        if (X1 != null) {
          magnitude_type err = OM->orthogError(*S,*X1);
          sout << "   || <S,X1> || before     : " << err << endl;
        }
        if (X2 != null) {
          magnitude_type err = OM->orthogError(*S,*X2);
          sout << "   || <S,X2> || before     : " << err << endl;
        }
 
        for (int t = 0; t < numtests; ++t)
          {
            Array< RCP< const MV > > theX;
            Array< RCP< mat_type > > C;
            if ( (t % 3) == 0 ) {
              // neither X1 nor X2
              // C and theX are already empty
            }
            else if ( (t % 3) == 1 ) {
              // X1
              theX = tuple(X1);
              C = tuple( rcp(new mat_type(sizeX1,sizeS)) );
            }
            else if ( (t % 3) == 2 ) {
              // X2
              theX = tuple(X2);
              C = tuple( rcp(new mat_type(sizeX2,sizeS)) );
            }
            else {
              // X1 and X2, and the reverse.
              theX = tuple(X1,X2);
              C = tuple( rcp(new mat_type(sizeX1,sizeS)),
                         rcp(new mat_type(sizeX2,sizeS)) );
            }
 
            try {
              // call routine
              // if (t && 3) == 3, {
              //    call with reversed input: X2 X1
              // }
              // test all outputs for correctness
              // test all outputs for equivalence
 
              // here is where the outputs go
              Array< RCP< MV > > S_outs;
              Array< Array< RCP< mat_type > > > C_outs;
              RCP< MV > Scopy;
 
              // copies of S,MS
              Scopy = MVT::CloneCopy(*S);
              // randomize this data, it should be overwritten
              for (size_type i = 0; i < C.size(); ++i) {
                Teuchos::randomSyncedMatrix(*C[i]);
              }
              // Run test.
              // Note that Anasazi and Belos differ, among other places,
              // in the order of arguments to project().
              OM->project(*Scopy,C,theX);
              // we allocate S and MS for each test, so we can save these as views
              // however, save copies of the C
              S_outs.push_back( Scopy );
              C_outs.push_back( Array< RCP< mat_type > >(0) );
              if (C.size() > 0) {
                C_outs.back().push_back( rcp( new mat_type(*C[0]) ) );
              }
              if (C.size() > 1) {
                C_outs.back().push_back( rcp( new mat_type(*C[1]) ) );
              }
 
              // do we run the reversed input?
              if ( (t % 3) == 3 ) {
                // copies of S,MS
                Scopy = MVT::CloneCopy(*S);
                // randomize this data, it should be overwritten
                for (size_type i = 0; i < C.size(); ++i) {
                  Teuchos::randomSyncedMatrix(*C[i]);
                }
                // flip the inputs
                theX = tuple( theX[1], theX[0] );
                // Run test.
                // Note that Anasazi and Belos differ, among other places,
                // in the order of arguments to project().
                OM->project(*Scopy,C,theX);
                // we allocate S and MS for each test, so we can save these as views
                // however, save copies of the C
                S_outs.push_back( Scopy );
                // we are in a special case: P_X1 and P_X2, so we know we applied
                // two projectors, and therefore have two C[i]
                C_outs.push_back( Array<RCP<mat_type > >() );
                // reverse the Cs to compensate for the reverse projectors
                C_outs.back().push_back( rcp( new mat_type(*C[1]) ) );
                C_outs.back().push_back( rcp( new mat_type(*C[0]) ) );
                // flip the inputs back
                theX = tuple( theX[1], theX[0] );
              }
 
              // test all outputs for correctness
              for (size_type o = 0; o < S_outs.size(); ++o) {
                // S_in = X1*C1 + C2*C2 + S_out
                {
                  RCP<MV> tmp = MVT::CloneCopy(*S_outs[o]);
                  if (C_outs[o].size() > 0) {
                    MVT::MvTimesMatAddMv(ONE,*X1,*C_outs[o][0],ONE,*tmp);
                    if (C_outs[o].size() > 1) {
                      MVT::MvTimesMatAddMv(ONE,*X2,*C_outs[o][1],ONE,*tmp);
                    }
                  }
                  magnitude_type err = MVDiff(*tmp,*S);
                  if (err > ATOL*TOL) {
                    sout << "         vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv         tolerance exceeded! test failed!" << endl;
                    numerr++;
                  }
                  sout << "  " << t << "|| S_in - X1*C1 - X2*C2 - S_out || : " << err << endl;
                }
                // <X1,S> == 0
                if (theX.size() > 0 && theX[0] != null) {
                  magnitude_type err = OM->orthogError(*theX[0],*S_outs[o]);
                  if (err > TOL) {
                    sout << "         vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv         tolerance exceeded! test failed!" << endl;
                    numerr++;
                  }
                  sout << "  " << t << "|| <X[0],S> || after      : " << err << endl;
                }
                // <X2,S> == 0
                if (theX.size() > 1 && theX[1] != null) {
                  magnitude_type err = OM->orthogError(*theX[1],*S_outs[o]);
                  if (err > TOL) {
                    sout << "         vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv         tolerance exceeded! test failed!" << endl;
                    numerr++;
                  }
                  sout << "  " << t << "|| <X[1],S> || after      : " << err << endl;
                }
              }
 
              // test all outputs for equivalence
              // check all combinations:
              //    output 0 == output 1
              //    output 0 == output 2
              //    output 1 == output 2
              for (size_type o1=0; o1<S_outs.size(); o1++) {
                for (size_type o2=o1+1; o2<S_outs.size(); o2++) {
                  // don't need to check MS_outs because we check
                  //   S_outs and MS_outs = M*S_outs
                  // don't need to check C_outs either
                  //
                  // check that S_outs[o1] == S_outs[o2]
                  magnitude_type err = MVDiff(*S_outs[o1],*S_outs[o2]);
                  if (err > TOL) {
                    sout << "    vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv         tolerance exceeded! test failed!" << endl;
                    numerr++;
                  }
                }
              }
 
            }
            catch (Belos::OrthoError& e) {
              sout << "   -------------------------------------------         project() threw exception" << endl;
              sout << "   Error: " << e.what() << endl;
              numerr++;
            }
          } // test for
 
        MsgType type = Debug;
        if (numerr>0) type = Errors;
        MyOM->stream(type) << sout.str();
        MyOM->stream(type) << endl;
 
        return numerr;
      }
 
 
    };
 
 
 
  } // namespace Test
} // namespace Belos