TMetricTest.cpp

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/* *****************************************************************
    MESQUITE -- The Mesh Quality Improvement Toolkit

    Copyright 2006 Sandia National Laboratories.  Developed at the
    University of Wisconsin--Madison under SNL contract number
    624796.  The U.S. Government and the University of Wisconsin
    retain certain rights to this software.

    This library is free software; you can redistribute it and/or
    modify it under the terms of the GNU Lesser General Public
    License as published by the Free Software Foundation; either
    version 2.1 of the License, or (at your option) any later version.

    This library is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
    Lesser General Public License for more details.

    You should have received a copy of the GNU Lesser General Public License
    (lgpl.txt) along with this library; if not, write to the Free Software
    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA

    (2010) kraftche@cae.wisc.edu

  ***************************************************************** */

/** \file TMetricTest.cpp
 *  \brief Unit tests TMetric base class functionality
 *  \author Jason Kraftcheck
 */

#include "Mesquite.hpp"
#include "TMetric.hpp"
#include "UnitUtil.hpp"
#include "MsqError.hpp"

using namespace MBMesquite;

class TMetricTest : public CppUnit::TestFixture
{
    CPPUNIT_TEST_SUITE( TMetricTest );
    CPPUNIT_TEST( test_numerical_gradient_2D );
    CPPUNIT_TEST( test_numerical_hessian_2D );
    CPPUNIT_TEST( test_numerical_gradient_3D );
    CPPUNIT_TEST( test_numerical_hessian_3D );
    CPPUNIT_TEST_SUITE_END();

  public:
    void test_numerical_gradient_2D();
    void test_numerical_hessian_2D();
    void test_numerical_gradient_3D();
    void test_numerical_hessian_3D();
};

CPPUNIT_TEST_SUITE_NAMED_REGISTRATION( TMetricTest, "Unit" );
CPPUNIT_TEST_SUITE_NAMED_REGISTRATION( TMetricTest, "TMetricTest" );

// implement metric such that the expected derivatives
// at each location r,c in dm/dT are 3r+c+1
class GradTestMetricRel : public TMetric
{
  public:
    std::string get_name() const
    {
        return "GradTest";
    }

    static double grad( int r, int c )
    {
        return 3 * r + c + 1;
    }

    bool evaluate( const MsqMatrix< 2, 2 >& T, double& result, MsqError& )
    {
        result = 0;
        for( int r = 0; r < 2; ++r )
            for( int c = 0; c < 2; ++c )
                result += grad( r, c ) * T( r, c );
        return true;
    }

    bool evaluate( const MsqMatrix< 3, 3 >& T, double& result, MsqError& )
    {
        result = 0;
        for( int r = 0; r < 3; ++r )
            for( int c = 0; c < 3; ++c )
                result += grad( r, c ) * T( r, c );
        return true;
    }
};

// implement metric that is the |2T - T^t|^2
// such that the Hessian is the constant:
//  _          _
// | 2  0  0  0 |
// | 0 10 -8  0 |
// | 0 -8 10  0 |
// |_0  0  0  2_|
//  _                         _
// | 2  0  0  0  0  0  0  0  0 |
// | 0 10  0 -8  0  0  0  0  0 |
// | 0  0 10  0  0  0 -8  0  0 |
// | 0 -8  0 10  0  0  0  0  0 |
// | 0  0  0  0  2  0  0  0  0 |
// | 0  0  0  0  0 10  0 -8  0 |
// | 0  0 -8  0  0  0 10  0  0 |
// | 0  0  0  0  0 -8  0 10  0 |
// |_0  0  0  0  0  0  0  0  2_|
class HessTestMetricRel : public TMetric
{
  public:
    std::string get_name() const
    {
        return "HessTest";
    }

    bool evaluate( const MsqMatrix< 2, 2 >& T, double& result, MsqError& )
    {
        result = sqr_Frobenius( 2 * T - transpose( T ) );
        return true;
    }
    bool evaluate( const MsqMatrix< 3, 3 >& T, double& result, MsqError& )
    {
        result = sqr_Frobenius( 2 * T - transpose( T ) );
        return true;
    }
    bool evaluate_with_grad( const MsqMatrix< 2, 2 >& T, double& result, MsqMatrix< 2, 2 >& wrt_T, MsqError& )
    {
        result = sqr_Frobenius( 2 * T - transpose( T ) );
        wrt_T  = 10 * T - 8 * transpose( T );
        return true;
    }
    bool evaluate_with_grad( const MsqMatrix< 3, 3 >& T, double& result, MsqMatrix< 3, 3 >& wrt_T, MsqError& )
    {
        result = sqr_Frobenius( 2 * T - transpose( T ) );
        wrt_T  = 10 * T - 8 * transpose( T );
        return true;
    }
};

/** Simple target metric for testing second partial derivatives.
 *  \f$\mu(T) = |T|\f$
 *  \f$\frac{\partial\mu}{\partial T} = \frac{1}{|T|} T \f$
 *  \f$\frac{\partial^{2}\mu}{\partial t_{i,i}^2} = \frac{1}{|T|} - \frac{t_{i,i}^2}{|T|^3}\f$
 *  \f$\frac{\partial^{2}\mu}{\partial t_{i,j} \partial t_{k,l} (i \ne k or j \ne l)} =
 * -\frac{t_{i,j} a_{k,l}}{|T|^3}\f$
 */
class HessTestMetricRel_2 : public TMetric
{
  public:
    std::string get_name() const
    {
        return "HessTest2";
    }

    bool evaluate( const MsqMatrix< 2, 2 >& T, double& result, MsqError& err )
    {
        result = Frobenius( T );
        return true;
    }

    bool evaluate_with_grad( const MsqMatrix< 2, 2 >& T, double& result, MsqMatrix< 2, 2 >& d, MsqError& err )
    {
        result = Frobenius( T );
        d      = T / result;
        return true;
    }

    bool evaluate_with_hess( const MsqMatrix< 2, 2 >& T, double& result, MsqMatrix< 2, 2 >& d, MsqMatrix< 2, 2 > d2[3],
                             MsqError& err )
    {
        result = Frobenius( T );
        d      = T / result;
        int h  = 0;
        for( int r = 0; r < 2; ++r )
        {
            int i = h;
            for( int c = r; c < 2; ++c )
                d2[h++] = transpose( T.row( r ) ) * T.row( c ) / -( result * result * result );
            d2[i] += MsqMatrix< 2, 2 >( 1.0 / result );
        }
        return true;
    }

    bool evaluate( const MsqMatrix< 3, 3 >& T, double& result, MsqError& err )
    {
        result = Frobenius( T );
        return true;
    }

    bool evaluate_with_grad( const MsqMatrix< 3, 3 >& T, double& result, MsqMatrix< 3, 3 >& d, MsqError& err )
    {
        result = Frobenius( T );
        d      = T / result;
        return true;
    }

    bool evaluate_with_hess( const MsqMatrix< 3, 3 >& T, double& result, MsqMatrix< 3, 3 >& d, MsqMatrix< 3, 3 > d2[6],
                             MsqError& err )
    {
        result = Frobenius( T );
        d      = T / result;
        int h  = 0;
        for( int r = 0; r < 3; ++r )
        {
            int i = h;
            for( int c = r; c < 3; ++c )
                d2[h++] = transpose( T.row( r ) ) * T.row( c ) / -( result * result * result );
            d2[i] += MsqMatrix< 3, 3 >( 1.0 / result );
        }
        return true;
    }
};

void TMetricTest::test_numerical_gradient_2D()
{
    GradTestMetricRel metric;
    HessTestMetricRel_2 metric2;
    const double Avals[] = { 1, 2, 2, 5 };
    const double Bvals[] = { -0.1, -0.15, -0.25, -0.8 };
    const MsqMatrix< 2, 2 > A( Avals );
    const MsqMatrix< 2, 2 > B( Bvals );

    MsqError err;
    MsqMatrix< 2, 2 > d;
    bool valid;
    double val, gval;

    MsqMatrix< 2, 2 > expected;
    for( int r = 0; r < 2; ++r )
        for( int c = 0; c < 2; ++c )
            expected( r, c ) = metric.grad( r, c );

    valid = metric.evaluate( A, val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( A, gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    valid = metric.evaluate( B, val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( B, gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    valid = metric.evaluate( inverse( A ), val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( inverse( A ), gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    valid = metric.evaluate( inverse( B ), val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( inverse( B ), gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    valid = metric.evaluate( A * inverse( B ), val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( A * inverse( B ), gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    valid = metric.evaluate( B * inverse( A ), val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( B * inverse( A ), gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    MsqMatrix< 2, 2 > da;
    valid = metric2.evaluate_with_grad( A, val, da, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric2.TMetric::evaluate_with_grad( A, gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( da, d, 1e-6 );

    valid = metric2.evaluate_with_grad( B, val, da, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric2.TMetric::evaluate_with_grad( B, gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( da, d, 1e-6 );
}

void TMetricTest::test_numerical_gradient_3D()
{
    GradTestMetricRel metric;
    HessTestMetricRel_2 metric2;
    const double Avals[] = { 1, 2, 3, 4, 1, 4, 3, 2, 1 };
    const double Bvals[] = { 0.1, 0.15, 0.05, 0.2, -0.1, -0.15, -0.05, -0.2, 2 };
    const MsqMatrix< 3, 3 > A( Avals );
    const MsqMatrix< 3, 3 > B( Bvals );

    MsqError err;
    MsqMatrix< 3, 3 > d;
    bool valid;
    double val, gval;

    MsqMatrix< 3, 3 > expected;
    for( int r = 0; r < 3; ++r )
        for( int c = 0; c < 3; ++c )
            expected( r, c ) = metric.grad( r, c );

    valid = metric.evaluate( A, val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( A, gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    valid = metric.evaluate( B, val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( B, gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    valid = metric.evaluate( inverse( A ), val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( inverse( A ), gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    valid = metric.evaluate( inverse( B ), val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( inverse( B ), gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    valid = metric.evaluate( A * inverse( B ), val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( A * inverse( B ), gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    valid = metric.evaluate( B * inverse( A ), val, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_grad( B * inverse( A ), gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( expected, d, 1e-6 );

    MsqMatrix< 3, 3 > da;
    valid = metric2.evaluate_with_grad( A, val, da, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric2.TMetric::evaluate_with_grad( A, gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( da, d, 1e-6 );

    valid = metric2.evaluate_with_grad( B, val, da, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric2.TMetric::evaluate_with_grad( B, gval, d, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, gval, 1e-6 );
    ASSERT_MATRICES_EQUAL( da, d, 1e-6 );
}

void TMetricTest::test_numerical_hessian_2D()
{
    HessTestMetricRel metric;
    HessTestMetricRel_2 metric2;
    const double Avals[] = { 1, 2, 2, 5 };
    const double Bvals[] = { -0.1, -0.15, -0.25, -0.8 };
    const MsqMatrix< 2, 2 > A( Avals );
    const MsqMatrix< 2, 2 > B( Bvals );

    MsqError err;
    MsqMatrix< 2, 2 > g, gh;
    MsqMatrix< 2, 2 > h[3];
    bool valid;
    double val, hval;

    const double h_00[] = { 2, 0, 0, 10 };
    const double h_01[] = { 0, 0, -8, 0 };
    const double h_11[] = { 10, 0, 0, 2 };
    MsqMatrix< 2, 2 > h00( h_00 ), h01( h_01 ), h11( h_11 );

    valid = metric.evaluate_with_grad( A, val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( A, hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[2], 1e-6 );

    valid = metric.evaluate_with_grad( B, val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( B, hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[2], 1e-6 );

    valid = metric.evaluate_with_grad( inverse( A ), val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( inverse( A ), hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[2], 1e-6 );

    valid = metric.evaluate_with_grad( inverse( B ), val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( inverse( B ), hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[2], 1e-6 );

    valid = metric.evaluate_with_grad( A * inverse( B ), val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( A * inverse( B ), hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[2], 1e-6 );

    valid = metric.evaluate_with_grad( B * inverse( A ), val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( B * inverse( A ), hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[2], 1e-6 );

    MsqMatrix< 2, 2 > ah[3];
    valid = metric2.evaluate_with_hess( A, val, g, ah, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric2.TMetric::evaluate_with_hess( A, hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[0], h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[1], h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[2], h[2], 1e-6 );

    valid = metric2.evaluate_with_hess( B, val, g, ah, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric2.TMetric::evaluate_with_hess( B, hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[0], h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[1], h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[2], h[2], 1e-6 );
}

void TMetricTest::test_numerical_hessian_3D()
{
    HessTestMetricRel metric;
    HessTestMetricRel_2 metric2;
    const double Avals[] = { 1, 2, 3, 4, 1, 4, 3, 2, 1 };
    const double Bvals[] = { 0.1, 0.15, 0.05, 0.2, -0.1, -0.15, -0.05, -0.2, 2 };
    const MsqMatrix< 3, 3 > A( Avals );
    const MsqMatrix< 3, 3 > B( Bvals );

    MsqError err;
    MsqMatrix< 3, 3 > g, gh;
    MsqMatrix< 3, 3 > h[6];
    bool valid;
    double val, hval;

    const double h_00[] = { 2, 0, 0, 0, 10, 0, 0, 0, 10 };
    const double h_01[] = { 0, 0, 0, -8, 0, 0, 0, 0, 0 };
    const double h_02[] = { 0, 0, 0, 0, 0, 0, -8, 0, 0 };
    const double h_11[] = { 10, 0, 0, 0, 2, 0, 0, 0, 10 };
    const double h_12[] = { 0, 0, 0, 0, 0, 0, 0, -8, 0 };
    const double h_22[] = { 10, 0, 0, 0, 10, 0, 0, 0, 2 };
    MsqMatrix< 3, 3 > h00( h_00 ), h01( h_01 ), h02( h_02 ), h11( h_11 ), h12( h_12 ), h22( h_22 );

    valid = metric.evaluate_with_grad( A, val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( A, hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h02, h[2], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[3], 1e-6 );
    ASSERT_MATRICES_EQUAL( h12, h[4], 1e-6 );
    ASSERT_MATRICES_EQUAL( h22, h[5], 1e-6 );

    valid = metric.evaluate_with_grad( B, val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( B, hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h02, h[2], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[3], 1e-6 );
    ASSERT_MATRICES_EQUAL( h12, h[4], 1e-6 );
    ASSERT_MATRICES_EQUAL( h22, h[5], 1e-6 );

    valid = metric.evaluate_with_grad( inverse( A ), val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( inverse( A ), hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h02, h[2], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[3], 1e-6 );
    ASSERT_MATRICES_EQUAL( h12, h[4], 1e-6 );
    ASSERT_MATRICES_EQUAL( h22, h[5], 1e-6 );

    valid = metric.evaluate_with_grad( inverse( B ), val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( inverse( B ), hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h02, h[2], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[3], 1e-6 );
    ASSERT_MATRICES_EQUAL( h12, h[4], 1e-6 );
    ASSERT_MATRICES_EQUAL( h22, h[5], 1e-6 );

    valid = metric.evaluate_with_grad( A * inverse( B ), val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( A * inverse( B ), hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h02, h[2], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[3], 1e-6 );
    ASSERT_MATRICES_EQUAL( h12, h[4], 1e-6 );
    ASSERT_MATRICES_EQUAL( h22, h[5], 1e-6 );

    valid = metric.evaluate_with_grad( B * inverse( A ), val, g, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric.evaluate_with_hess( B * inverse( A ), hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( h00, h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( h01, h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( h02, h[2], 1e-6 );
    ASSERT_MATRICES_EQUAL( h11, h[3], 1e-6 );
    ASSERT_MATRICES_EQUAL( h12, h[4], 1e-6 );
    ASSERT_MATRICES_EQUAL( h22, h[5], 1e-6 );

    MsqMatrix< 3, 3 > ah[6];
    valid = metric2.evaluate_with_hess( A, val, g, ah, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric2.TMetric::evaluate_with_hess( A, hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[0], h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[1], h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[2], h[2], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[3], h[3], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[4], h[4], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[5], h[5], 1e-6 );

    valid = metric2.evaluate_with_hess( B, val, g, ah, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    valid = metric2.TMetric::evaluate_with_hess( B, hval, gh, h, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT( valid );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( val, hval, 1e-6 );
    ASSERT_MATRICES_EQUAL( g, gh, 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[0], h[0], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[1], h[1], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[2], h[2], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[3], h[3], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[4], h[4], 1e-6 );
    ASSERT_MATRICES_EQUAL( ah[5], h[5], 1e-6 );
}