AveragingQMTest.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
    retian 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

    (2006) kraftche@cae.wisc.edu

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

/*! \file AveragingQMTest.cpp

Unit testing for the AveragingQM class
\author Jasno Kraftcheck
*/
#include "Mesquite.hpp"
#include "AveragingQM.hpp"
#include "IdealElements.hpp"
#include "UnitUtil.hpp"
#include "TopologyInfo.hpp"
#include "PatchData.hpp"
#include "cppunit/extensions/HelperMacros.h"

using namespace MBMesquite;

class AveragingQMTest : public CppUnit::TestFixture
{
    CPPUNIT_TEST_SUITE( AveragingQMTest );
    CPPUNIT_TEST( test_average_metrics_liner );
    CPPUNIT_TEST( test_average_metrics_rms );
    CPPUNIT_TEST( test_average_metrics_hms );
    CPPUNIT_TEST( test_average_metrics_minimum );
    CPPUNIT_TEST( test_average_metrics_maximum );
    CPPUNIT_TEST( test_average_metrics_harmonic );
    CPPUNIT_TEST( test_average_metrics_geometric );
    CPPUNIT_TEST( test_average_metrics_sum );
    CPPUNIT_TEST( test_average_metrics_sum_squared );
    CPPUNIT_TEST( test_average_metrics_standard_deviation );
    CPPUNIT_TEST( test_average_metrics_max_over_min );
    CPPUNIT_TEST( test_average_metrics_max_minus_min );
    CPPUNIT_TEST( test_average_metrics_sum_of_ratios_squared );
    CPPUNIT_TEST( test_average_and_weights_linear );
    CPPUNIT_TEST( test_average_and_weights_rms );
    CPPUNIT_TEST( test_average_and_weights_hms );
    CPPUNIT_TEST( test_average_and_weights_minimum );
    CPPUNIT_TEST( test_average_and_weights_maximum );
    CPPUNIT_TEST( test_average_and_weights_harmonic );
    CPPUNIT_TEST( test_average_and_weights_geometric );
    CPPUNIT_TEST( test_average_and_weights_sum );
    CPPUNIT_TEST( test_average_and_weights_sum_squared );
    CPPUNIT_TEST( test_average_and_weights_standard_deviation );
    CPPUNIT_TEST( test_average_and_weights_max_over_min );
    CPPUNIT_TEST( test_average_and_weights_max_minus_min );
    CPPUNIT_TEST( test_average_and_weights_sum_of_ratios_squared );
    CPPUNIT_TEST( test_average_corner_gradients_linear );
    CPPUNIT_TEST( test_average_corner_gradients_rms );
    CPPUNIT_TEST( test_average_corner_gradients_hms );
    CPPUNIT_TEST( test_average_corner_gradients_minimum );
    CPPUNIT_TEST( test_average_corner_gradients_maximum );
    CPPUNIT_TEST( test_average_corner_gradients_harmonic );
    CPPUNIT_TEST( test_average_corner_gradients_geometric );
    CPPUNIT_TEST( test_average_corner_gradients_sum );
    CPPUNIT_TEST( test_average_corner_gradients_sum_squared );
    CPPUNIT_TEST( test_average_corner_gradients_standard_deviation );
    CPPUNIT_TEST( test_average_corner_gradients_max_over_min );
    CPPUNIT_TEST( test_average_corner_gradients_max_minus_min );
    CPPUNIT_TEST( test_average_corner_gradients_sum_of_ratios_squared );
    CPPUNIT_TEST( test_average_corner_hessians_linear );
    CPPUNIT_TEST( test_average_corner_hessians_rms );
    CPPUNIT_TEST( test_average_corner_hessians_hms );
    CPPUNIT_TEST( test_average_corner_hessians_minimum );
    CPPUNIT_TEST( test_average_corner_hessians_maximum );
    CPPUNIT_TEST( test_average_corner_hessians_harmonic );
    CPPUNIT_TEST( test_average_corner_hessians_geometric );
    CPPUNIT_TEST( test_average_corner_hessians_sum );
    CPPUNIT_TEST( test_average_corner_hessians_sum_squared );
    CPPUNIT_TEST( test_average_corner_hessians_standard_deviation );
    CPPUNIT_TEST( test_average_corner_hessians_max_over_min );
    CPPUNIT_TEST( test_average_corner_hessians_max_minus_min );
    CPPUNIT_TEST( test_average_corner_hessians_sum_of_ratios_squared );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_linear );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_rms );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_hms );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_minimum );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_maximum );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_harmonic );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_geometric );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_sum );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_sum_squared );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_standard_deviation );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_max_over_min );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_max_minus_min );
    CPPUNIT_TEST( test_average_corner_hessian_diagonals_sum_of_ratios_squared );
    CPPUNIT_TEST_SUITE_END();

    static const double VAL_LIST_1[5];
    static const double VAL_LIST_2[8];
    static const unsigned LEN_LIST_1;
    static const unsigned LEN_LIST_2;

    void check_average_and_weights_fails( QualityMetric::AveragingMethod scheme );

    void check_average_gradients( QualityMetric::AveragingMethod scheme );

    void check_average_gradient_fails( QualityMetric::AveragingMethod scheme );

    void check_pmean_hessian_diagonals( QualityMetric::AveragingMethod scheme,
                                        double inner_power,
                                        double outer_power,
                                        bool scale );

    void check_hessian_diagonal_fails( QualityMetric::AveragingMethod scheme );

    void check_pmean_hessian( QualityMetric::AveragingMethod scheme,
                              double inner_power,
                              double outer_power,
                              bool scale );

    void check_hessian_fails( QualityMetric::AveragingMethod scheme );

    void check_average_and_weights( const double* vals,
                                    unsigned n,
                                    QualityMetric::AveragingMethod method,
                                    const double* weights );

  public:
    void test_average_metrics_liner();
    void test_average_metrics_rms();
    void test_average_metrics_hms();
    void test_average_metrics_minimum();
    void test_average_metrics_maximum();
    void test_average_metrics_harmonic();
    void test_average_metrics_geometric();
    void test_average_metrics_sum();
    void test_average_metrics_sum_squared();
    void test_average_metrics_standard_deviation();
    void test_average_metrics_max_over_min();
    void test_average_metrics_max_minus_min();
    void test_average_metrics_sum_of_ratios_squared();
    void test_average_and_weights_linear();
    void test_average_and_weights_rms();
    void test_average_and_weights_hms();
    void test_average_and_weights_minimum();
    void test_average_and_weights_maximum();
    void test_average_and_weights_harmonic();
    void test_average_and_weights_geometric();
    void test_average_and_weights_sum();
    void test_average_and_weights_sum_squared();
    void test_average_and_weights_standard_deviation();
    void test_average_and_weights_max_over_min();
    void test_average_and_weights_max_minus_min();
    void test_average_and_weights_sum_of_ratios_squared();
    void test_average_corner_gradients_linear();
    void test_average_corner_gradients_rms();
    void test_average_corner_gradients_hms();
    void test_average_corner_gradients_minimum();
    void test_average_corner_gradients_maximum();
    void test_average_corner_gradients_harmonic();
    void test_average_corner_gradients_geometric();
    void test_average_corner_gradients_sum();
    void test_average_corner_gradients_sum_squared();
    void test_average_corner_gradients_standard_deviation();
    void test_average_corner_gradients_max_over_min();
    void test_average_corner_gradients_max_minus_min();
    void test_average_corner_gradients_sum_of_ratios_squared();
    void test_average_corner_hessians_linear();
    void test_average_corner_hessians_rms();
    void test_average_corner_hessians_hms();
    void test_average_corner_hessians_minimum();
    void test_average_corner_hessians_maximum();
    void test_average_corner_hessians_harmonic();
    void test_average_corner_hessians_geometric();
    void test_average_corner_hessians_sum();
    void test_average_corner_hessians_sum_squared();
    void test_average_corner_hessians_standard_deviation();
    void test_average_corner_hessians_max_over_min();
    void test_average_corner_hessians_max_minus_min();
    void test_average_corner_hessians_sum_of_ratios_squared();
    void test_average_corner_hessian_diagonals_linear();
    void test_average_corner_hessian_diagonals_rms();
    void test_average_corner_hessian_diagonals_hms();
    void test_average_corner_hessian_diagonals_minimum();
    void test_average_corner_hessian_diagonals_maximum();
    void test_average_corner_hessian_diagonals_harmonic();
    void test_average_corner_hessian_diagonals_geometric();
    void test_average_corner_hessian_diagonals_sum();
    void test_average_corner_hessian_diagonals_sum_squared();
    void test_average_corner_hessian_diagonals_standard_deviation();
    void test_average_corner_hessian_diagonals_max_over_min();
    void test_average_corner_hessian_diagonals_max_minus_min();
    void test_average_corner_hessian_diagonals_sum_of_ratios_squared();
};

CPPUNIT_TEST_SUITE_NAMED_REGISTRATION( AveragingQMTest, "AveragingQMTest" );
CPPUNIT_TEST_SUITE_NAMED_REGISTRATION( AveragingQMTest, "Unit" );

const double AveragingQMTest::VAL_LIST_1[5] = { 1, 2, -1, -2, 5 };
const double AveragingQMTest::VAL_LIST_2[8] = {
    M_PI, std::exp( 1.0 ), -20, 8, M_PI / 4, std::log( 2.0 ), std::sqrt( 2.0 ), -1 };
const unsigned AveragingQMTest::LEN_LIST_1 = sizeof( VAL_LIST_1 ) / sizeof( double );
const unsigned AveragingQMTest::LEN_LIST_2 = sizeof( VAL_LIST_2 ) / sizeof( double );

static double pmean( const double* vals, unsigned n, double power )
{
    double result = 0;
    for( unsigned i = 0; i < n; ++i )
        result += std::pow( vals[i], power );
    return pow( result / n, 1.0 / power );
}

static double list_min( const double* vals, unsigned n )
{
    double result = vals[0];
    for( unsigned i = 1; i < n; ++i )
        if( vals[i] < result ) result = vals[i];
    return result;
}

static double list_max( const double* vals, unsigned n )
{
    double result = vals[0];
    for( unsigned i = 1; i < n; ++i )
        if( vals[i] > result ) result = vals[i];
    return result;
}

static double sum_sqr( const double* vals, unsigned n )
{
    double result = 0.0;
    for( unsigned i = 0; i < n; ++i )
        result += vals[i] * vals[i];
    return result;
}

static double sum_of_ratios_squared( const double* vals, unsigned n )
{
    double result = 0.0;
    for( unsigned i = 0; i < n; ++i )
        for( unsigned j = 0; j < n; ++j )
        {
            double ratio = vals[i] / vals[j];
            result += ratio * ratio;
        }
    return result / ( n * n );
}

static double geometric_mean( const double* vals, unsigned n )
{
    double result = 1.0;
    for( unsigned i = 0; i < n; ++i )
        result *= vals[i];
    return std::pow( result, 1.0 / n );
}

void AveragingQMTest::test_average_metrics_liner()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::LINEAR );
    double exp, act;

    exp = pmean( VAL_LIST_1, LEN_LIST_1, 1 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = pmean( VAL_LIST_2, LEN_LIST_2, 1 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_rms()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::RMS );
    double exp, act;

    exp = pmean( VAL_LIST_1, LEN_LIST_1, 2 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = pmean( VAL_LIST_2, LEN_LIST_2, 2 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_hms()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::HMS );
    double exp, act;

    exp = pmean( VAL_LIST_1, LEN_LIST_1, -2.0 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = pmean( VAL_LIST_2, LEN_LIST_2, -2.0 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_minimum()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::MINIMUM );
    double exp, act;

    exp = list_min( VAL_LIST_1, LEN_LIST_1 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = list_min( VAL_LIST_2, LEN_LIST_2 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_maximum()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::MAXIMUM );
    double exp, act;

    exp = list_max( VAL_LIST_1, LEN_LIST_1 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = list_max( VAL_LIST_2, LEN_LIST_2 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_harmonic()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::HARMONIC );
    double exp, act;

    exp = pmean( VAL_LIST_1, LEN_LIST_1, -1.0 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = pmean( VAL_LIST_2, LEN_LIST_2, -1.0 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_geometric()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::GEOMETRIC );
    double exp, act;

    exp = geometric_mean( VAL_LIST_1, LEN_LIST_1 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = geometric_mean( VAL_LIST_2, LEN_LIST_2 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_sum()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::SUM );
    double exp, act;
    unsigned i;

    for( exp = 0.0, i = 0; i < LEN_LIST_1; exp += VAL_LIST_1[i++] )
        ;
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    for( exp = 0.0, i = 0; i < LEN_LIST_2; exp += VAL_LIST_2[i++] )
        ;
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_sum_squared()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::SUM_SQUARED );
    double exp, act;

    exp = sum_sqr( VAL_LIST_1, LEN_LIST_1 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = sum_sqr( VAL_LIST_2, LEN_LIST_2 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_standard_deviation()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::STANDARD_DEVIATION );
    double exp, act, tmp;

    tmp = pmean( VAL_LIST_1, LEN_LIST_1, 1.0 );
    exp = sum_sqr( VAL_LIST_1, LEN_LIST_1 );
    exp = exp / LEN_LIST_1 - tmp * tmp;
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    tmp = pmean( VAL_LIST_2, LEN_LIST_2, 1.0 );
    exp = sum_sqr( VAL_LIST_2, LEN_LIST_2 );
    exp = exp / LEN_LIST_2 - tmp * tmp;
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_max_over_min()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::MAX_OVER_MIN );
    double exp, act;

    exp = list_max( VAL_LIST_1, LEN_LIST_1 ) / list_min( VAL_LIST_1, LEN_LIST_1 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = list_max( VAL_LIST_2, LEN_LIST_2 ) / list_min( VAL_LIST_2, LEN_LIST_2 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_max_minus_min()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::MAX_MINUS_MIN );
    double exp, act;

    exp = list_max( VAL_LIST_1, LEN_LIST_1 ) - list_min( VAL_LIST_1, LEN_LIST_1 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = list_max( VAL_LIST_2, LEN_LIST_2 ) - list_min( VAL_LIST_2, LEN_LIST_2 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::test_average_metrics_sum_of_ratios_squared()
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( QualityMetric::SUM_OF_RATIOS_SQUARED );
    double exp, act;

    exp = sum_of_ratios_squared( VAL_LIST_1, LEN_LIST_1 );
    act = aqm.average_metrics( VAL_LIST_1, LEN_LIST_1, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );

    exp = sum_of_ratios_squared( VAL_LIST_2, LEN_LIST_2 );
    act = aqm.average_metrics( VAL_LIST_2, LEN_LIST_2, err );
    ASSERT_NO_ERROR( err );
    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp, act, 1e-8 );
}

void AveragingQMTest::check_average_and_weights( const double* vals,
                                                 unsigned n,
                                                 QualityMetric::AveragingMethod method,
                                                 const double* weights )
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( method );

    CPPUNIT_ASSERT( n > 0 );
    std::vector< double > working( n );
    std::copy( vals, vals + n, working.begin() );

    double avg1, avg2;
    avg1 = aqm.average_metrics( vals, n, err );
    ASSERT_NO_ERROR( err );
    avg2 = aqm.average_metric_and_weights( arrptr( working ), n, err );
    ASSERT_NO_ERROR( err );

    CPPUNIT_ASSERT_DOUBLES_EQUAL( avg1, avg2, 1e-6 );
    for( unsigned i = 0; i < n; ++i )
        CPPUNIT_ASSERT_DOUBLES_EQUAL( weights[i], working[i], 1e-6 );
}

void AveragingQMTest::test_average_and_weights_linear()
{
    std::vector< double > weights1( LEN_LIST_1, 1.0 / LEN_LIST_1 ), weights2( LEN_LIST_2, 1.0 / LEN_LIST_2 );
    check_average_and_weights( VAL_LIST_1, LEN_LIST_1, QualityMetric::LINEAR, arrptr( weights1 ) );
    check_average_and_weights( VAL_LIST_2, LEN_LIST_2, QualityMetric::LINEAR, arrptr( weights2 ) );
}

void AveragingQMTest::test_average_and_weights_rms()
{
    std::vector< double > weights1( LEN_LIST_1 ), weights2( LEN_LIST_2 );
    unsigned i;
    double rms;

    rms = pmean( VAL_LIST_1, LEN_LIST_1, 2.0 );
    for( i = 0; i < LEN_LIST_1; ++i )
        weights1[i] = VAL_LIST_1[i] / ( LEN_LIST_1 * rms );
    check_average_and_weights( VAL_LIST_1, LEN_LIST_1, QualityMetric::RMS, arrptr( weights1 ) );

    rms = pmean( VAL_LIST_2, LEN_LIST_2, 2.0 );
    for( i = 0; i < LEN_LIST_2; ++i )
        weights2[i] = VAL_LIST_2[i] / ( LEN_LIST_2 * rms );
    check_average_and_weights( VAL_LIST_2, LEN_LIST_2, QualityMetric::RMS, arrptr( weights2 ) );
}

void AveragingQMTest::test_average_and_weights_hms()
{
    std::vector< double > weights1( LEN_LIST_1 ), weights2( LEN_LIST_2 );
    unsigned i;
    double hms, tmp;

    hms = pmean( VAL_LIST_1, LEN_LIST_1, -2.0 );
    for( i = 0; i < LEN_LIST_1; ++i )
    {
        tmp         = hms / VAL_LIST_1[i];
        weights1[i] = tmp * tmp * tmp / LEN_LIST_1;
    }
    check_average_and_weights( VAL_LIST_1, LEN_LIST_1, QualityMetric::HMS, arrptr( weights1 ) );

    hms = pmean( VAL_LIST_2, LEN_LIST_2, -2.0 );
    for( i = 0; i < LEN_LIST_2; ++i )
    {
        tmp         = hms / VAL_LIST_2[i];
        weights2[i] = tmp * tmp * tmp / LEN_LIST_2;
    }
    check_average_and_weights( VAL_LIST_2, LEN_LIST_2, QualityMetric::HMS, arrptr( weights2 ) );
}

void AveragingQMTest::test_average_and_weights_minimum()
{
    std::vector< double > weights1( LEN_LIST_1, 0.0 ), weights2( LEN_LIST_2, 0.0 );
    const double* ptr;

    ptr                        = std::min_element( VAL_LIST_1, VAL_LIST_1 + LEN_LIST_1 );
    weights1[ptr - VAL_LIST_1] = 1.0;
    check_average_and_weights( VAL_LIST_1, LEN_LIST_1, QualityMetric::MINIMUM, arrptr( weights1 ) );

    ptr                        = std::min_element( VAL_LIST_2, VAL_LIST_2 + LEN_LIST_2 );
    weights2[ptr - VAL_LIST_2] = 1.0;
    check_average_and_weights( VAL_LIST_2, LEN_LIST_2, QualityMetric::MINIMUM, arrptr( weights2 ) );
}

void AveragingQMTest::test_average_and_weights_maximum()
{
    std::vector< double > weights1( LEN_LIST_1, 0.0 ), weights2( LEN_LIST_2, 0.0 );
    const double* ptr;

    ptr                        = std::max_element( VAL_LIST_1, VAL_LIST_1 + LEN_LIST_1 );
    weights1[ptr - VAL_LIST_1] = 1.0;
    check_average_and_weights( VAL_LIST_1, LEN_LIST_1, QualityMetric::MAXIMUM, arrptr( weights1 ) );

    ptr                        = std::max_element( VAL_LIST_2, VAL_LIST_2 + LEN_LIST_2 );
    weights2[ptr - VAL_LIST_2] = 1.0;
    check_average_and_weights( VAL_LIST_2, LEN_LIST_2, QualityMetric::MAXIMUM, arrptr( weights2 ) );
}

void AveragingQMTest::test_average_and_weights_harmonic()
{
    std::vector< double > weights1( LEN_LIST_1 ), weights2( LEN_LIST_2 );
    unsigned i;
    double h, tmp;

    h = pmean( VAL_LIST_1, LEN_LIST_1, -1.0 );
    for( i = 0; i < LEN_LIST_1; ++i )
    {
        tmp         = h / VAL_LIST_1[i];
        weights1[i] = tmp * tmp / LEN_LIST_1;
    }
    check_average_and_weights( VAL_LIST_1, LEN_LIST_1, QualityMetric::HARMONIC, arrptr( weights1 ) );

    h = pmean( VAL_LIST_2, LEN_LIST_2, -1.0 );
    for( i = 0; i < LEN_LIST_2; ++i )
    {
        tmp         = h / VAL_LIST_2[i];
        weights2[i] = tmp * tmp / LEN_LIST_2;
    }
    check_average_and_weights( VAL_LIST_2, LEN_LIST_2, QualityMetric::HARMONIC, arrptr( weights2 ) );
}

void AveragingQMTest::test_average_and_weights_geometric()
{
    std::vector< double > weights1( LEN_LIST_1 ), weights2( LEN_LIST_2 );
    unsigned i;
    double g;

    g = geometric_mean( VAL_LIST_1, LEN_LIST_1 );
    for( i = 0; i < LEN_LIST_1; ++i )
    {
        weights1[i] = g / VAL_LIST_1[i] / LEN_LIST_1;
    }
    check_average_and_weights( VAL_LIST_1, LEN_LIST_1, QualityMetric::GEOMETRIC, arrptr( weights1 ) );

    g = geometric_mean( VAL_LIST_2, LEN_LIST_2 );
    for( i = 0; i < LEN_LIST_2; ++i )
    {
        weights2[i] = g / VAL_LIST_2[i] / LEN_LIST_2;
    }
    check_average_and_weights( VAL_LIST_2, LEN_LIST_2, QualityMetric::GEOMETRIC, arrptr( weights2 ) );
}

void AveragingQMTest::test_average_and_weights_sum()
{
    std::vector< double > weights1( LEN_LIST_1, 1.0 ), weights2( LEN_LIST_2, 1.0 );
    check_average_and_weights( VAL_LIST_1, LEN_LIST_1, QualityMetric::SUM, arrptr( weights1 ) );
    check_average_and_weights( VAL_LIST_2, LEN_LIST_2, QualityMetric::SUM, arrptr( weights2 ) );
}

void AveragingQMTest::test_average_and_weights_sum_squared()
{
    std::vector< double > weights1( LEN_LIST_1 ), weights2( LEN_LIST_2 );
    unsigned i;

    for( i = 0; i < LEN_LIST_1; ++i )
        weights1[i] = 2 * VAL_LIST_1[i];
    check_average_and_weights( VAL_LIST_1, LEN_LIST_1, QualityMetric::SUM_SQUARED, arrptr( weights1 ) );

    for( i = 0; i < LEN_LIST_2; ++i )
        weights2[i] = 2 * VAL_LIST_2[i];
    check_average_and_weights( VAL_LIST_2, LEN_LIST_2, QualityMetric::SUM_SQUARED, arrptr( weights2 ) );
}

void AveragingQMTest::check_average_and_weights_fails( QualityMetric::AveragingMethod scheme )
{
    MsqError err;
    AveragingQM aqm( scheme );
    double w[] = { 1.0, 2.0 };
    aqm.average_metric_and_weights( w, 2, err );
    CPPUNIT_ASSERT( err );
}

void AveragingQMTest::test_average_and_weights_standard_deviation()
{
    check_average_and_weights_fails( QualityMetric::STANDARD_DEVIATION );
}

void AveragingQMTest::test_average_and_weights_max_over_min()
{
    check_average_and_weights_fails( QualityMetric::MAX_OVER_MIN );
}

void AveragingQMTest::test_average_and_weights_max_minus_min()
{
    check_average_and_weights_fails( QualityMetric::MAX_MINUS_MIN );
}

void AveragingQMTest::test_average_and_weights_sum_of_ratios_squared()
{
    check_average_and_weights_fails( QualityMetric::SUM_OF_RATIOS_SQUARED );
}

void AveragingQMTest::check_average_gradients( QualityMetric::AveragingMethod scheme )
{
    double vals1[] = { 1, -2, -1, 0.5 };
    double vals2[] = { 1, -2, -1, 0.5 };
    Vector3D grads[12], vtxgrads[4];
    for( unsigned i = 0; i < 12; ++i )
        grads[i] = Vector3D( (double)i );

    uint32_t fixed = 0x2;  // mark second vertex as fixed
    Vector3D exp;
    MsqError err;
    AveragingQM aqm( scheme );

    double a1 = aqm.average_metric_and_weights( vals1, 4, err );
    ASSERT_NO_ERROR( err );

    double a2 = aqm.average_corner_gradients( QUADRILATERAL, fixed, 4, vals2, grads, vtxgrads, err );
    ASSERT_NO_ERROR( err );

    CPPUNIT_ASSERT_DOUBLES_EQUAL( a1, a2, 1e-8 );

    exp = vals1[0] * grads[0] + vals1[1] * grads[5] + vals1[3] * grads[10];
    CPPUNIT_ASSERT_VECTORS_EQUAL( exp, vtxgrads[0], 1e-6 );

    exp = vals1[1] * grads[4] + vals1[2] * grads[6] + vals1[3] * grads[11];
    CPPUNIT_ASSERT_VECTORS_EQUAL( exp, vtxgrads[2], 1e-6 );

    exp = vals1[0] * grads[2] + vals1[2] * grads[7] + vals1[3] * grads[9];
    CPPUNIT_ASSERT_VECTORS_EQUAL( exp, vtxgrads[3], 1e-6 );
}

void AveragingQMTest::test_average_corner_gradients_linear()
{
    check_average_gradients( QualityMetric::LINEAR );
}

void AveragingQMTest::test_average_corner_gradients_rms()
{
    check_average_gradients( QualityMetric::RMS );
}

void AveragingQMTest::test_average_corner_gradients_hms()
{
    check_average_gradients( QualityMetric::HMS );
}

void AveragingQMTest::test_average_corner_gradients_minimum()
{
    check_average_gradients( QualityMetric::MINIMUM );
}

void AveragingQMTest::test_average_corner_gradients_maximum()
{
    check_average_gradients( QualityMetric::MAXIMUM );
}

void AveragingQMTest::test_average_corner_gradients_harmonic()
{
    check_average_gradients( QualityMetric::HARMONIC );
}

void AveragingQMTest::test_average_corner_gradients_geometric()
{
    check_average_gradients( QualityMetric::GEOMETRIC );
}

void AveragingQMTest::test_average_corner_gradients_sum()
{
    check_average_gradients( QualityMetric::SUM );
}

void AveragingQMTest::test_average_corner_gradients_sum_squared()
{
    check_average_gradients( QualityMetric::SUM_SQUARED );
}

void AveragingQMTest::check_average_gradient_fails( QualityMetric::AveragingMethod scheme )
{
    double vals[] = { 1, 2, -1, 0.5 };
    Vector3D grads[12], vtxgrads[4];
    for( unsigned i = 0; i < 12; ++i )
        grads[i] = Vector3D( (double)i );

    MsqError err;
    AveragingQM aqm( scheme );

    aqm.average_corner_gradients( QUADRILATERAL, 0, 4, vals, grads, vtxgrads, err );
    CPPUNIT_ASSERT( err );
}

void AveragingQMTest::test_average_corner_gradients_standard_deviation()
{
    check_average_gradient_fails( QualityMetric::STANDARD_DEVIATION );
}

void AveragingQMTest::test_average_corner_gradients_max_over_min()
{
    check_average_gradient_fails( QualityMetric::MAX_OVER_MIN );
}

void AveragingQMTest::test_average_corner_gradients_max_minus_min()
{
    check_average_gradient_fails( QualityMetric::MAX_MINUS_MIN );
}

void AveragingQMTest::test_average_corner_gradients_sum_of_ratios_squared()
{
    check_average_gradient_fails( QualityMetric::SUM_OF_RATIOS_SQUARED );
}

static double pmean_of_triangle_corner_hessians( double inner_power,
                                                 double outer_power,
                                                 const double* v,
                                                 const Vector3D* cg,
                                                 const Matrix3D* ch,
                                                 Vector3D* tg,
                                                 Matrix3D* th,
                                                 bool scale )
{
    Matrix3D op;
    double gf[3], hf[3];
    double nm, m = 0;
    double den = scale ? 3.0 : 1.0;
    for( unsigned i = 0; i < 3; ++i )
    {
        nm = pow( v[i], inner_power );
        m += nm;

        gf[i] = inner_power * nm / v[i] / den;
        hf[i] = ( inner_power - 1 ) * gf[i] / v[i];
    }

    nm = m / den;

    tg[0] = gf[0] * cg[0] + gf[1] * cg[5] + gf[2] * cg[7];
    tg[1] = gf[0] * cg[1] + gf[1] * cg[3] + gf[2] * cg[8];
    tg[2] = gf[0] * cg[2] + gf[1] * cg[4] + gf[2] * cg[6];

    th[0] = hf[0] * op.outer_product( cg[0], cg[0] ) + gf[0] * ch[0] + hf[1] * op.outer_product( cg[5], cg[5] ) +
            gf[1] * ch[11] + hf[2] * op.outer_product( cg[7], cg[7] ) + gf[2] * ch[15];
    th[3] = hf[0] * op.outer_product( cg[1], cg[1] ) + gf[0] * ch[3] + hf[1] * op.outer_product( cg[3], cg[3] ) +
            gf[1] * ch[6] + hf[2] * op.outer_product( cg[8], cg[8] ) + gf[2] * ch[17];
    th[5] = hf[0] * op.outer_product( cg[2], cg[2] ) + gf[0] * ch[5] + hf[1] * op.outer_product( cg[4], cg[4] ) +
            gf[1] * ch[9] + hf[2] * op.outer_product( cg[6], cg[6] ) + gf[2] * ch[12];
    th[1] = hf[0] * op.outer_product( cg[0], cg[1] ) + gf[0] * ch[1] + hf[1] * op.outer_product( cg[5], cg[3] ) +
            gf[1] * transpose( ch[8] ) + hf[2] * op.outer_product( cg[7], cg[8] ) + gf[2] * ch[16];
    th[2] = hf[0] * op.outer_product( cg[0], cg[2] ) + gf[0] * ch[2] + hf[1] * op.outer_product( cg[5], cg[4] ) +
            gf[1] * transpose( ch[10] ) + hf[2] * op.outer_product( cg[7], cg[6] ) + gf[2] * transpose( ch[13] );
    th[4] = hf[0] * op.outer_product( cg[1], cg[2] ) + gf[0] * ch[4] + hf[1] * op.outer_product( cg[3], cg[4] ) +
            gf[1] * ch[7] + hf[2] * op.outer_product( cg[8], cg[6] ) + gf[2] * transpose( ch[14] );

    m        = pow( nm, outer_power );
    double g = m * outer_power / nm;
    double h = ( outer_power - 1.0 ) * g / nm;
    for( unsigned r = 0; r < 3; ++r )
    {
        for( unsigned c = r; c < 3; ++c )
        {
            *th = g * *th + h * op.outer_product( tg[r], tg[c] );
            ++th;
        }
        tg[r] *= g;
    }

    return m;
}

void AveragingQMTest::check_pmean_hessian_diagonals( QualityMetric::AveragingMethod scheme,
                                                     double inner_power,
                                                     double outer_power,
                                                     bool scale )
{
    MsqPrintError err( std::cout );
    AveragingQM aqm( scheme );
    uint32_t fixed = 0;  // mark no vertices as fixed

    // define corner values, gradients and Hessians for a triangle
    double vals[]           = { 1, 2, 0.5 };
    const Vector3D grads[9] = { Vector3D( 0, 1, 1 ), Vector3D( 0, 2, 2 ), Vector3D( 0, 3, 3 ),
                                Vector3D( 0, 4, 4 ), Vector3D( 0, 5, 5 ), Vector3D( 0, 6, 6 ),
                                Vector3D( 0, 7, 7 ), Vector3D( 0, 8, 8 ), Vector3D( 0, 9, 9 ) };
    Matrix3D hesss[18]      = { Matrix3D( 1.0 ), Matrix3D( 2.0 ), Matrix3D( -1. ), Matrix3D( 0.5 ),  Matrix3D( 6.0 ),
                           Matrix3D( 1.1 ), Matrix3D( 1.2 ), Matrix3D( 0.3 ), Matrix3D( 0.0 ),  Matrix3D( 22. ),
                           Matrix3D( -2. ), Matrix3D( 4.3 ), Matrix3D( 0.8 ), Matrix3D( 1, 7 ), Matrix3D( -.2 ),
                           Matrix3D( 0.5 ), Matrix3D( 7.0 ), Matrix3D( 1.5 ) };
    // change some values so we catch transpositional errors
    for( unsigned i = 0; i < sizeof( hesss ) / sizeof( hesss[0] ); ++i )
        hesss[i][2][2] = 0.0;

    // Calculate expected values
    Vector3D aqm_grad[3], exp_grad[3];
    Matrix3D exp_hess[6];
    double aqm_avg, exp_avg;
    exp_avg =
        pmean_of_triangle_corner_hessians( inner_power, outer_power, vals, grads, hesss, exp_grad, exp_hess, scale );

    // Test calculation of diagonal from full corner hessian data.
    SymMatrix3D aqm_hess[3] = { 1, 2, 3 };
    aqm_avg = aqm.average_corner_hessian_diagonals( TRIANGLE, fixed, 3, vals, grads, hesss, aqm_grad, aqm_hess, err );
    ASSERT_NO_ERROR( err );

    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp_avg, aqm_avg, 1e-6 );

    CPPUNIT_ASSERT_VECTORS_EQUAL( exp_grad[0], aqm_grad[0], 1e-6 );
    CPPUNIT_ASSERT_VECTORS_EQUAL( exp_grad[1], aqm_grad[1], 1e-6 );
    CPPUNIT_ASSERT_VECTORS_EQUAL( exp_grad[2], aqm_grad[2], 1e-6 );

    CPPUNIT_ASSERT_MATRICES_EQUAL( exp_hess[0], Matrix3D( aqm_hess[0] ), 1e-6 );
    CPPUNIT_ASSERT_MATRICES_EQUAL( exp_hess[3], Matrix3D( aqm_hess[1] ), 1e-6 );
    CPPUNIT_ASSERT_MATRICES_EQUAL( exp_hess[5], Matrix3D( aqm_hess[2] ), 1e-6 );

    // Test calculation of diagonal from diagonal corner hessian data.
    SymMatrix3D diags[9] = { hesss[0].upper(),  hesss[3].upper(),  hesss[5].upper(),
                             hesss[6].upper(),  hesss[9].upper(),  hesss[11].upper(),
                             hesss[12].upper(), hesss[15].upper(), hesss[17].upper() };
    aqm_avg = aqm.average_corner_hessian_diagonals( TRIANGLE, fixed, 3, vals, grads, diags, aqm_grad, aqm_hess, err );
    ASSERT_NO_ERROR( err );

    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp_avg, aqm_avg, 1e-6 );

    CPPUNIT_ASSERT_VECTORS_EQUAL( exp_grad[0], aqm_grad[0], 1e-6 );
    CPPUNIT_ASSERT_VECTORS_EQUAL( exp_grad[1], aqm_grad[1], 1e-6 );
    CPPUNIT_ASSERT_VECTORS_EQUAL( exp_grad[2], aqm_grad[2], 1e-6 );

    CPPUNIT_ASSERT_MATRICES_EQUAL( exp_hess[0], Matrix3D( aqm_hess[0] ), 1e-6 );
    CPPUNIT_ASSERT_MATRICES_EQUAL( exp_hess[3], Matrix3D( aqm_hess[1] ), 1e-6 );
    CPPUNIT_ASSERT_MATRICES_EQUAL( exp_hess[5], Matrix3D( aqm_hess[2] ), 1e-6 );
}

void AveragingQMTest::check_hessian_diagonal_fails( QualityMetric::AveragingMethod scheme )
{
    // define corner values, gradients and Hessians for a triangle
    double vals[]            = { 1, 2, 0.5 };
    const Vector3D grads[9]  = { Vector3D( 1.0 ), Vector3D( 2.0 ), Vector3D( 3.0 ), Vector3D( 4.0 ), Vector3D( 5.0 ),
                                Vector3D( 6.0 ), Vector3D( 7.0 ), Vector3D( 8.0 ), Vector3D( 9.0 ) };
    const Matrix3D hesss[18] = { Matrix3D( 1.0 ), Matrix3D( 2.0 ), Matrix3D( -1. ), Matrix3D( 0.5 ),  Matrix3D( 6.0 ),
                                 Matrix3D( 1.1 ), Matrix3D( 1.2 ), Matrix3D( 0.3 ), Matrix3D( 0.0 ),  Matrix3D( 22. ),
                                 Matrix3D( -2. ), Matrix3D( 4.3 ), Matrix3D( 0.8 ), Matrix3D( 1, 7 ), Matrix3D( -.2 ),
                                 Matrix3D( 0.5 ), Matrix3D( 7.0 ), Matrix3D( 1.5 ) };

    Vector3D vtx_grad[3];
    SymMatrix3D vtx_hess[6];
    MsqError err;
    AveragingQM aqm( scheme );
    uint32_t fixed = 0x4;  // mark third vertex as fixed

    aqm.average_corner_hessian_diagonals( TRIANGLE, fixed, 3, vals, grads, hesss, vtx_grad, vtx_hess, err );
    CPPUNIT_ASSERT( err );

    const SymMatrix3D diags[9] = { SymMatrix3D( 1.0 ), SymMatrix3D( 0.5 ), SymMatrix3D( 1.0 ),
                                   SymMatrix3D( 1.2 ), SymMatrix3D( 22. ), SymMatrix3D( 4.3 ),
                                   SymMatrix3D( 0.8 ), SymMatrix3D( 0.5 ), SymMatrix3D( 1.5 ) };

    aqm.average_corner_hessian_diagonals( TRIANGLE, fixed, 3, vals, grads, diags, vtx_grad, vtx_hess, err );
    CPPUNIT_ASSERT( err );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_linear()
{
    check_pmean_hessian_diagonals( QualityMetric::LINEAR, 1, 1, true );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_rms()
{
    check_pmean_hessian_diagonals( QualityMetric::RMS, 2, 0.5, true );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_hms()
{
    check_pmean_hessian_diagonals( QualityMetric::HMS, -2, -0.5, true );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_minimum()
{
    check_hessian_diagonal_fails( QualityMetric::MINIMUM );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_maximum()
{
    check_hessian_diagonal_fails( QualityMetric::MAXIMUM );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_harmonic()
{
    check_pmean_hessian_diagonals( QualityMetric::HARMONIC, -1, -1, true );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_geometric()
{
    check_hessian_diagonal_fails( QualityMetric::GEOMETRIC );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_sum()
{
    check_pmean_hessian_diagonals( QualityMetric::SUM, 1, 1, false );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_sum_squared()
{
    check_pmean_hessian_diagonals( QualityMetric::SUM_SQUARED, 2, 1, false );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_standard_deviation()
{
    check_hessian_diagonal_fails( QualityMetric::STANDARD_DEVIATION );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_max_over_min()
{
    check_hessian_diagonal_fails( QualityMetric::MAX_OVER_MIN );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_max_minus_min()
{
    check_hessian_diagonal_fails( QualityMetric::MAX_MINUS_MIN );
}

void AveragingQMTest::test_average_corner_hessian_diagonals_sum_of_ratios_squared()
{
    check_hessian_diagonal_fails( QualityMetric::SUM_OF_RATIOS_SQUARED );
}

void AveragingQMTest::check_pmean_hessian( QualityMetric::AveragingMethod scheme,
                                           double inner_power,
                                           double outer_power,
                                           bool scale )
{
    // define corner values, gradients and Hessians for a triangle
    double vals[]           = { 1, 2, 0.5 };
    const Vector3D grads[9] = { Vector3D( 0, 1, 1 ), Vector3D( 0, 2, 2 ), Vector3D( 0, 3, 3 ),
                                Vector3D( 0, 4, 4 ), Vector3D( 0, 5, 5 ), Vector3D( 0, 6, 6 ),
                                Vector3D( 0, 7, 7 ), Vector3D( 0, 8, 8 ), Vector3D( 0, 9, 9 ) };
    Matrix3D hesss[18]      = { Matrix3D( 1.0 ), Matrix3D( 2.0 ), Matrix3D( -1. ), Matrix3D( 0.5 ),  Matrix3D( 6.0 ),
                           Matrix3D( 1.1 ), Matrix3D( 1.2 ), Matrix3D( 0.3 ), Matrix3D( 0.0 ),  Matrix3D( 22. ),
                           Matrix3D( -2. ), Matrix3D( 4.3 ), Matrix3D( 0.8 ), Matrix3D( 1, 7 ), Matrix3D( -.2 ),
                           Matrix3D( 0.5 ), Matrix3D( 7.0 ), Matrix3D( 1.5 ) };
    // change some values so we catch transpositional errors
    for( unsigned i = 0; i < sizeof( hesss ) / sizeof( hesss[0] ); ++i )
        hesss[i][2][2] = 0.0;

    Vector3D aqm_grad[3], exp_grad[3];
    Matrix3D aqm_hess[6], exp_hess[6];
    ;
    MsqPrintError err( std::cout );
    AveragingQM aqm( scheme );
    uint32_t fixed = 0x4;  // mark third vertex as fixed
    double aqm_avg, exp_avg;

    exp_avg =
        pmean_of_triangle_corner_hessians( inner_power, outer_power, vals, grads, hesss, exp_grad, exp_hess, scale );

    aqm_avg = aqm.average_corner_hessians( TRIANGLE, fixed, 3, vals, grads, hesss, aqm_grad, aqm_hess, err );
    ASSERT_NO_ERROR( err );

    CPPUNIT_ASSERT_DOUBLES_EQUAL( exp_avg, aqm_avg, 1e-6 );

    // only check gradient terms for free vertices
    CPPUNIT_ASSERT_VECTORS_EQUAL( exp_grad[0], aqm_grad[0], 1e-6 );
    CPPUNIT_ASSERT_VECTORS_EQUAL( exp_grad[1], aqm_grad[1], 1e-6 );

    // only check Hessian terms for free vertices
    CPPUNIT_ASSERT_MATRICES_EQUAL( exp_hess[0], aqm_hess[0], 1e-6 );
    CPPUNIT_ASSERT_MATRICES_EQUAL( exp_hess[1], aqm_hess[1], 1e-6 );
    CPPUNIT_ASSERT_MATRICES_EQUAL( exp_hess[3], aqm_hess[3], 1e-6 );
}

void AveragingQMTest::check_hessian_fails( QualityMetric::AveragingMethod scheme )
{
    // define corner values, gradients and Hessians for a triangle
    double vals[]            = { 1, 2, 0.5 };
    const Vector3D grads[9]  = { Vector3D( 1.0 ), Vector3D( 2.0 ), Vector3D( 3.0 ), Vector3D( 4.0 ), Vector3D( 5.0 ),
                                Vector3D( 6.0 ), Vector3D( 7.0 ), Vector3D( 8.0 ), Vector3D( 9.0 ) };
    const Matrix3D hesss[18] = { Matrix3D( 1.0 ), Matrix3D( 2.0 ), Matrix3D( -1. ), Matrix3D( 0.5 ),  Matrix3D( 6.0 ),
                                 Matrix3D( 1.1 ), Matrix3D( 1.2 ), Matrix3D( 0.3 ), Matrix3D( 0.0 ),  Matrix3D( 22. ),
                                 Matrix3D( -2. ), Matrix3D( 4.3 ), Matrix3D( 0.8 ), Matrix3D( 1, 7 ), Matrix3D( -.2 ),
                                 Matrix3D( 0.5 ), Matrix3D( 7.0 ), Matrix3D( 1.5 ) };

    Vector3D vtx_grad[3];
    Matrix3D vtx_hess[6];
    MsqError err;
    AveragingQM aqm( scheme );
    uint32_t fixed = 0x4;  // mark third vertex as fixed

    aqm.average_corner_hessians( TRIANGLE, fixed, 3, vals, grads, hesss, vtx_grad, vtx_hess, err );
    CPPUNIT_ASSERT( err );
}

void AveragingQMTest::test_average_corner_hessians_linear()
{
    check_pmean_hessian( QualityMetric::LINEAR, 1, 1, true );
}

void AveragingQMTest::test_average_corner_hessians_rms()
{
    check_pmean_hessian( QualityMetric::RMS, 2, 0.5, true );
}

void AveragingQMTest::test_average_corner_hessians_hms()
{
    check_pmean_hessian( QualityMetric::HMS, -2, -0.5, true );
}

void AveragingQMTest::test_average_corner_hessians_minimum()
{
    check_hessian_fails( QualityMetric::MINIMUM );
}

void AveragingQMTest::test_average_corner_hessians_maximum()
{
    check_hessian_fails( QualityMetric::MAXIMUM );
}

void AveragingQMTest::test_average_corner_hessians_harmonic()
{
    check_pmean_hessian( QualityMetric::HARMONIC, -1, -1, true );
}

void AveragingQMTest::test_average_corner_hessians_geometric()
{
    check_hessian_fails( QualityMetric::GEOMETRIC );
}

void AveragingQMTest::test_average_corner_hessians_sum()
{
    check_pmean_hessian( QualityMetric::SUM, 1, 1, false );
}

void AveragingQMTest::test_average_corner_hessians_sum_squared()
{
    check_pmean_hessian( QualityMetric::SUM_SQUARED, 2, 1, false );
}

void AveragingQMTest::test_average_corner_hessians_standard_deviation()
{
    check_hessian_fails( QualityMetric::STANDARD_DEVIATION );
}

void AveragingQMTest::test_average_corner_hessians_max_over_min()
{
    check_hessian_fails( QualityMetric::MAX_OVER_MIN );
}

void AveragingQMTest::test_average_corner_hessians_max_minus_min()
{
    check_hessian_fails( QualityMetric::MAX_MINUS_MIN );
}

void AveragingQMTest::test_average_corner_hessians_sum_of_ratios_squared()
{
    check_hessian_fails( QualityMetric::SUM_OF_RATIOS_SQUARED );
}