MeanRatioFunctions.hpp

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

    Copyright 2004 Sandia Corporation and Argonne National
    Laboratory.  Under the terms of Contract DE-AC04-94AL85000
    with Sandia Corporation, the U.S. Government retains certain
    rights in 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
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    You should have received a copy of the GNU Lesser General Public License
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    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA

    [email protected], [email protected], [email protected],
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  ***************************************************************** */
// -*- Mode : c++; tab-width: 3; c-tab-always-indent: t; indent-tabs-mode: nil; c-basic-offset: 3
// -*-

/*! \file MeanRatioFunctions.hpp

Header that defines generalized Mean Ratio function, gradient, and hessian
evaluations.

\author Todd Munson
\date   2003-06-11
 */

#ifndef MeanRatioFunctions_hpp
#define MeanRatioFunctions_hpp

#include <cmath>
#include "Mesquite.hpp"
#include "Vector3D.hpp"
#include "Matrix3D.hpp"
#include "Exponent.hpp"

namespace MBMesquite
{
/***************************************************************************/
/* Gradient calculation courtesy of Paul Hovland.  The code was modified   */
/* to reduce the number of flops and intermediate variables, and improve   */
/* the locality of reference.                                              */
/***************************************************************************/
/* The Hessian calculation is computes everyting in (x,y,z) blocks and     */
/* stores only the upper triangular blocks.  The results are put into      */
/* (c1,c2,c3,c4) order prior to returning the Hessian matrix.              */
/***************************************************************************/
/* The form of the function, gradient, and Hessian follows:                */
/*   o(x) = a * pow(f(A(x)), b) * pow(g(A(x)), c)                          */
/* where A(x) is the incidence matrix generated by:                        */
/*           [x1-x0 x2-x0 x3-x0]                                           */
/*    A(x) = [y1-y0 y2-y0 y3-y0] * inv(W)                                  */
/*           [z1-z0 z2-z0 z3-z0]                                           */
/* and f() is the squared Frobenius norm of A(x), and g() is the           */
/* determinant of A(x).                                                    */
/*                                                                         */
/* The gradient is calculated as follows:                                  */
/*   alpha := a*b*pow(f(A(x)),b-1)*pow(g(A(x)),c)                          */
/*   beta  := a*c*pow(f(A(x)),b)*pow(g(A(x)),c-1)                          */
/*                                                                         */
/*   do/dx = (alpha * (df/dA) + beta * (dg/dA)) (dA/dx)                    */
/*                                                                         */
/*   (df/dA)_i = 2*A_i                                                     */
/*   (dg/dA)_i = A_j*A_k - A_l*A_m for some {j,k,l,m}                      */
/*                                                                         */
/*   d^2o/dx^2 = (dA/dx)' * ((d alpha/dA) * (df/dA) +                      */
/*                           (d  beta/dA) * (dg/dA)                        */
/*                                  alpha * (d^2f/dA^2)                    */
/*                                   beta * (d^2g/dA^2)) * (dA/dx)         */
/*                                                                         */
/*   Note: since A(x) is a linear function, there are no terms involving   */
/*   d^2A/dx^2 since this matrix is zero.                                  */
/*                                                                         */
/*   gamma := a*b*c*pow(f(A(x)),b-1)*pow(g(A(x)),c-1)                      */
/*   delta := a*c*(c-1)*pow(f(A(x)),b)*pow(g(A(x)),c-2)                    */
/*   psi   := a*b*(b-1)*pow(f(A(x)),b-2)*pow(g(A(x)),c)                    */
/*                                                                         */
/*   d^2o/dx^2 = (dA/dx)' * (gamma*((dg/dA)'*(df/dA) + (df/dA)'*(dg/dA)) + */
/*                           delta* (dg/dA)'*(dg/dA) +                     */
/*                             psi* (df/dA)'*(df/dA) +                     */
/*                           alpha*(d^2f/dA^2) +                           */
/*                            beta*(d^2g/dA^2)) * (dA/dx)                  */
/*                                                                         */
/*   Note: (df/dA) and (dg/dA) are row vectors and we only calculate the   */
/*   upper triangular part of the inner matrices.                          */
/*                                                                         */
/*   For regular tetrahedral elements, we have the following:              */
/*                                                                         */
/*           [-1         1        0         0         ]                    */
/*       M = [-sqrt(3)  -sqrt(3)  2*sqrt(3) 0         ]                    */
/*           [-sqrt(6)  -sqrt(6)  -sqrt(6)  3*sqrt(6) ]                    */
/*                                                                         */
/*           [M 0 0]                                                       */
/*   dA/dx = [0 M 0]                                                       */
/*           [0 0 M]                                                       */
/***************************************************************************/

/*****************************************************************************/
/* Not all compilers substitute out constants (especially the square root).  */
/* Therefore, they are substituted out manually.  The values below were      */
/* calculated on a solaris machine using long doubles. I believe they are    */
/* accurate.                                                                 */
/*****************************************************************************/

#define isqrt3  5.77350269189625797959429519858e-01 /*  1.0/sqrt(3.0)*/
#define tisqrt3 1.15470053837925159591885903972e+00 /*  2.0/sqrt(3.0)*/
#define isqrt6  4.08248290463863052509822647505e-01 /*  1.0/sqrt(6.0)*/
#define tisqrt6 1.22474487139158915752946794252e+00 /*  3.0/sqrt(6.0)*/

/*****************************************************************************/
/* The following set of functions reference triangular elements to an        */
/* equilateral triangle in the plane defined by the normal.  They are        */
/* used when assessing the quality of a triangular element.  A zero          */
/* return value indicates success, while a nonzero value indicates failure.  */
/*****************************************************************************/

/*****************************************************************************/
/* Function evaluation requires 44 flops.                                    */
/*   Reductions possible when b == 1 or c == 1                               */
/*****************************************************************************/
inline bool m_fcn_2e( double& obj,
                      const Vector3D x[3],
                      const Vector3D& n,
                      const double a,
                      const Exponent& b,
                      const Exponent& c )
{
    double matr[9], f;
    double g;

    /* Calculate M = [A*inv(W) n] */
    matr[0] = x[1][0] - x[0][0];
    matr[1] = ( 2.0 * x[2][0] - x[1][0] - x[0][0] ) * isqrt3;
    matr[2] = n[0];

    matr[3] = x[1][1] - x[0][1];
    matr[4] = ( 2.0 * x[2][1] - x[1][1] - x[0][1] ) * isqrt3;
    matr[5] = n[1];

    matr[6] = x[1][2] - x[0][2];
    matr[7] = ( 2.0 * x[2][2] - x[1][2] - x[0][2] ) * isqrt3;
    matr[8] = n[2];

    /* Calculate det(M). */
    g = matr[0] * ( matr[4] * matr[8] - matr[5] * matr[7] ) + matr[3] * ( matr[2] * matr[7] - matr[1] * matr[8] ) +
        matr[6] * ( matr[1] * matr[5] - matr[2] * matr[4] );
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[3] * matr[3] + matr[4] * matr[4] + matr[6] * matr[6] +
        matr[7] * matr[7];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );
    return true;
}

/*****************************************************************************/
/* Gradient evaluation requires 88 flops.                                    */
/*   Reductions possible when b == 1 or c == 1                               */
/*****************************************************************************/
inline bool g_fcn_2e( double& obj,
                      Vector3D g_obj[3],
                      const Vector3D x[3],
                      const Vector3D& n,
                      const double a,
                      const Exponent& b,
                      const Exponent& c )
{
    double matr[9], f;
    double adj_m[9], g;  // adj_m[2,5,8] not used
    double loc1, loc2, loc3;

    /* Calculate M = [A*inv(W) n] */
    matr[0] = x[1][0] - x[0][0];
    matr[1] = ( 2.0 * x[2][0] - x[1][0] - x[0][0] ) * isqrt3;
    matr[2] = n[0];

    matr[3] = x[1][1] - x[0][1];
    matr[4] = ( 2.0 * x[2][1] - x[1][1] - x[0][1] ) * isqrt3;
    matr[5] = n[1];

    matr[6] = x[1][2] - x[0][2];
    matr[7] = ( 2.0 * x[2][2] - x[1][2] - x[0][2] ) * isqrt3;
    matr[8] = n[2];

    /* Calculate det([n M]). */
    loc1 = matr[4] * matr[8] - matr[5] * matr[7];
    loc2 = matr[2] * matr[7] - matr[1] * matr[8];
    loc3 = matr[1] * matr[5] - matr[2] * matr[4];
    g    = matr[0] * loc1 + matr[3] * loc2 + matr[6] * loc3;
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[3] * matr[3] + matr[4] * matr[4] + matr[6] * matr[6] +
        matr[7] * matr[7];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    adj_m[0] = matr[0] * f + loc1 * g;
    adj_m[3] = matr[3] * f + loc2 * g;
    adj_m[6] = matr[6] * f + loc3 * g;

    loc1 = matr[0] * g;
    loc2 = matr[3] * g;
    loc3 = matr[6] * g;

    adj_m[1] = matr[1] * f + loc3 * matr[5] - loc2 * matr[8];
    adj_m[4] = matr[4] * f + loc1 * matr[8] - loc3 * matr[2];
    adj_m[7] = matr[7] * f + loc2 * matr[2] - loc1 * matr[5];

    loc1        = isqrt3 * adj_m[1];
    g_obj[0][0] = -adj_m[0] - loc1;
    g_obj[1][0] = adj_m[0] - loc1;
    g_obj[2][0] = 2.0 * loc1;

    loc1        = isqrt3 * adj_m[4];
    g_obj[0][1] = -adj_m[3] - loc1;
    g_obj[1][1] = adj_m[3] - loc1;
    g_obj[2][1] = 2.0 * loc1;

    loc1        = isqrt3 * adj_m[7];
    g_obj[0][2] = -adj_m[6] - loc1;
    g_obj[1][2] = adj_m[6] - loc1;
    g_obj[2][2] = 2.0 * loc1;
    return true;
}

/*****************************************************************************/
/* Hessian evaluation requires 316 flops.                                    */
/*   Reductions possible when b == 1 or c == 1                               */
/*****************************************************************************/
inline bool h_fcn_2e( double& obj,
                      Vector3D g_obj[3],
                      Matrix3D h_obj[6],
                      const Vector3D x[3],
                      const Vector3D& n,
                      const double a,
                      const Exponent& b,
                      const Exponent& c )
{
    double matr[9], f;
    double adj_m[9], g;  // adj_m[2,5,8] not used
    double dg[9];        // dg[2,5,8] not used
    double loc0, loc1, loc2, loc3, loc4;
    double A[12], J_A[6], J_B[9], J_C[9], cross;  // only 2x2 corners used

    /* Calculate M = [A*inv(W) n] */
    matr[0] = x[1][0] - x[0][0];
    matr[1] = ( 2.0 * x[2][0] - x[1][0] - x[0][0] ) * isqrt3;
    matr[2] = n[0];

    matr[3] = x[1][1] - x[0][1];
    matr[4] = ( 2.0 * x[2][1] - x[1][1] - x[0][1] ) * isqrt3;
    matr[5] = n[1];

    matr[6] = x[1][2] - x[0][2];
    matr[7] = ( 2.0 * x[2][2] - x[1][2] - x[0][2] ) * isqrt3;
    matr[8] = n[2];

    /* Calculate det([n M]). */
    dg[0] = matr[4] * matr[8] - matr[5] * matr[7];
    dg[3] = matr[2] * matr[7] - matr[1] * matr[8];
    dg[6] = matr[1] * matr[5] - matr[2] * matr[4];
    g     = matr[0] * dg[0] + matr[3] * dg[3] + matr[6] * dg[6];
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[3] * matr[3] + matr[4] * matr[4] + matr[6] * matr[6] +
        matr[7] * matr[7];

    loc3 = f;
    loc4 = g;

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    dg[1] = matr[5] * matr[6] - matr[3] * matr[8];
    dg[4] = matr[0] * matr[8] - matr[2] * matr[6];
    dg[7] = matr[2] * matr[3] - matr[0] * matr[5];

    adj_m[0] = matr[0] * f + dg[0] * g;
    adj_m[1] = matr[1] * f + dg[1] * g;
    adj_m[3] = matr[3] * f + dg[3] * g;
    adj_m[4] = matr[4] * f + dg[4] * g;
    adj_m[6] = matr[6] * f + dg[6] * g;
    adj_m[7] = matr[7] * f + dg[7] * g;

    loc1        = isqrt3 * adj_m[1];
    g_obj[0][0] = -adj_m[0] - loc1;
    g_obj[1][0] = adj_m[0] - loc1;
    g_obj[2][0] = 2.0 * loc1;

    loc1        = isqrt3 * adj_m[4];
    g_obj[0][1] = -adj_m[3] - loc1;
    g_obj[1][1] = adj_m[3] - loc1;
    g_obj[2][1] = 2.0 * loc1;

    loc1        = isqrt3 * adj_m[7];
    g_obj[0][2] = -adj_m[6] - loc1;
    g_obj[1][2] = adj_m[6] - loc1;
    g_obj[2][2] = 2.0 * loc1;

    /* Calculate the hessian of the objective.                   */
    loc0  = f;                            /* Constant on nabla^2 f       */
    loc1  = g;                            /* Constant on nabla^2 g       */
    cross = f * c.value() / loc4;         /* Constant on nabla g nabla f */
    f     = f * ( b.value() - 1 ) / loc3; /* Constant on nabla f nabla f */
    g     = g * ( c.value() - 1 ) / loc4; /* Constant on nabla g nabla g */
    f *= 2.0;                             /* Modification for nabla f    */

    /* First block of rows */
    loc3 = matr[0] * f + dg[0] * cross;
    loc4 = dg[0] * g + matr[0] * cross;

    J_A[0] = loc0 + loc3 * matr[0] + loc4 * dg[0];
    J_A[1] = loc3 * matr[1] + loc4 * dg[1];
    J_B[0] = loc3 * matr[3] + loc4 * dg[3];
    J_B[1] = loc3 * matr[4] + loc4 * dg[4];
    J_C[0] = loc3 * matr[6] + loc4 * dg[6];
    J_C[1] = loc3 * matr[7] + loc4 * dg[7];

    loc3 = matr[1] * f + dg[1] * cross;
    loc4 = dg[1] * g + matr[1] * cross;

    J_A[3] = loc0 + loc3 * matr[1] + loc4 * dg[1];
    J_B[3] = loc3 * matr[3] + loc4 * dg[3];
    J_B[4] = loc3 * matr[4] + loc4 * dg[4];
    J_C[3] = loc3 * matr[6] + loc4 * dg[6];
    J_C[4] = loc3 * matr[7] + loc4 * dg[7];

    /* First diagonal block */
    loc2 = isqrt3 * J_A[1];
    A[0] = -J_A[0] - loc2;
    A[1] = J_A[0] - loc2;

    loc2 = isqrt3 * J_A[3];
    A[4] = -J_A[1] - loc2;
    A[5] = J_A[1] - loc2;
    A[6] = 2.0 * loc2;

    loc2           = isqrt3 * A[4];
    h_obj[0][0][0] = -A[0] - loc2;
    h_obj[1][0][0] = A[0] - loc2;
    h_obj[2][0][0] = 2.0 * loc2;

    loc2           = isqrt3 * A[5];
    h_obj[3][0][0] = A[1] - loc2;
    h_obj[4][0][0] = 2.0 * loc2;

    h_obj[5][0][0] = tisqrt3 * A[6];

    /* First off-diagonal block */
    loc2 = matr[8] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    loc2 = isqrt3 * J_B[3];
    A[0] = -J_B[0] - loc2;
    A[1] = J_B[0] - loc2;
    A[2] = 2.0 * loc2;

    loc2 = isqrt3 * J_B[4];
    A[4] = -J_B[1] - loc2;
    A[5] = J_B[1] - loc2;
    A[6] = 2.0 * loc2;

    loc2           = isqrt3 * A[4];
    h_obj[0][0][1] = -A[0] - loc2;
    h_obj[1][0][1] = A[0] - loc2;
    h_obj[2][0][1] = 2.0 * loc2;

    loc2           = isqrt3 * A[5];
    h_obj[1][1][0] = -A[1] - loc2;
    h_obj[3][0][1] = A[1] - loc2;
    h_obj[4][0][1] = 2.0 * loc2;

    loc2           = isqrt3 * A[6];
    h_obj[2][1][0] = -A[2] - loc2;
    h_obj[4][1][0] = A[2] - loc2;
    h_obj[5][0][1] = 2.0 * loc2;

    /* Second off-diagonal block */
    loc2 = matr[5] * loc1;
    J_C[1] -= loc2;
    J_C[3] += loc2;

    loc2 = isqrt3 * J_C[3];
    A[0] = -J_C[0] - loc2;
    A[1] = J_C[0] - loc2;
    A[2] = 2.0 * loc2;

    loc2 = isqrt3 * J_C[4];
    A[4] = -J_C[1] - loc2;
    A[5] = J_C[1] - loc2;
    A[6] = 2.0 * loc2;

    loc2           = isqrt3 * A[4];
    h_obj[0][0][2] = -A[0] - loc2;
    h_obj[1][0][2] = A[0] - loc2;
    h_obj[2][0][2] = 2.0 * loc2;

    loc2           = isqrt3 * A[5];
    h_obj[1][2][0] = -A[1] - loc2;
    h_obj[3][0][2] = A[1] - loc2;
    h_obj[4][0][2] = 2.0 * loc2;

    loc2           = isqrt3 * A[6];
    h_obj[2][2][0] = -A[2] - loc2;
    h_obj[4][2][0] = A[2] - loc2;
    h_obj[5][0][2] = 2.0 * loc2;

    /* Second block of rows */
    loc3 = matr[3] * f + dg[3] * cross;
    loc4 = dg[3] * g + matr[3] * cross;

    J_A[0] = loc0 + loc3 * matr[3] + loc4 * dg[3];
    J_A[1] = loc3 * matr[4] + loc4 * dg[4];
    J_B[0] = loc3 * matr[6] + loc4 * dg[6];
    J_B[1] = loc3 * matr[7] + loc4 * dg[7];

    loc3 = matr[4] * f + dg[4] * cross;
    loc4 = dg[4] * g + matr[4] * cross;

    J_A[3] = loc0 + loc3 * matr[4] + loc4 * dg[4];
    J_B[3] = loc3 * matr[6] + loc4 * dg[6];
    J_B[4] = loc3 * matr[7] + loc4 * dg[7];

    /* Second diagonal block */
    loc2 = isqrt3 * J_A[1];
    A[0] = -J_A[0] - loc2;
    A[1] = J_A[0] - loc2;

    loc2 = isqrt3 * J_A[3];
    A[4] = -J_A[1] - loc2;
    A[5] = J_A[1] - loc2;
    A[6] = 2.0 * loc2;

    loc2           = isqrt3 * A[4];
    h_obj[0][1][1] = -A[0] - loc2;
    h_obj[1][1][1] = A[0] - loc2;
    h_obj[2][1][1] = 2.0 * loc2;

    loc2           = isqrt3 * A[5];
    h_obj[3][1][1] = A[1] - loc2;
    h_obj[4][1][1] = 2.0 * loc2;

    h_obj[5][1][1] = tisqrt3 * A[6];

    /* Third off-diagonal block */
    loc2 = matr[2] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    loc2 = isqrt3 * J_B[3];
    A[0] = -J_B[0] - loc2;
    A[1] = J_B[0] - loc2;
    A[2] = 2.0 * loc2;

    loc2 = isqrt3 * J_B[4];
    A[4] = -J_B[1] - loc2;
    A[5] = J_B[1] - loc2;
    A[6] = 2.0 * loc2;

    loc2           = isqrt3 * A[4];
    h_obj[0][1][2] = -A[0] - loc2;
    h_obj[1][1][2] = A[0] - loc2;
    h_obj[2][1][2] = 2.0 * loc2;

    loc2           = isqrt3 * A[5];
    h_obj[1][2][1] = -A[1] - loc2;
    h_obj[3][1][2] = A[1] - loc2;
    h_obj[4][1][2] = 2.0 * loc2;

    loc2           = isqrt3 * A[6];
    h_obj[2][2][1] = -A[2] - loc2;
    h_obj[4][2][1] = A[2] - loc2;
    h_obj[5][1][2] = 2.0 * loc2;

    /* Third block of rows */
    loc3 = matr[6] * f + dg[6] * cross;
    loc4 = dg[6] * g + matr[6] * cross;

    J_A[0] = loc0 + loc3 * matr[6] + loc4 * dg[6];
    J_A[1] = loc3 * matr[7] + loc4 * dg[7];

    loc3 = matr[7] * f + dg[7] * cross;
    loc4 = dg[7] * g + matr[7] * cross;

    J_A[3] = loc0 + loc3 * matr[7] + loc4 * dg[7];

    /* Third diagonal block */
    loc2 = isqrt3 * J_A[1];
    A[0] = -J_A[0] - loc2;
    A[1] = J_A[0] - loc2;

    loc2 = isqrt3 * J_A[3];
    A[4] = -J_A[1] - loc2;
    A[5] = J_A[1] - loc2;
    A[6] = 2.0 * loc2;

    loc2           = isqrt3 * A[4];
    h_obj[0][2][2] = -A[0] - loc2;
    h_obj[1][2][2] = A[0] - loc2;
    h_obj[2][2][2] = 2.0 * loc2;

    loc2           = isqrt3 * A[5];
    h_obj[3][2][2] = A[1] - loc2;
    h_obj[4][2][2] = 2.0 * loc2;

    h_obj[5][2][2] = tisqrt3 * A[6];

    // completes diagonal blocks.
    h_obj[0].fill_lower_triangle();
    h_obj[3].fill_lower_triangle();
    h_obj[5].fill_lower_triangle();
    return true;
}

/*****************************************************************************/
/* The following set of functions reference triangular elements to an        */
/* right triangle in the plane defined by the normal.  They are used when    */
/* assessing the quality of a quadrilateral elements.  A zero return value   */
/* indicates success, while a nonzero value indicates failure.               */
/*****************************************************************************/

/*****************************************************************************/
/* Function evaluation -- requires 41 flops.                                 */
/*   Reductions possible when b == 1, c == 1, or d == 1                      */
/*****************************************************************************/
inline bool m_fcn_2i( double& obj,
                      const Vector3D x[3],
                      const Vector3D& n,
                      const double a,
                      const Exponent& b,
                      const Exponent& c,
                      const Vector3D& d )
{
    double matr[9];
    double f;
    double g;

    /* Calculate M = A*inv(W). */
    matr[0] = d[0] * ( x[1][0] - x[0][0] );
    matr[1] = d[1] * ( x[2][0] - x[0][0] );
    matr[2] = n[0];

    matr[3] = d[0] * ( x[1][1] - x[0][1] );
    matr[4] = d[1] * ( x[2][1] - x[0][1] );
    matr[5] = n[1];

    matr[6] = d[0] * ( x[1][2] - x[0][2] );
    matr[7] = d[1] * ( x[2][2] - x[0][2] );
    matr[8] = n[2];

    /* Calculate det(M). */
    g = matr[0] * ( matr[4] * matr[8] - matr[5] * matr[7] ) + matr[3] * ( matr[2] * matr[7] - matr[1] * matr[8] ) +
        matr[6] * ( matr[1] * matr[5] - matr[2] * matr[4] );
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[3] * matr[3] + matr[4] * matr[4] + matr[6] * matr[6] +
        matr[7] * matr[7];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );
    return true;
}

/*****************************************************************************/
/* Gradient requires 82 flops.                                               */
/*   Reductions possible when b == 1, c == 1, or d == 1                      */
/*****************************************************************************/
inline bool g_fcn_2i( double& obj,
                      Vector3D g_obj[3],
                      const Vector3D x[3],
                      const Vector3D& n,
                      const double a,
                      const Exponent& b,
                      const Exponent& c,
                      const Vector3D& d )
{
    double matr[9], f;
    double adj_m[9], g;  // adj_m[2,5,8] not used
    double loc1, loc2, loc3;

    /* Calculate M = [A*inv(W) n] */
    matr[0] = d[0] * ( x[1][0] - x[0][0] );
    matr[1] = d[1] * ( x[2][0] - x[0][0] );
    matr[2] = n[0];

    matr[3] = d[0] * ( x[1][1] - x[0][1] );
    matr[4] = d[1] * ( x[2][1] - x[0][1] );
    matr[5] = n[1];

    matr[6] = d[0] * ( x[1][2] - x[0][2] );
    matr[7] = d[1] * ( x[2][2] - x[0][2] );
    matr[8] = n[2];

    /* Calculate det([n M]). */
    loc1 = matr[4] * matr[8] - matr[5] * matr[7];
    loc2 = matr[2] * matr[7] - matr[1] * matr[8];
    loc3 = matr[1] * matr[5] - matr[2] * matr[4];
    g    = matr[0] * loc1 + matr[3] * loc2 + matr[6] * loc3;
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[3] * matr[3] + matr[4] * matr[4] + matr[6] * matr[6] +
        matr[7] * matr[7];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    adj_m[0] = d[0] * ( matr[0] * f + loc1 * g );
    adj_m[3] = d[0] * ( matr[3] * f + loc2 * g );
    adj_m[6] = d[0] * ( matr[6] * f + loc3 * g );

    loc1 = matr[0] * g;
    loc2 = matr[3] * g;
    loc3 = matr[6] * g;

    adj_m[1] = d[1] * ( matr[1] * f + loc3 * matr[5] - loc2 * matr[8] );
    adj_m[4] = d[1] * ( matr[4] * f + loc1 * matr[8] - loc3 * matr[2] );
    adj_m[7] = d[1] * ( matr[7] * f + loc2 * matr[2] - loc1 * matr[5] );

    g_obj[0][0] = -adj_m[0] - adj_m[1];
    g_obj[1][0] = adj_m[0];
    g_obj[2][0] = adj_m[1];

    g_obj[0][1] = -adj_m[3] - adj_m[4];
    g_obj[1][1] = adj_m[3];
    g_obj[2][1] = adj_m[4];

    g_obj[0][2] = -adj_m[6] - adj_m[7];
    g_obj[1][2] = adj_m[6];
    g_obj[2][2] = adj_m[7];
    return true;
}

/*****************************************************************************/
/* Hessian requires 253 flops.                                               */
/*   Reductions possible when b == 1, c == 1, or d == 1                      */
/*****************************************************************************/
inline bool h_fcn_2i( double& obj,
                      Vector3D g_obj[3],
                      Matrix3D h_obj[6],
                      const Vector3D x[3],
                      const Vector3D& n,
                      const double a,
                      const Exponent& b,
                      const Exponent& c,
                      const Vector3D& d )
{
    double matr[9], f;
    double adj_m[9], g;  // adj_m[2,5,8] not used
    double dg[9];        // dg[2,5,8] not used
    double loc0, loc1, loc2, loc3, loc4;
    double A[12], J_A[6], J_B[9], J_C[9], cross;  // only 2x2 corners used

    const double scale[3] = { d[0] * d[0], d[0] * d[1], d[1] * d[1] };

    /* Calculate M = [A*inv(W) n] */
    matr[0] = d[0] * ( x[1][0] - x[0][0] );
    matr[1] = d[1] * ( x[2][0] - x[0][0] );
    matr[2] = n[0];

    matr[3] = d[0] * ( x[1][1] - x[0][1] );
    matr[4] = d[1] * ( x[2][1] - x[0][1] );
    matr[5] = n[1];

    matr[6] = d[0] * ( x[1][2] - x[0][2] );
    matr[7] = d[1] * ( x[2][2] - x[0][2] );
    matr[8] = n[2];

    /* Calculate det([n M]). */
    dg[0] = matr[4] * matr[8] - matr[5] * matr[7];
    dg[3] = matr[2] * matr[7] - matr[1] * matr[8];
    dg[6] = matr[1] * matr[5] - matr[2] * matr[4];
    g     = matr[0] * dg[0] + matr[3] * dg[3] + matr[6] * dg[6];
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[3] * matr[3] + matr[4] * matr[4] + matr[6] * matr[6] +
        matr[7] * matr[7];

    loc3 = f;
    loc4 = g;

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    dg[1] = matr[5] * matr[6] - matr[3] * matr[8];
    dg[4] = matr[0] * matr[8] - matr[2] * matr[6];
    dg[7] = matr[2] * matr[3] - matr[0] * matr[5];

    adj_m[0] = d[0] * ( matr[0] * f + dg[0] * g );
    adj_m[1] = d[1] * ( matr[1] * f + dg[1] * g );
    adj_m[3] = d[0] * ( matr[3] * f + dg[3] * g );
    adj_m[4] = d[1] * ( matr[4] * f + dg[4] * g );
    adj_m[6] = d[0] * ( matr[6] * f + dg[6] * g );
    adj_m[7] = d[1] * ( matr[7] * f + dg[7] * g );

    g_obj[0][0] = -adj_m[0] - adj_m[1];
    g_obj[1][0] = adj_m[0];
    g_obj[2][0] = adj_m[1];

    g_obj[0][1] = -adj_m[3] - adj_m[4];
    g_obj[1][1] = adj_m[3];
    g_obj[2][1] = adj_m[4];

    g_obj[0][2] = -adj_m[6] - adj_m[7];
    g_obj[1][2] = adj_m[6];
    g_obj[2][2] = adj_m[7];

    /* Calculate the hessian of the objective.                   */
    loc0  = f;                            /* Constant on nabla^2 f       */
    loc1  = g;                            /* Constant on nabla^2 g       */
    cross = f * c.value() / loc4;         /* Constant on nabla g nabla f */
    f     = f * ( b.value() - 1 ) / loc3; /* Constant on nabla f nabla f */
    g     = g * ( c.value() - 1 ) / loc4; /* Constant on nabla g nabla g */
    f *= 2.0;                             /* Modification for nabla f    */

    /* First block of rows */
    loc3 = matr[0] * f + dg[0] * cross;
    loc4 = dg[0] * g + matr[0] * cross;

    J_A[0] = loc0 + loc3 * matr[0] + loc4 * dg[0];
    J_A[1] = loc3 * matr[1] + loc4 * dg[1];
    J_B[0] = loc3 * matr[3] + loc4 * dg[3];
    J_B[1] = loc3 * matr[4] + loc4 * dg[4];
    J_C[0] = loc3 * matr[6] + loc4 * dg[6];
    J_C[1] = loc3 * matr[7] + loc4 * dg[7];

    loc3 = matr[1] * f + dg[1] * cross;
    loc4 = dg[1] * g + matr[1] * cross;

    J_A[3] = loc0 + loc3 * matr[1] + loc4 * dg[1];
    J_B[3] = loc3 * matr[3] + loc4 * dg[3];
    J_B[4] = loc3 * matr[4] + loc4 * dg[4];
    J_C[3] = loc3 * matr[6] + loc4 * dg[6];
    J_C[4] = loc3 * matr[7] + loc4 * dg[7];

    /* First diagonal block */
    J_A[0] *= scale[0];
    J_A[1] *= scale[1];
    J_A[3] *= scale[2];

    A[0] = -J_A[0] - J_A[1];
    A[4] = -J_A[1] - J_A[3];

    h_obj[0][0][0] = -A[0] - A[4];
    h_obj[1][0][0] = A[0];
    h_obj[2][0][0] = A[4];

    h_obj[3][0][0] = J_A[0];
    h_obj[4][0][0] = J_A[1];

    h_obj[5][0][0] = J_A[3];

    /* First off-diagonal block */
    loc2 = matr[8] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    J_B[0] *= scale[0];
    J_B[1] *= scale[1];
    J_B[3] *= scale[1];
    J_B[4] *= scale[2];

    A[0] = -J_B[0] - J_B[3];
    A[4] = -J_B[1] - J_B[4];

    h_obj[0][0][1] = -A[0] - A[4];
    h_obj[1][0][1] = A[0];
    h_obj[2][0][1] = A[4];

    h_obj[1][1][0] = -J_B[0] - J_B[1];
    h_obj[3][0][1] = J_B[0];
    h_obj[4][0][1] = J_B[1];

    h_obj[2][1][0] = -J_B[3] - J_B[4];
    h_obj[4][1][0] = J_B[3];
    h_obj[5][0][1] = J_B[4];

    /* Second off-diagonal block */
    loc2 = matr[5] * loc1;
    J_C[1] -= loc2;
    J_C[3] += loc2;

    J_C[0] *= scale[0];
    J_C[1] *= scale[1];
    J_C[3] *= scale[1];
    J_C[4] *= scale[2];

    A[0] = -J_C[0] - J_C[3];
    A[4] = -J_C[1] - J_C[4];

    h_obj[0][0][2] = -A[0] - A[4];
    h_obj[1][0][2] = A[0];
    h_obj[2][0][2] = A[4];

    h_obj[1][2][0] = -J_C[0] - J_C[1];
    h_obj[3][0][2] = J_C[0];
    h_obj[4][0][2] = J_C[1];

    h_obj[2][2][0] = -J_C[3] - J_C[4];
    h_obj[4][2][0] = J_C[3];
    h_obj[5][0][2] = J_C[4];

    /* Second block of rows */
    loc3 = matr[3] * f + dg[3] * cross;
    loc4 = dg[3] * g + matr[3] * cross;

    J_A[0] = loc0 + loc3 * matr[3] + loc4 * dg[3];
    J_A[1] = loc3 * matr[4] + loc4 * dg[4];
    J_B[0] = loc3 * matr[6] + loc4 * dg[6];
    J_B[1] = loc3 * matr[7] + loc4 * dg[7];

    loc3 = matr[4] * f + dg[4] * cross;
    loc4 = dg[4] * g + matr[4] * cross;

    J_A[3] = loc0 + loc3 * matr[4] + loc4 * dg[4];
    J_B[3] = loc3 * matr[6] + loc4 * dg[6];
    J_B[4] = loc3 * matr[7] + loc4 * dg[7];

    /* Second diagonal block */
    J_A[0] *= scale[0];
    J_A[1] *= scale[1];
    J_A[3] *= scale[2];

    A[0] = -J_A[0] - J_A[1];
    A[4] = -J_A[1] - J_A[3];

    h_obj[0][1][1] = -A[0] - A[4];
    h_obj[1][1][1] = A[0];
    h_obj[2][1][1] = A[4];

    h_obj[3][1][1] = J_A[0];
    h_obj[4][1][1] = J_A[1];

    h_obj[5][1][1] = J_A[3];

    /* Third off-diagonal block */
    loc2 = matr[2] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    J_B[0] *= scale[0];
    J_B[1] *= scale[1];
    J_B[3] *= scale[1];
    J_B[4] *= scale[2];

    A[0] = -J_B[0] - J_B[3];
    A[4] = -J_B[1] - J_B[4];

    h_obj[0][1][2] = -A[0] - A[4];
    h_obj[1][1][2] = A[0];
    h_obj[2][1][2] = A[4];

    h_obj[1][2][1] = -J_B[0] - J_B[1];
    h_obj[3][1][2] = J_B[0];
    h_obj[4][1][2] = J_B[1];

    h_obj[2][2][1] = -J_B[3] - J_B[4];
    h_obj[4][2][1] = J_B[3];
    h_obj[5][1][2] = J_B[4];

    /* Third block of rows */
    loc3 = matr[6] * f + dg[6] * cross;
    loc4 = dg[6] * g + matr[6] * cross;

    J_A[0] = loc0 + loc3 * matr[6] + loc4 * dg[6];
    J_A[1] = loc3 * matr[7] + loc4 * dg[7];

    loc3 = matr[7] * f + dg[7] * cross;
    loc4 = dg[7] * g + matr[7] * cross;

    J_A[3] = loc0 + loc3 * matr[7] + loc4 * dg[7];

    /* Third diagonal block */
    J_A[0] *= scale[0];
    J_A[1] *= scale[1];
    J_A[3] *= scale[2];

    A[0] = -J_A[0] - J_A[1];
    A[4] = -J_A[1] - J_A[3];

    h_obj[0][2][2] = -A[0] - A[4];
    h_obj[1][2][2] = A[0];
    h_obj[2][2][2] = A[4];

    h_obj[3][2][2] = J_A[0];
    h_obj[4][2][2] = J_A[1];

    h_obj[5][2][2] = J_A[3];

    // completes diagonal blocks.
    h_obj[0].fill_lower_triangle();
    h_obj[3].fill_lower_triangle();
    h_obj[5].fill_lower_triangle();
    return true;
}

/*****************************************************************************/
/* The following set of functions reference tetrahedral elements to a        */
/* regular tetrahedron.  They are used when assessing the quality of a       */
/* tetrahedral element.  A zero return value indicates success, while        */
/* a nonzero value indicates failure.                                        */
/*****************************************************************************/

/*****************************************************************************/
/* Function evaluation requires 62 flops.                                    */
/*   Reductions possible when b == 1 or c == 1                               */
/*****************************************************************************/
inline bool m_fcn_3e( double& obj, const Vector3D x[4], const double a, const Exponent& b, const Exponent& c )
{
    double matr[9], f;
    double g;

    /* Calculate M = A*inv(W). */
    f       = x[1][0] + x[0][0];
    matr[0] = x[1][0] - x[0][0];
    matr[1] = ( 2.0 * x[2][0] - f ) * isqrt3;
    matr[2] = ( 3.0 * x[3][0] - x[2][0] - f ) * isqrt6;

    f       = x[1][1] + x[0][1];
    matr[3] = x[1][1] - x[0][1];
    matr[4] = ( 2.0 * x[2][1] - f ) * isqrt3;
    matr[5] = ( 3.0 * x[3][1] - x[2][1] - f ) * isqrt6;

    f       = x[1][2] + x[0][2];
    matr[6] = x[1][2] - x[0][2];
    matr[7] = ( 2.0 * x[2][2] - f ) * isqrt3;
    matr[8] = ( 3.0 * x[3][2] - x[2][2] - f ) * isqrt6;

    /* Calculate det(M). */
    g = matr[0] * ( matr[4] * matr[8] - matr[5] * matr[7] ) + matr[1] * ( matr[5] * matr[6] - matr[3] * matr[8] ) +
        matr[2] * ( matr[3] * matr[7] - matr[4] * matr[6] );
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );
    return true;
}

/*****************************************************************************/
/* Gradient evaluation requires 133 flops.                                   */
/*   Reductions possible when b == 1 or c == 1                               */
/*****************************************************************************/
inline bool g_fcn_3e( double& obj,
                      Vector3D g_obj[4],
                      const Vector3D x[4],
                      const double a,
                      const Exponent& b,
                      const Exponent& c )
{
    double matr[9], f;
    double adj_m[9], g;
    double loc1, loc2, loc3;

    /* Calculate M = A*inv(W). */
    f       = x[1][0] + x[0][0];
    matr[0] = x[1][0] - x[0][0];
    matr[1] = ( 2.0 * x[2][0] - f ) * isqrt3;
    matr[2] = ( 3.0 * x[3][0] - x[2][0] - f ) * isqrt6;

    f       = x[1][1] + x[0][1];
    matr[3] = x[1][1] - x[0][1];
    matr[4] = ( 2.0 * x[2][1] - f ) * isqrt3;
    matr[5] = ( 3.0 * x[3][1] - x[2][1] - f ) * isqrt6;

    f       = x[1][2] + x[0][2];
    matr[6] = x[1][2] - x[0][2];
    matr[7] = ( 2.0 * x[2][2] - f ) * isqrt3;
    matr[8] = ( 3.0 * x[3][2] - x[2][2] - f ) * isqrt6;

    /* Calculate det(M). */
    loc1 = matr[4] * matr[8] - matr[5] * matr[7];
    loc2 = matr[5] * matr[6] - matr[3] * matr[8];
    loc3 = matr[3] * matr[7] - matr[4] * matr[6];
    g    = matr[0] * loc1 + matr[1] * loc2 + matr[2] * loc3;
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    adj_m[0] = matr[0] * f + loc1 * g;
    adj_m[1] = matr[1] * f + loc2 * g;
    adj_m[2] = matr[2] * f + loc3 * g;

    loc1 = matr[0] * g;
    loc2 = matr[1] * g;
    loc3 = matr[2] * g;

    adj_m[3] = matr[3] * f + loc3 * matr[7] - loc2 * matr[8];
    adj_m[4] = matr[4] * f + loc1 * matr[8] - loc3 * matr[6];
    adj_m[5] = matr[5] * f + loc2 * matr[6] - loc1 * matr[7];

    adj_m[6] = matr[6] * f + loc2 * matr[5] - loc3 * matr[4];
    adj_m[7] = matr[7] * f + loc3 * matr[3] - loc1 * matr[5];
    adj_m[8] = matr[8] * f + loc1 * matr[4] - loc2 * matr[3];

    loc1        = isqrt3 * adj_m[1];
    loc2        = isqrt6 * adj_m[2];
    loc3        = loc1 + loc2;
    g_obj[0][0] = -adj_m[0] - loc3;
    g_obj[1][0] = adj_m[0] - loc3;
    g_obj[2][0] = 2.0 * loc1 - loc2;
    g_obj[3][0] = 3.0 * loc2;

    loc1        = isqrt3 * adj_m[4];
    loc2        = isqrt6 * adj_m[5];
    loc3        = loc1 + loc2;
    g_obj[0][1] = -adj_m[3] - loc3;
    g_obj[1][1] = adj_m[3] - loc3;
    g_obj[2][1] = 2.0 * loc1 - loc2;
    g_obj[3][1] = 3.0 * loc2;

    loc1        = isqrt3 * adj_m[7];
    loc2        = isqrt6 * adj_m[8];
    loc3        = loc1 + loc2;
    g_obj[0][2] = -adj_m[6] - loc3;
    g_obj[1][2] = adj_m[6] - loc3;
    g_obj[2][2] = 2.0 * loc1 - loc2;
    g_obj[3][2] = 3.0 * loc2;
    return true;
}

/*****************************************************************************/
/* Hessian evaluation requires 634 flops.                                    */
/*   Reductions possible when b == 1 or c == 1                               */
/*****************************************************************************/
inline bool h_fcn_3e( double& obj,
                      Vector3D g_obj[4],
                      Matrix3D h_obj[10],
                      const Vector3D x[4],
                      const double a,
                      const Exponent& b,
                      const Exponent& c )
{
    double matr[9], f;
    double adj_m[9], g;
    double dg[9], loc0, loc1, loc2, loc3, loc4;
    double A[12], J_A[6], J_B[9], J_C[9], cross;

    /* Calculate M = A*inv(W). */
    f       = x[1][0] + x[0][0];
    matr[0] = x[1][0] - x[0][0];
    matr[1] = ( 2.0 * x[2][0] - f ) * isqrt3;
    matr[2] = ( 3.0 * x[3][0] - x[2][0] - f ) * isqrt6;

    f       = x[1][1] + x[0][1];
    matr[3] = x[1][1] - x[0][1];
    matr[4] = ( 2.0 * x[2][1] - f ) * isqrt3;
    matr[5] = ( 3.0 * x[3][1] - x[2][1] - f ) * isqrt6;

    f       = x[1][2] + x[0][2];
    matr[6] = x[1][2] - x[0][2];
    matr[7] = ( 2.0 * x[2][2] - f ) * isqrt3;
    matr[8] = ( 3.0 * x[3][2] - x[2][2] - f ) * isqrt6;

    /* Calculate det(M). */
    dg[0] = matr[4] * matr[8] - matr[5] * matr[7];
    dg[1] = matr[5] * matr[6] - matr[3] * matr[8];
    dg[2] = matr[3] * matr[7] - matr[4] * matr[6];
    g     = matr[0] * dg[0] + matr[1] * dg[1] + matr[2] * dg[2];
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    loc3 = f;
    loc4 = g;

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    dg[3] = matr[2] * matr[7] - matr[1] * matr[8];
    dg[4] = matr[0] * matr[8] - matr[2] * matr[6];
    dg[5] = matr[1] * matr[6] - matr[0] * matr[7];
    dg[6] = matr[1] * matr[5] - matr[2] * matr[4];
    dg[7] = matr[2] * matr[3] - matr[0] * matr[5];
    dg[8] = matr[0] * matr[4] - matr[1] * matr[3];

    adj_m[0] = matr[0] * f + dg[0] * g;
    adj_m[1] = matr[1] * f + dg[1] * g;
    adj_m[2] = matr[2] * f + dg[2] * g;
    adj_m[3] = matr[3] * f + dg[3] * g;
    adj_m[4] = matr[4] * f + dg[4] * g;
    adj_m[5] = matr[5] * f + dg[5] * g;
    adj_m[6] = matr[6] * f + dg[6] * g;
    adj_m[7] = matr[7] * f + dg[7] * g;
    adj_m[8] = matr[8] * f + dg[8] * g;

    loc0        = isqrt3 * adj_m[1];
    loc1        = isqrt6 * adj_m[2];
    loc2        = loc0 + loc1;
    g_obj[0][0] = -adj_m[0] - loc2;
    g_obj[1][0] = adj_m[0] - loc2;
    g_obj[2][0] = 2.0 * loc0 - loc1;
    g_obj[3][0] = 3.0 * loc1;

    loc0        = isqrt3 * adj_m[4];
    loc1        = isqrt6 * adj_m[5];
    loc2        = loc0 + loc1;
    g_obj[0][1] = -adj_m[3] - loc2;
    g_obj[1][1] = adj_m[3] - loc2;
    g_obj[2][1] = 2.0 * loc0 - loc1;
    g_obj[3][1] = 3.0 * loc1;

    loc0        = isqrt3 * adj_m[7];
    loc1        = isqrt6 * adj_m[8];
    loc2        = loc0 + loc1;
    g_obj[0][2] = -adj_m[6] - loc2;
    g_obj[1][2] = adj_m[6] - loc2;
    g_obj[2][2] = 2.0 * loc0 - loc1;
    g_obj[3][2] = 3.0 * loc1;

    /* Calculate the hessian of the objective.                   */
    loc0  = f;                            /* Constant on nabla^2 f       */
    loc1  = g;                            /* Constant on nabla^2 g       */
    cross = f * c.value() / loc4;         /* Constant on nabla g nabla f */
    f     = f * ( b.value() - 1 ) / loc3; /* Constant on nabla f nabla f */
    g     = g * ( c.value() - 1 ) / loc4; /* Constant on nabla g nabla g */
    f *= 2.0;                             /* Modification for nabla f    */

    /* First block of rows */
    loc3 = matr[0] * f + dg[0] * cross;
    loc4 = dg[0] * g + matr[0] * cross;

    J_A[0] = loc0 + loc3 * matr[0] + loc4 * dg[0];
    J_A[1] = loc3 * matr[1] + loc4 * dg[1];
    J_A[2] = loc3 * matr[2] + loc4 * dg[2];
    J_B[0] = loc3 * matr[3] + loc4 * dg[3];
    J_B[1] = loc3 * matr[4] + loc4 * dg[4];
    J_B[2] = loc3 * matr[5] + loc4 * dg[5];
    J_C[0] = loc3 * matr[6] + loc4 * dg[6];
    J_C[1] = loc3 * matr[7] + loc4 * dg[7];
    J_C[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[1] * f + dg[1] * cross;
    loc4 = dg[1] * g + matr[1] * cross;

    J_A[3] = loc0 + loc3 * matr[1] + loc4 * dg[1];
    J_A[4] = loc3 * matr[2] + loc4 * dg[2];
    J_B[3] = loc3 * matr[3] + loc4 * dg[3];
    J_B[4] = loc3 * matr[4] + loc4 * dg[4];
    J_B[5] = loc3 * matr[5] + loc4 * dg[5];
    J_C[3] = loc3 * matr[6] + loc4 * dg[6];
    J_C[4] = loc3 * matr[7] + loc4 * dg[7];
    J_C[5] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[2] * f + dg[2] * cross;
    loc4 = dg[2] * g + matr[2] * cross;

    J_A[5] = loc0 + loc3 * matr[2] + loc4 * dg[2];
    J_B[6] = loc3 * matr[3] + loc4 * dg[3];
    J_B[7] = loc3 * matr[4] + loc4 * dg[4];
    J_B[8] = loc3 * matr[5] + loc4 * dg[5];
    J_C[6] = loc3 * matr[6] + loc4 * dg[6];
    J_C[7] = loc3 * matr[7] + loc4 * dg[7];
    J_C[8] = loc3 * matr[8] + loc4 * dg[8];

    /* First diagonal block */
    loc2 = isqrt3 * J_A[1];
    loc3 = isqrt6 * J_A[2];
    loc4 = loc2 + loc3;

    A[0] = -J_A[0] - loc4;
    A[1] = J_A[0] - loc4;

    loc2 = isqrt3 * J_A[3];
    loc3 = isqrt6 * J_A[4];
    loc4 = loc2 + loc3;

    A[4] = -J_A[1] - loc4;
    A[5] = J_A[1] - loc4;
    A[6] = 2.0 * loc2 - loc3;

    loc2 = isqrt3 * J_A[4];
    loc3 = isqrt6 * J_A[5];
    loc4 = loc2 + loc3;

    A[8]  = -J_A[2] - loc4;
    A[9]  = J_A[2] - loc4;
    A[10] = 2.0 * loc2 - loc3;
    A[11] = 3.0 * loc3;

    loc2 = isqrt3 * A[4];
    loc3 = isqrt6 * A[8];
    loc4 = loc2 + loc3;

    h_obj[0][0][0] = -A[0] - loc4;
    h_obj[1][0][0] = A[0] - loc4;
    h_obj[2][0][0] = 2.0 * loc2 - loc3;
    h_obj[3][0][0] = 3.0 * loc3;

    loc2 = isqrt3 * A[5];
    loc3 = isqrt6 * A[9];

    h_obj[4][0][0] = A[1] - loc2 - loc3;
    h_obj[5][0][0] = 2.0 * loc2 - loc3;
    h_obj[6][0][0] = 3.0 * loc3;

    loc3           = isqrt6 * A[10];
    h_obj[7][0][0] = tisqrt3 * A[6] - loc3;
    h_obj[8][0][0] = 3.0 * loc3;

    h_obj[9][0][0] = tisqrt6 * A[11];

    /* First off-diagonal block */
    loc2 = matr[8] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    loc2 = matr[7] * loc1;
    J_B[2] -= loc2;
    J_B[6] += loc2;

    loc2 = matr[6] * loc1;
    J_B[5] += loc2;
    J_B[7] -= loc2;

    loc2 = isqrt3 * J_B[3];
    loc3 = isqrt6 * J_B[6];
    loc4 = loc2 + loc3;

    A[0] = -J_B[0] - loc4;
    A[1] = J_B[0] - loc4;
    A[2] = 2.0 * loc2 - loc3;
    A[3] = 3.0 * loc3;

    loc2 = isqrt3 * J_B[4];
    loc3 = isqrt6 * J_B[7];
    loc4 = loc2 + loc3;

    A[4] = -J_B[1] - loc4;
    A[5] = J_B[1] - loc4;
    A[6] = 2.0 * loc2 - loc3;
    A[7] = 3.0 * loc3;

    loc2 = isqrt3 * J_B[5];
    loc3 = isqrt6 * J_B[8];
    loc4 = loc2 + loc3;

    A[8]  = -J_B[2] - loc4;
    A[9]  = J_B[2] - loc4;
    A[10] = 2.0 * loc2 - loc3;
    A[11] = 3.0 * loc3;

    loc2 = isqrt3 * A[4];
    loc3 = isqrt6 * A[8];
    loc4 = loc2 + loc3;

    h_obj[0][0][1] = -A[0] - loc4;
    h_obj[1][0][1] = A[0] - loc4;
    h_obj[2][0][1] = 2.0 * loc2 - loc3;
    h_obj[3][0][1] = 3.0 * loc3;

    loc2 = isqrt3 * A[5];
    loc3 = isqrt6 * A[9];
    loc4 = loc2 + loc3;

    h_obj[1][1][0] = -A[1] - loc4;
    h_obj[4][0][1] = A[1] - loc4;
    h_obj[5][0][1] = 2.0 * loc2 - loc3;
    h_obj[6][0][1] = 3.0 * loc3;

    loc2 = isqrt3 * A[6];
    loc3 = isqrt6 * A[10];
    loc4 = loc2 + loc3;

    h_obj[2][1][0] = -A[2] - loc4;
    h_obj[5][1][0] = A[2] - loc4;
    h_obj[7][0][1] = 2.0 * loc2 - loc3;
    h_obj[8][0][1] = 3.0 * loc3;

    loc2 = isqrt3 * A[7];
    loc3 = isqrt6 * A[11];
    loc4 = loc2 + loc3;

    h_obj[3][1][0] = -A[3] - loc4;
    h_obj[6][1][0] = A[3] - loc4;
    h_obj[8][1][0] = 2.0 * loc2 - loc3;
    h_obj[9][0][1] = 3.0 * loc3;

    /* Second off-diagonal block */
    loc2 = matr[5] * loc1;
    J_C[1] -= loc2;
    J_C[3] += loc2;

    loc2 = matr[4] * loc1;
    J_C[2] += loc2;
    J_C[6] -= loc2;

    loc2 = matr[3] * loc1;
    J_C[5] -= loc2;
    J_C[7] += loc2;

    loc2 = isqrt3 * J_C[3];
    loc3 = isqrt6 * J_C[6];
    loc4 = loc2 + loc3;

    A[0] = -J_C[0] - loc4;
    A[1] = J_C[0] - loc4;
    A[2] = 2.0 * loc2 - loc3;
    A[3] = 3.0 * loc3;

    loc2 = isqrt3 * J_C[4];
    loc3 = isqrt6 * J_C[7];
    loc4 = loc2 + loc3;

    A[4] = -J_C[1] - loc4;
    A[5] = J_C[1] - loc4;
    A[6] = 2.0 * loc2 - loc3;
    A[7] = 3.0 * loc3;

    loc2 = isqrt3 * J_C[5];
    loc3 = isqrt6 * J_C[8];
    loc4 = loc2 + loc3;

    A[8]  = -J_C[2] - loc4;
    A[9]  = J_C[2] - loc4;
    A[10] = 2.0 * loc2 - loc3;
    A[11] = 3.0 * loc3;

    loc2 = isqrt3 * A[4];
    loc3 = isqrt6 * A[8];
    loc4 = loc2 + loc3;

    h_obj[0][0][2] = -A[0] - loc4;
    h_obj[1][0][2] = A[0] - loc4;
    h_obj[2][0][2] = 2.0 * loc2 - loc3;
    h_obj[3][0][2] = 3.0 * loc3;

    loc2 = isqrt3 * A[5];
    loc3 = isqrt6 * A[9];
    loc4 = loc2 + loc3;

    h_obj[1][2][0] = -A[1] - loc4;
    h_obj[4][0][2] = A[1] - loc4;
    h_obj[5][0][2] = 2.0 * loc2 - loc3;
    h_obj[6][0][2] = 3.0 * loc3;

    loc2 = isqrt3 * A[6];
    loc3 = isqrt6 * A[10];
    loc4 = loc2 + loc3;

    h_obj[2][2][0] = -A[2] - loc4;
    h_obj[5][2][0] = A[2] - loc4;
    h_obj[7][0][2] = 2.0 * loc2 - loc3;
    h_obj[8][0][2] = 3.0 * loc3;

    loc2 = isqrt3 * A[7];
    loc3 = isqrt6 * A[11];
    loc4 = loc2 + loc3;

    h_obj[3][2][0] = -A[3] - loc4;
    h_obj[6][2][0] = A[3] - loc4;
    h_obj[8][2][0] = 2.0 * loc2 - loc3;
    h_obj[9][0][2] = 3.0 * loc3;

    /* Second block of rows */
    loc3 = matr[3] * f + dg[3] * cross;
    loc4 = dg[3] * g + matr[3] * cross;

    J_A[0] = loc0 + loc3 * matr[3] + loc4 * dg[3];
    J_A[1] = loc3 * matr[4] + loc4 * dg[4];
    J_A[2] = loc3 * matr[5] + loc4 * dg[5];
    J_B[0] = loc3 * matr[6] + loc4 * dg[6];
    J_B[1] = loc3 * matr[7] + loc4 * dg[7];
    J_B[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[4] * f + dg[4] * cross;
    loc4 = dg[4] * g + matr[4] * cross;

    J_A[3] = loc0 + loc3 * matr[4] + loc4 * dg[4];
    J_A[4] = loc3 * matr[5] + loc4 * dg[5];
    J_B[3] = loc3 * matr[6] + loc4 * dg[6];
    J_B[4] = loc3 * matr[7] + loc4 * dg[7];
    J_B[5] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[5] * f + dg[5] * cross;
    loc4 = dg[5] * g + matr[5] * cross;

    J_A[5] = loc0 + loc3 * matr[5] + loc4 * dg[5];
    J_B[6] = loc3 * matr[6] + loc4 * dg[6];
    J_B[7] = loc3 * matr[7] + loc4 * dg[7];
    J_B[8] = loc3 * matr[8] + loc4 * dg[8];

    /* Second diagonal block */
    loc2 = isqrt3 * J_A[1];
    loc3 = isqrt6 * J_A[2];
    loc4 = loc2 + loc3;

    A[0] = -J_A[0] - loc4;
    A[1] = J_A[0] - loc4;

    loc2 = isqrt3 * J_A[3];
    loc3 = isqrt6 * J_A[4];
    loc4 = loc2 + loc3;

    A[4] = -J_A[1] - loc4;
    A[5] = J_A[1] - loc4;
    A[6] = 2.0 * loc2 - loc3;

    loc2 = isqrt3 * J_A[4];
    loc3 = isqrt6 * J_A[5];
    loc4 = loc2 + loc3;

    A[8]  = -J_A[2] - loc4;
    A[9]  = J_A[2] - loc4;
    A[10] = 2.0 * loc2 - loc3;
    A[11] = 3.0 * loc3;

    loc2 = isqrt3 * A[4];
    loc3 = isqrt6 * A[8];
    loc4 = loc2 + loc3;

    h_obj[0][1][1] = -A[0] - loc4;
    h_obj[1][1][1] = A[0] - loc4;
    h_obj[2][1][1] = 2.0 * loc2 - loc3;
    h_obj[3][1][1] = 3.0 * loc3;

    loc2 = isqrt3 * A[5];
    loc3 = isqrt6 * A[9];

    h_obj[4][1][1] = A[1] - loc2 - loc3;
    h_obj[5][1][1] = 2.0 * loc2 - loc3;
    h_obj[6][1][1] = 3.0 * loc3;

    loc3           = isqrt6 * A[10];
    h_obj[7][1][1] = tisqrt3 * A[6] - loc3;
    h_obj[8][1][1] = 3.0 * loc3;

    h_obj[9][1][1] = tisqrt6 * A[11];

    /* Third off-diagonal block */
    loc2 = matr[2] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    loc2 = matr[1] * loc1;
    J_B[2] -= loc2;
    J_B[6] += loc2;

    loc2 = matr[0] * loc1;
    J_B[5] += loc2;
    J_B[7] -= loc2;

    loc2 = isqrt3 * J_B[3];
    loc3 = isqrt6 * J_B[6];
    loc4 = loc2 + loc3;

    A[0] = -J_B[0] - loc4;
    A[1] = J_B[0] - loc4;
    A[2] = 2.0 * loc2 - loc3;
    A[3] = 3.0 * loc3;

    loc2 = isqrt3 * J_B[4];
    loc3 = isqrt6 * J_B[7];
    loc4 = loc2 + loc3;

    A[4] = -J_B[1] - loc4;
    A[5] = J_B[1] - loc4;
    A[6] = 2.0 * loc2 - loc3;
    A[7] = 3.0 * loc3;

    loc2 = isqrt3 * J_B[5];
    loc3 = isqrt6 * J_B[8];
    loc4 = loc2 + loc3;

    A[8]  = -J_B[2] - loc4;
    A[9]  = J_B[2] - loc4;
    A[10] = 2.0 * loc2 - loc3;
    A[11] = 3.0 * loc3;

    loc2 = isqrt3 * A[4];
    loc3 = isqrt6 * A[8];
    loc4 = loc2 + loc3;

    h_obj[0][1][2] = -A[0] - loc4;
    h_obj[1][1][2] = A[0] - loc4;
    h_obj[2][1][2] = 2.0 * loc2 - loc3;
    h_obj[3][1][2] = 3.0 * loc3;

    loc2 = isqrt3 * A[5];
    loc3 = isqrt6 * A[9];
    loc4 = loc2 + loc3;

    h_obj[1][2][1] = -A[1] - loc4;
    h_obj[4][1][2] = A[1] - loc4;
    h_obj[5][1][2] = 2.0 * loc2 - loc3;
    h_obj[6][1][2] = 3.0 * loc3;

    loc2 = isqrt3 * A[6];
    loc3 = isqrt6 * A[10];
    loc4 = loc2 + loc3;

    h_obj[2][2][1] = -A[2] - loc4;
    h_obj[5][2][1] = A[2] - loc4;
    h_obj[7][1][2] = 2.0 * loc2 - loc3;
    h_obj[8][1][2] = 3.0 * loc3;

    loc2 = isqrt3 * A[7];
    loc3 = isqrt6 * A[11];
    loc4 = loc2 + loc3;

    h_obj[3][2][1] = -A[3] - loc4;
    h_obj[6][2][1] = A[3] - loc4;
    h_obj[8][2][1] = 2.0 * loc2 - loc3;
    h_obj[9][1][2] = 3.0 * loc3;

    /* Third block of rows */
    loc3 = matr[6] * f + dg[6] * cross;
    loc4 = dg[6] * g + matr[6] * cross;

    J_A[0] = loc0 + loc3 * matr[6] + loc4 * dg[6];
    J_A[1] = loc3 * matr[7] + loc4 * dg[7];
    J_A[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[7] * f + dg[7] * cross;
    loc4 = dg[7] * g + matr[7] * cross;

    J_A[3] = loc0 + loc3 * matr[7] + loc4 * dg[7];
    J_A[4] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[8] * f + dg[8] * cross;
    loc4 = dg[8] * g + matr[8] * cross;

    J_A[5] = loc0 + loc3 * matr[8] + loc4 * dg[8];

    /* Third diagonal block */
    loc2 = isqrt3 * J_A[1];
    loc3 = isqrt6 * J_A[2];
    loc4 = loc2 + loc3;

    A[0] = -J_A[0] - loc4;
    A[1] = J_A[0] - loc4;

    loc2 = isqrt3 * J_A[3];
    loc3 = isqrt6 * J_A[4];
    loc4 = loc2 + loc3;

    A[4] = -J_A[1] - loc4;
    A[5] = J_A[1] - loc4;
    A[6] = 2.0 * loc2 - loc3;

    loc2 = isqrt3 * J_A[4];
    loc3 = isqrt6 * J_A[5];
    loc4 = loc2 + loc3;

    A[8]  = -J_A[2] - loc4;
    A[9]  = J_A[2] - loc4;
    A[10] = 2.0 * loc2 - loc3;
    A[11] = 3.0 * loc3;

    loc2 = isqrt3 * A[4];
    loc3 = isqrt6 * A[8];
    loc4 = loc2 + loc3;

    h_obj[0][2][2] = -A[0] - loc4;
    h_obj[1][2][2] = A[0] - loc4;
    h_obj[2][2][2] = 2.0 * loc2 - loc3;
    h_obj[3][2][2] = 3.0 * loc3;

    loc2 = isqrt3 * A[5];
    loc3 = isqrt6 * A[9];

    h_obj[4][2][2] = A[1] - loc2 - loc3;
    h_obj[5][2][2] = 2.0 * loc2 - loc3;
    h_obj[6][2][2] = 3.0 * loc3;

    loc3           = isqrt6 * A[10];
    h_obj[7][2][2] = tisqrt3 * A[6] - loc3;
    h_obj[8][2][2] = 3.0 * loc3;

    h_obj[9][2][2] = tisqrt6 * A[11];

    // completes diagonal blocks.
    h_obj[0].fill_lower_triangle();
    h_obj[4].fill_lower_triangle();
    h_obj[7].fill_lower_triangle();
    h_obj[9].fill_lower_triangle();
    return true;
}

/*****************************************************************************/
/* The following set of functions reference tetrahedral elements to a        */
/* equilateral tetrahedron.  These functions are specialized toward          */
/* obtaining the gradient and Hessian with respect to a single vertex.       */
/*****************************************************************************/

inline bool g_fcn_3e_v3( double& obj,
                         Vector3D& g_obj,
                         const Vector3D x[4],
                         const double a,
                         const Exponent& b,
                         const Exponent& c )
{
    double matr[9], f, g;
    double loc1, loc2, loc3;

    /* Calculate M = A*inv(W). */
    f       = x[1][0] + x[0][0];
    matr[0] = x[1][0] - x[0][0];
    matr[1] = ( 2.0 * x[2][0] - f ) * isqrt3;
    matr[2] = ( 3.0 * x[3][0] - x[2][0] - f ) * isqrt6;

    f       = x[1][1] + x[0][1];
    matr[3] = x[1][1] - x[0][1];
    matr[4] = ( 2.0 * x[2][1] - f ) * isqrt3;
    matr[5] = ( 3.0 * x[3][1] - x[2][1] - f ) * isqrt6;

    f       = x[1][2] + x[0][2];
    matr[6] = x[1][2] - x[0][2];
    matr[7] = ( 2.0 * x[2][2] - f ) * isqrt3;
    matr[8] = ( 3.0 * x[3][2] - x[2][2] - f ) * isqrt6;

    /* Calculate det(M). */
    loc1 = matr[4] * matr[8] - matr[5] * matr[7];
    loc2 = matr[5] * matr[6] - matr[3] * matr[8];
    loc3 = matr[3] * matr[7] - matr[4] * matr[6];
    g    = matr[0] * loc1 + matr[1] * loc2 + matr[2] * loc3;
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    g_obj[0] = tisqrt6 * ( matr[2] * f + loc3 * g );

    loc1 = matr[0] * g;
    loc2 = matr[1] * g;

    g_obj[1] = tisqrt6 * ( matr[5] * f + loc2 * matr[6] - loc1 * matr[7] );
    g_obj[2] = tisqrt6 * ( matr[8] * f + loc1 * matr[4] - loc2 * matr[3] );
    return true;
}

inline bool g_fcn_3e_v0( double& obj,
                         Vector3D& g_obj,
                         const Vector3D x[4],
                         const double a,
                         const Exponent& b,
                         const Exponent& c )
{
    static Vector3D my_x[4];

    my_x[0] = x[1];
    my_x[1] = x[3];
    my_x[2] = x[2];
    my_x[3] = x[0];
    return g_fcn_3e_v3( obj, g_obj, my_x, a, b, c );
}

inline bool g_fcn_3e_v1( double& obj,
                         Vector3D& g_obj,
                         const Vector3D x[4],
                         const double a,
                         const Exponent& b,
                         const Exponent& c )
{
    static Vector3D my_x[4];

    my_x[0] = x[0];
    my_x[1] = x[2];
    my_x[2] = x[3];
    my_x[3] = x[1];
    return g_fcn_3e_v3( obj, g_obj, my_x, a, b, c );
}

inline bool g_fcn_3e_v2( double& obj,
                         Vector3D& g_obj,
                         const Vector3D x[4],
                         const double a,
                         const Exponent& b,
                         const Exponent& c )
{
    static Vector3D my_x[4];

    my_x[0] = x[1];
    my_x[1] = x[0];
    my_x[2] = x[3];
    my_x[3] = x[2];
    return g_fcn_3e_v3( obj, g_obj, my_x, a, b, c );
}

inline bool h_fcn_3e_v3( double& obj,
                         Vector3D& g_obj,
                         Matrix3D& h_obj,
                         const Vector3D x[4],
                         const double a,
                         const Exponent& b,
                         const Exponent& c )
{
    double matr[9], f, g;
    double dg[9], loc0, /*loc1,*/ loc3, loc4;
    double cross;

    /* Calculate M = A*inv(W). */
    f       = x[1][0] + x[0][0];
    matr[0] = x[1][0] - x[0][0];
    matr[1] = ( 2.0 * x[2][0] - f ) * isqrt3;
    matr[2] = ( 3.0 * x[3][0] - x[2][0] - f ) * isqrt6;

    f       = x[1][1] + x[0][1];
    matr[3] = x[1][1] - x[0][1];
    matr[4] = ( 2.0 * x[2][1] - f ) * isqrt3;
    matr[5] = ( 3.0 * x[3][1] - x[2][1] - f ) * isqrt6;

    f       = x[1][2] + x[0][2];
    matr[6] = x[1][2] - x[0][2];
    matr[7] = ( 2.0 * x[2][2] - f ) * isqrt3;
    matr[8] = ( 3.0 * x[3][2] - x[2][2] - f ) * isqrt6;

    /* Calculate det(M). */
    dg[0] = matr[4] * matr[8] - matr[5] * matr[7];
    dg[1] = matr[5] * matr[6] - matr[3] * matr[8];
    dg[2] = matr[3] * matr[7] - matr[4] * matr[6];
    g     = matr[0] * dg[0] + matr[1] * dg[1] + matr[2] * dg[2];
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    loc3 = f;
    loc4 = g;

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    dg[5] = matr[1] * matr[6] - matr[0] * matr[7];
    dg[8] = matr[0] * matr[4] - matr[1] * matr[3];

    g_obj[0] = tisqrt6 * ( matr[2] * f + dg[2] * g );
    g_obj[1] = tisqrt6 * ( matr[5] * f + dg[5] * g );
    g_obj[2] = tisqrt6 * ( matr[8] * f + dg[8] * g );

    /* Calculate the hessian of the objective.                   */
    loc0 = f; /* Constant on nabla^2 f       */
    //  loc1 = g;           /* Constant on nabla^2 g       */
    cross = f * c.value() / loc4;         /* Constant on nabla g nabla f */
    f     = f * ( b.value() - 1 ) / loc3; /* Constant on nabla f nabla f */
    g     = g * ( c.value() - 1 ) / loc4; /* Constant on nabla g nabla g */
    f *= 2.0;                             /* Modification for nabla f    */

    /* First block of rows */
    loc3 = matr[2] * f + dg[2] * cross;
    loc4 = dg[2] * g + matr[2] * cross;

    h_obj[0][0] = 1.5 * ( loc0 + loc3 * matr[2] + loc4 * dg[2] );
    h_obj[0][1] = 1.5 * ( loc3 * matr[5] + loc4 * dg[5] );
    h_obj[0][2] = 1.5 * ( loc3 * matr[8] + loc4 * dg[8] );

    /* Second block of rows */
    loc3 = matr[5] * f + dg[5] * cross;
    loc4 = dg[5] * g + matr[5] * cross;

    h_obj[1][1] = 1.5 * ( loc0 + loc3 * matr[5] + loc4 * dg[5] );
    h_obj[1][2] = 1.5 * ( loc3 * matr[8] + loc4 * dg[8] );

    /* Third block of rows */
    loc3 = matr[8] * f + dg[8] * cross;
    loc4 = dg[8] * g + matr[8] * cross;

    h_obj[2][2] = 1.5 * ( loc0 + loc3 * matr[8] + loc4 * dg[8] );

    // completes diagonal block.
    h_obj.fill_lower_triangle();
    return true;
}

inline bool h_fcn_3e_v0( double& obj,
                         Vector3D& g_obj,
                         Matrix3D& h_obj,
                         const Vector3D x[4],
                         const double a,
                         const Exponent& b,
                         const Exponent& c )
{
    static Vector3D my_x[4];

    my_x[0] = x[1];
    my_x[1] = x[3];
    my_x[2] = x[2];
    my_x[3] = x[0];
    return h_fcn_3e_v3( obj, g_obj, h_obj, my_x, a, b, c );
}

inline bool h_fcn_3e_v1( double& obj,
                         Vector3D& g_obj,
                         Matrix3D& h_obj,
                         const Vector3D x[4],
                         const double a,
                         const Exponent& b,
                         const Exponent& c )
{
    static Vector3D my_x[4];

    my_x[0] = x[0];
    my_x[1] = x[2];
    my_x[2] = x[3];
    my_x[3] = x[1];
    return h_fcn_3e_v3( obj, g_obj, h_obj, my_x, a, b, c );
}

inline bool h_fcn_3e_v2( double& obj,
                         Vector3D& g_obj,
                         Matrix3D& h_obj,
                         const Vector3D x[4],
                         const double a,
                         const Exponent& b,
                         const Exponent& c )
{
    static Vector3D my_x[4];

    my_x[0] = x[1];
    my_x[1] = x[0];
    my_x[2] = x[3];
    my_x[3] = x[2];
    return h_fcn_3e_v3( obj, g_obj, h_obj, my_x, a, b, c );
}

/*****************************************************************************/
/* The following set of functions reference tetrahedral elements to a        */
/* right tetrahedron.  They are used when assessing the quality of a         */
/* hexahedral element.  A zero return value indicates success, while         */
/* a nonzero value indicates failure.                                        */
/*****************************************************************************/

/*****************************************************************************/
/* Function evaluation requires 53 flops.                                    */
/*   Reductions possible when b == 1, c == 1, or d == 1                      */
/*****************************************************************************/
inline bool m_fcn_3i( double& obj,
                      const Vector3D x[4],
                      const double a,
                      const Exponent& b,
                      const Exponent& c,
                      const Vector3D& d )
{
    double matr[9], f;
    double g;

    /* Calculate M = A*inv(W). */
    matr[0] = d[0] * ( x[1][0] - x[0][0] );
    matr[1] = d[1] * ( x[2][0] - x[0][0] );
    matr[2] = d[2] * ( x[3][0] - x[0][0] );

    matr[3] = d[0] * ( x[1][1] - x[0][1] );
    matr[4] = d[1] * ( x[2][1] - x[0][1] );
    matr[5] = d[2] * ( x[3][1] - x[0][1] );

    matr[6] = d[0] * ( x[1][2] - x[0][2] );
    matr[7] = d[1] * ( x[2][2] - x[0][2] );
    matr[8] = d[2] * ( x[3][2] - x[0][2] );

    /* Calculate det(M). */
    g = matr[0] * ( matr[4] * matr[8] - matr[5] * matr[7] ) + matr[1] * ( matr[5] * matr[6] - matr[3] * matr[8] ) +
        matr[2] * ( matr[3] * matr[7] - matr[4] * matr[6] );
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );
    return true;
}

/*****************************************************************************/
/* Gradient evaluation requires 115 flops.                                   */
/*   Reductions possible when b == 1, c == 1, or d == 1                      */
/*****************************************************************************/
inline bool g_fcn_3i( double& obj,
                      Vector3D g_obj[4],
                      const Vector3D x[4],
                      const double a,
                      const Exponent& b,
                      const Exponent& c,
                      const Vector3D& d )
{
    double matr[9], f;
    double adj_m[9], g;
    double loc1, loc2, loc3;

    /* Calculate M = A*inv(W). */
    matr[0] = d[0] * ( x[1][0] - x[0][0] );
    matr[1] = d[1] * ( x[2][0] - x[0][0] );
    matr[2] = d[2] * ( x[3][0] - x[0][0] );

    matr[3] = d[0] * ( x[1][1] - x[0][1] );
    matr[4] = d[1] * ( x[2][1] - x[0][1] );
    matr[5] = d[2] * ( x[3][1] - x[0][1] );

    matr[6] = d[0] * ( x[1][2] - x[0][2] );
    matr[7] = d[1] * ( x[2][2] - x[0][2] );
    matr[8] = d[2] * ( x[3][2] - x[0][2] );

    /* Calculate det(M). */
    loc1 = matr[4] * matr[8] - matr[5] * matr[7];
    loc2 = matr[5] * matr[6] - matr[3] * matr[8];
    loc3 = matr[3] * matr[7] - matr[4] * matr[6];
    g    = matr[0] * loc1 + matr[1] * loc2 + matr[2] * loc3;
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    adj_m[0] = d[0] * ( matr[0] * f + loc1 * g );
    adj_m[1] = d[1] * ( matr[1] * f + loc2 * g );
    adj_m[2] = d[2] * ( matr[2] * f + loc3 * g );

    loc1 = matr[0] * g;
    loc2 = matr[1] * g;
    loc3 = matr[2] * g;

    adj_m[3] = d[0] * ( matr[3] * f + loc3 * matr[7] - loc2 * matr[8] );
    adj_m[4] = d[1] * ( matr[4] * f + loc1 * matr[8] - loc3 * matr[6] );
    adj_m[5] = d[2] * ( matr[5] * f + loc2 * matr[6] - loc1 * matr[7] );

    adj_m[6] = d[0] * ( matr[6] * f + loc2 * matr[5] - loc3 * matr[4] );
    adj_m[7] = d[1] * ( matr[7] * f + loc3 * matr[3] - loc1 * matr[5] );
    adj_m[8] = d[2] * ( matr[8] * f + loc1 * matr[4] - loc2 * matr[3] );

    g_obj[0][0] = -adj_m[0] - adj_m[1] - adj_m[2];
    g_obj[1][0] = adj_m[0];
    g_obj[2][0] = adj_m[1];
    g_obj[3][0] = adj_m[2];

    g_obj[0][1] = -adj_m[3] - adj_m[4] - adj_m[5];
    g_obj[1][1] = adj_m[3];
    g_obj[2][1] = adj_m[4];
    g_obj[3][1] = adj_m[5];

    g_obj[0][2] = -adj_m[6] - adj_m[7] - adj_m[8];
    g_obj[1][2] = adj_m[6];
    g_obj[2][2] = adj_m[7];
    g_obj[3][2] = adj_m[8];
    return true;
}

/*****************************************************************************/
/* Hessian evaluation requires 469 flops.                                    */
/*   Reductions possible when b == 1, c == 1, or d == 1                      */
/*****************************************************************************/
inline int h_fcn_3i( double& obj,
                     Vector3D g_obj[4],
                     Matrix3D h_obj[10],
                     const Vector3D x[4],
                     const double a,
                     const Exponent& b,
                     const Exponent& c,
                     const Vector3D& d )
{
    double matr[9], f;
    double adj_m[9], g;
    double dg[9], loc0, loc1, loc2, loc3, loc4;
    double A[3], J_A[6], J_B[9], J_C[9], cross;

    const double scale[6] = { d[0] * d[0], d[0] * d[1], d[0] * d[2], d[1] * d[1], d[1] * d[2], d[2] * d[2] };

    /* Calculate M = A*inv(W). */
    matr[0] = d[0] * ( x[1][0] - x[0][0] );
    matr[1] = d[1] * ( x[2][0] - x[0][0] );
    matr[2] = d[2] * ( x[3][0] - x[0][0] );

    matr[3] = d[0] * ( x[1][1] - x[0][1] );
    matr[4] = d[1] * ( x[2][1] - x[0][1] );
    matr[5] = d[2] * ( x[3][1] - x[0][1] );

    matr[6] = d[0] * ( x[1][2] - x[0][2] );
    matr[7] = d[1] * ( x[2][2] - x[0][2] );
    matr[8] = d[2] * ( x[3][2] - x[0][2] );

    /* Calculate det(M). */
    dg[0] = matr[4] * matr[8] - matr[5] * matr[7];
    dg[1] = matr[5] * matr[6] - matr[3] * matr[8];
    dg[2] = matr[3] * matr[7] - matr[4] * matr[6];
    g     = matr[0] * dg[0] + matr[1] * dg[1] + matr[2] * dg[2];
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    loc3 = f;
    loc4 = g;

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    dg[3] = matr[2] * matr[7] - matr[1] * matr[8];
    dg[4] = matr[0] * matr[8] - matr[2] * matr[6];
    dg[5] = matr[1] * matr[6] - matr[0] * matr[7];
    dg[6] = matr[1] * matr[5] - matr[2] * matr[4];
    dg[7] = matr[2] * matr[3] - matr[0] * matr[5];
    dg[8] = matr[0] * matr[4] - matr[1] * matr[3];

    adj_m[0] = d[0] * ( matr[0] * f + dg[0] * g );
    adj_m[1] = d[1] * ( matr[1] * f + dg[1] * g );
    adj_m[2] = d[2] * ( matr[2] * f + dg[2] * g );
    adj_m[3] = d[0] * ( matr[3] * f + dg[3] * g );
    adj_m[4] = d[1] * ( matr[4] * f + dg[4] * g );
    adj_m[5] = d[2] * ( matr[5] * f + dg[5] * g );
    adj_m[6] = d[0] * ( matr[6] * f + dg[6] * g );
    adj_m[7] = d[1] * ( matr[7] * f + dg[7] * g );
    adj_m[8] = d[2] * ( matr[8] * f + dg[8] * g );

    g_obj[0][0] = -adj_m[0] - adj_m[1] - adj_m[2];
    g_obj[1][0] = adj_m[0];
    g_obj[2][0] = adj_m[1];
    g_obj[3][0] = adj_m[2];

    g_obj[0][1] = -adj_m[3] - adj_m[4] - adj_m[5];
    g_obj[1][1] = adj_m[3];
    g_obj[2][1] = adj_m[4];
    g_obj[3][1] = adj_m[5];

    g_obj[0][2] = -adj_m[6] - adj_m[7] - adj_m[8];
    g_obj[1][2] = adj_m[6];
    g_obj[2][2] = adj_m[7];
    g_obj[3][2] = adj_m[8];

    /* Calculate the hessian of the objective.                   */
    loc0  = f;                            /* Constant on nabla^2 f       */
    loc1  = g;                            /* Constant on nabla^2 g       */
    cross = f * c.value() / loc4;         /* Constant on nabla g nabla f */
    f     = f * ( b.value() - 1 ) / loc3; /* Constant on nabla f nabla f */
    g     = g * ( c.value() - 1 ) / loc4; /* Constant on nabla g nabla g */
    f *= 2.0;                             /* Modification for nabla f    */

    /* First block of rows */
    loc3 = matr[0] * f + dg[0] * cross;
    loc4 = dg[0] * g + matr[0] * cross;

    J_A[0] = loc0 + loc3 * matr[0] + loc4 * dg[0];
    J_A[1] = loc3 * matr[1] + loc4 * dg[1];
    J_A[2] = loc3 * matr[2] + loc4 * dg[2];
    J_B[0] = loc3 * matr[3] + loc4 * dg[3];
    J_B[1] = loc3 * matr[4] + loc4 * dg[4];
    J_B[2] = loc3 * matr[5] + loc4 * dg[5];
    J_C[0] = loc3 * matr[6] + loc4 * dg[6];
    J_C[1] = loc3 * matr[7] + loc4 * dg[7];
    J_C[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[1] * f + dg[1] * cross;
    loc4 = dg[1] * g + matr[1] * cross;

    J_A[3] = loc0 + loc3 * matr[1] + loc4 * dg[1];
    J_A[4] = loc3 * matr[2] + loc4 * dg[2];
    J_B[3] = loc3 * matr[3] + loc4 * dg[3];
    J_B[4] = loc3 * matr[4] + loc4 * dg[4];
    J_B[5] = loc3 * matr[5] + loc4 * dg[5];
    J_C[3] = loc3 * matr[6] + loc4 * dg[6];
    J_C[4] = loc3 * matr[7] + loc4 * dg[7];
    J_C[5] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[2] * f + dg[2] * cross;
    loc4 = dg[2] * g + matr[2] * cross;

    J_A[5] = loc0 + loc3 * matr[2] + loc4 * dg[2];
    J_B[6] = loc3 * matr[3] + loc4 * dg[3];
    J_B[7] = loc3 * matr[4] + loc4 * dg[4];
    J_B[8] = loc3 * matr[5] + loc4 * dg[5];
    J_C[6] = loc3 * matr[6] + loc4 * dg[6];
    J_C[7] = loc3 * matr[7] + loc4 * dg[7];
    J_C[8] = loc3 * matr[8] + loc4 * dg[8];

    /* First diagonal block */
    J_A[0] *= scale[0];
    J_A[1] *= scale[1];
    J_A[2] *= scale[2];
    J_A[3] *= scale[3];
    J_A[4] *= scale[4];
    J_A[5] *= scale[5];

    A[0] = -J_A[0] - J_A[1] - J_A[2];
    A[1] = -J_A[1] - J_A[3] - J_A[4];
    A[2] = -J_A[2] - J_A[4] - J_A[5];

    h_obj[0][0][0] = -A[0] - A[1] - A[2];
    h_obj[1][0][0] = A[0];
    h_obj[2][0][0] = A[1];
    h_obj[3][0][0] = A[2];

    h_obj[4][0][0] = J_A[0];
    h_obj[5][0][0] = J_A[1];
    h_obj[6][0][0] = J_A[2];

    h_obj[7][0][0] = J_A[3];
    h_obj[8][0][0] = J_A[4];

    h_obj[9][0][0] = J_A[5];

    /* First off-diagonal block */
    loc2 = matr[8] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    loc2 = matr[7] * loc1;
    J_B[2] -= loc2;
    J_B[6] += loc2;

    loc2 = matr[6] * loc1;
    J_B[5] += loc2;
    J_B[7] -= loc2;

    J_B[0] *= scale[0];
    J_B[1] *= scale[1];
    J_B[2] *= scale[2];
    J_B[3] *= scale[1];
    J_B[4] *= scale[3];
    J_B[5] *= scale[4];
    J_B[6] *= scale[2];
    J_B[7] *= scale[4];
    J_B[8] *= scale[5];

    A[0] = -J_B[0] - J_B[3] - J_B[6];
    A[1] = -J_B[1] - J_B[4] - J_B[7];
    A[2] = -J_B[2] - J_B[5] - J_B[8];

    h_obj[0][0][1] = -A[0] - A[1] - A[2];
    h_obj[1][0][1] = A[0];
    h_obj[2][0][1] = A[1];
    h_obj[3][0][1] = A[2];

    h_obj[1][1][0] = -J_B[0] - J_B[1] - J_B[2];
    h_obj[4][0][1] = J_B[0];
    h_obj[5][0][1] = J_B[1];
    h_obj[6][0][1] = J_B[2];

    h_obj[2][1][0] = -J_B[3] - J_B[4] - J_B[5];
    h_obj[5][1][0] = J_B[3];
    h_obj[7][0][1] = J_B[4];
    h_obj[8][0][1] = J_B[5];

    h_obj[3][1][0] = -J_B[6] - J_B[7] - J_B[8];
    h_obj[6][1][0] = J_B[6];
    h_obj[8][1][0] = J_B[7];
    h_obj[9][0][1] = J_B[8];

    /* Second off-diagonal block */
    loc2 = matr[5] * loc1;
    J_C[1] -= loc2;
    J_C[3] += loc2;

    loc2 = matr[4] * loc1;
    J_C[2] += loc2;
    J_C[6] -= loc2;

    loc2 = matr[3] * loc1;
    J_C[5] -= loc2;
    J_C[7] += loc2;

    J_C[0] *= scale[0];
    J_C[1] *= scale[1];
    J_C[2] *= scale[2];
    J_C[3] *= scale[1];
    J_C[4] *= scale[3];
    J_C[5] *= scale[4];
    J_C[6] *= scale[2];
    J_C[7] *= scale[4];
    J_C[8] *= scale[5];

    A[0] = -J_C[0] - J_C[3] - J_C[6];
    A[1] = -J_C[1] - J_C[4] - J_C[7];
    A[2] = -J_C[2] - J_C[5] - J_C[8];

    h_obj[0][0][2] = -A[0] - A[1] - A[2];
    h_obj[1][0][2] = A[0];
    h_obj[2][0][2] = A[1];
    h_obj[3][0][2] = A[2];

    h_obj[1][2][0] = -J_C[0] - J_C[1] - J_C[2];
    h_obj[4][0][2] = J_C[0];
    h_obj[5][0][2] = J_C[1];
    h_obj[6][0][2] = J_C[2];

    h_obj[2][2][0] = -J_C[3] - J_C[4] - J_C[5];
    h_obj[5][2][0] = J_C[3];
    h_obj[7][0][2] = J_C[4];
    h_obj[8][0][2] = J_C[5];

    h_obj[3][2][0] = -J_C[6] - J_C[7] - J_C[8];
    h_obj[6][2][0] = J_C[6];
    h_obj[8][2][0] = J_C[7];
    h_obj[9][0][2] = J_C[8];

    /* Second block of rows */
    loc3 = matr[3] * f + dg[3] * cross;
    loc4 = dg[3] * g + matr[3] * cross;

    J_A[0] = loc0 + loc3 * matr[3] + loc4 * dg[3];
    J_A[1] = loc3 * matr[4] + loc4 * dg[4];
    J_A[2] = loc3 * matr[5] + loc4 * dg[5];
    J_B[0] = loc3 * matr[6] + loc4 * dg[6];
    J_B[1] = loc3 * matr[7] + loc4 * dg[7];
    J_B[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[4] * f + dg[4] * cross;
    loc4 = dg[4] * g + matr[4] * cross;

    J_A[3] = loc0 + loc3 * matr[4] + loc4 * dg[4];
    J_A[4] = loc3 * matr[5] + loc4 * dg[5];
    J_B[3] = loc3 * matr[6] + loc4 * dg[6];
    J_B[4] = loc3 * matr[7] + loc4 * dg[7];
    J_B[5] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[5] * f + dg[5] * cross;
    loc4 = dg[5] * g + matr[5] * cross;

    J_A[5] = loc0 + loc3 * matr[5] + loc4 * dg[5];
    J_B[6] = loc3 * matr[6] + loc4 * dg[6];
    J_B[7] = loc3 * matr[7] + loc4 * dg[7];
    J_B[8] = loc3 * matr[8] + loc4 * dg[8];

    /* Second diagonal block */
    J_A[0] *= scale[0];
    J_A[1] *= scale[1];
    J_A[2] *= scale[2];
    J_A[3] *= scale[3];
    J_A[4] *= scale[4];
    J_A[5] *= scale[5];

    A[0] = -J_A[0] - J_A[1] - J_A[2];
    A[1] = -J_A[1] - J_A[3] - J_A[4];
    A[2] = -J_A[2] - J_A[4] - J_A[5];

    h_obj[0][1][1] = -A[0] - A[1] - A[2];
    h_obj[1][1][1] = A[0];
    h_obj[2][1][1] = A[1];
    h_obj[3][1][1] = A[2];

    h_obj[4][1][1] = J_A[0];
    h_obj[5][1][1] = J_A[1];
    h_obj[6][1][1] = J_A[2];

    h_obj[7][1][1] = J_A[3];
    h_obj[8][1][1] = J_A[4];

    h_obj[9][1][1] = J_A[5];

    /* Third off-diagonal block */
    loc2 = matr[2] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    loc2 = matr[1] * loc1;
    J_B[2] -= loc2;
    J_B[6] += loc2;

    loc2 = matr[0] * loc1;
    J_B[5] += loc2;
    J_B[7] -= loc2;

    J_B[0] *= scale[0];
    J_B[1] *= scale[1];
    J_B[2] *= scale[2];
    J_B[3] *= scale[1];
    J_B[4] *= scale[3];
    J_B[5] *= scale[4];
    J_B[6] *= scale[2];
    J_B[7] *= scale[4];
    J_B[8] *= scale[5];

    A[0] = -J_B[0] - J_B[3] - J_B[6];
    A[1] = -J_B[1] - J_B[4] - J_B[7];
    A[2] = -J_B[2] - J_B[5] - J_B[8];

    h_obj[0][1][2] = -A[0] - A[1] - A[2];
    h_obj[1][1][2] = A[0];
    h_obj[2][1][2] = A[1];
    h_obj[3][1][2] = A[2];

    h_obj[1][2][1] = -J_B[0] - J_B[1] - J_B[2];
    h_obj[4][1][2] = J_B[0];
    h_obj[5][1][2] = J_B[1];
    h_obj[6][1][2] = J_B[2];

    h_obj[2][2][1] = -J_B[3] - J_B[4] - J_B[5];
    h_obj[5][2][1] = J_B[3];
    h_obj[7][1][2] = J_B[4];
    h_obj[8][1][2] = J_B[5];

    h_obj[3][2][1] = -J_B[6] - J_B[7] - J_B[8];
    h_obj[6][2][1] = J_B[6];
    h_obj[8][2][1] = J_B[7];
    h_obj[9][1][2] = J_B[8];

    /* Third block of rows */
    loc3 = matr[6] * f + dg[6] * cross;
    loc4 = dg[6] * g + matr[6] * cross;

    J_A[0] = loc0 + loc3 * matr[6] + loc4 * dg[6];
    J_A[1] = loc3 * matr[7] + loc4 * dg[7];
    J_A[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[7] * f + dg[7] * cross;
    loc4 = dg[7] * g + matr[7] * cross;

    J_A[3] = loc0 + loc3 * matr[7] + loc4 * dg[7];
    J_A[4] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[8] * f + dg[8] * cross;
    loc4 = dg[8] * g + matr[8] * cross;

    J_A[5] = loc0 + loc3 * matr[8] + loc4 * dg[8];

    /* Third diagonal block */
    J_A[0] *= scale[0];
    J_A[1] *= scale[1];
    J_A[2] *= scale[2];
    J_A[3] *= scale[3];
    J_A[4] *= scale[4];
    J_A[5] *= scale[5];

    A[0] = -J_A[0] - J_A[1] - J_A[2];
    A[1] = -J_A[1] - J_A[3] - J_A[4];
    A[2] = -J_A[2] - J_A[4] - J_A[5];

    h_obj[0][2][2] = -A[0] - A[1] - A[2];
    h_obj[1][2][2] = A[0];
    h_obj[2][2][2] = A[1];
    h_obj[3][2][2] = A[2];

    h_obj[4][2][2] = J_A[0];
    h_obj[5][2][2] = J_A[1];
    h_obj[6][2][2] = J_A[2];

    h_obj[7][2][2] = J_A[3];
    h_obj[8][2][2] = J_A[4];

    h_obj[9][2][2] = J_A[5];

    // completes diagonal blocks.
    h_obj[0].fill_lower_triangle();
    h_obj[4].fill_lower_triangle();
    h_obj[7].fill_lower_triangle();
    h_obj[9].fill_lower_triangle();
    return true;
}

/*********************************************************************
 * Reference tetrahedral elements to halves of an ideal pyramid.
 * Vertices should be ordered such that the edge between the 2nd and
 * 3rd vertices is the one longer edge in the reference tetrahedron
 * (the diagonal of the base of the pyramid of which the tetrahedron
 * is one half of).
 *      1  0  1/2                     1  0 -1/sqrt(2)
 * W =  0  1  1/2           inv(W) =  0  1 -1/sqrt(2)
 *      0  0  1/sqrt(2)               0  0  sqrt(2)
 *
 *********************************************************************/
inline bool m_fcn_3p( double& obj, const Vector3D x[4], const double a, const Exponent& b, const Exponent& c )
{
    const double h = 0.5; /* h = 1 / (2*height) */

    double matr[9], f;
    double g;

    /* Calculate M = A*inv(W). */
    matr[0] = x[1][0] - x[0][0];
    matr[1] = x[2][0] - x[0][0];
    matr[2] = ( 2.0 * x[3][0] - x[1][0] - x[2][0] ) * h;

    matr[3] = x[1][1] - x[0][1];
    matr[4] = x[2][1] - x[0][1];
    matr[5] = ( 2.0 * x[3][1] - x[1][1] - x[2][1] ) * h;

    matr[6] = x[1][2] - x[0][2];
    matr[7] = x[2][2] - x[0][2];
    matr[8] = ( 2.0 * x[3][2] - x[1][2] - x[2][2] ) * h;

    /* Calculate det(M). */
    g = matr[0] * ( matr[4] * matr[8] - matr[5] * matr[7] ) + matr[1] * ( matr[5] * matr[6] - matr[3] * matr[8] ) +
        matr[2] * ( matr[3] * matr[7] - matr[4] * matr[6] );
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );
    return true;
}

inline bool g_fcn_3p( double& obj,
                      Vector3D g_obj[4],
                      const Vector3D x[4],
                      const double a,
                      const Exponent& b,
                      const Exponent& c )
{
    const double h  = 0.5; /* h = 1 / (2*height) */
    const double th = 1.0; /* h = 1 / (height)   */

    double matr[9], f;
    double adj_m[9], g;
    double loc1, loc2, loc3;

    /* Calculate M = A*inv(W). */
    matr[0] = x[1][0] - x[0][0];
    matr[1] = x[2][0] - x[0][0];
    matr[2] = ( 2.0 * x[3][0] - x[1][0] - x[2][0] ) * h;

    matr[3] = x[1][1] - x[0][1];
    matr[4] = x[2][1] - x[0][1];
    matr[5] = ( 2.0 * x[3][1] - x[1][1] - x[2][1] ) * h;

    matr[6] = x[1][2] - x[0][2];
    matr[7] = x[2][2] - x[0][2];
    matr[8] = ( 2.0 * x[3][2] - x[1][2] - x[2][2] ) * h;

    /* Calculate det(M). */
    loc1 = matr[4] * matr[8] - matr[5] * matr[7];
    loc2 = matr[5] * matr[6] - matr[3] * matr[8];
    loc3 = matr[3] * matr[7] - matr[4] * matr[6];
    g    = matr[0] * loc1 + matr[1] * loc2 + matr[2] * loc3;
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    adj_m[0] = matr[0] * f + loc1 * g;
    adj_m[1] = matr[1] * f + loc2 * g;
    adj_m[2] = matr[2] * f + loc3 * g;

    loc1 = matr[0] * g;
    loc2 = matr[1] * g;
    loc3 = matr[2] * g;

    adj_m[3] = matr[3] * f + loc3 * matr[7] - loc2 * matr[8];
    adj_m[4] = matr[4] * f + loc1 * matr[8] - loc3 * matr[6];
    adj_m[5] = matr[5] * f + loc2 * matr[6] - loc1 * matr[7];

    adj_m[6] = matr[6] * f + loc2 * matr[5] - loc3 * matr[4];
    adj_m[7] = matr[7] * f + loc3 * matr[3] - loc1 * matr[5];
    adj_m[8] = matr[8] * f + loc1 * matr[4] - loc2 * matr[3];

    g_obj[0][0] = -adj_m[0] - adj_m[1];
    g_obj[1][0] = adj_m[0] - h * adj_m[2];
    g_obj[2][0] = adj_m[1] - h * adj_m[2];
    g_obj[3][0] = th * adj_m[2];

    g_obj[0][1] = -adj_m[3] - adj_m[4];
    g_obj[1][1] = adj_m[3] - h * adj_m[5];
    g_obj[2][1] = adj_m[4] - h * adj_m[5];
    g_obj[3][1] = th * adj_m[5];

    g_obj[0][2] = -adj_m[6] - adj_m[7];
    g_obj[1][2] = adj_m[6] - h * adj_m[8];
    g_obj[2][2] = adj_m[7] - h * adj_m[8];
    g_obj[3][2] = th * adj_m[8];
    return true;
}

inline bool h_fcn_3p( double& obj,
                      Vector3D g_obj[4],
                      Matrix3D h_obj[10],
                      const Vector3D x[4],
                      const double a,
                      const Exponent& b,
                      const Exponent& c )
{
    const double h  = 0.5; /* h = 1 / (2*height) */
    const double th = 1.0; /* h = 1 / (height)   */

    double matr[9], f;
    double adj_m[9], g;
    double dg[9], loc0, loc1, loc2, loc3, loc4;
    double A[12], J_A[6], J_B[9], J_C[9], cross;

    /* Calculate M = A*inv(W). */
    matr[0] = x[1][0] - x[0][0];
    matr[1] = x[2][0] - x[0][0];
    matr[2] = ( 2.0 * x[3][0] - x[1][0] - x[2][0] ) * h;

    matr[3] = x[1][1] - x[0][1];
    matr[4] = x[2][1] - x[0][1];
    matr[5] = ( 2.0 * x[3][1] - x[1][1] - x[2][1] ) * h;

    matr[6] = x[1][2] - x[0][2];
    matr[7] = x[2][2] - x[0][2];
    matr[8] = ( 2.0 * x[3][2] - x[1][2] - x[2][2] ) * h;

    /* Calculate det(M). */
    dg[0] = matr[4] * matr[8] - matr[5] * matr[7];
    dg[1] = matr[5] * matr[6] - matr[3] * matr[8];
    dg[2] = matr[3] * matr[7] - matr[4] * matr[6];
    g     = matr[0] * dg[0] + matr[1] * dg[1] + matr[2] * dg[2];
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    dg[3] = matr[2] * matr[7] - matr[1] * matr[8];
    dg[4] = matr[0] * matr[8] - matr[2] * matr[6];
    dg[5] = matr[1] * matr[6] - matr[0] * matr[7];
    dg[6] = matr[1] * matr[5] - matr[2] * matr[4];
    dg[7] = matr[2] * matr[3] - matr[0] * matr[5];
    dg[8] = matr[0] * matr[4] - matr[1] * matr[3];

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    loc3 = f;
    loc4 = g;

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    adj_m[0] = matr[0] * f + dg[0] * g;
    adj_m[1] = matr[1] * f + dg[1] * g;
    adj_m[2] = matr[2] * f + dg[2] * g;
    adj_m[3] = matr[3] * f + dg[3] * g;
    adj_m[4] = matr[4] * f + dg[4] * g;
    adj_m[5] = matr[5] * f + dg[5] * g;
    adj_m[6] = matr[6] * f + dg[6] * g;
    adj_m[7] = matr[7] * f + dg[7] * g;
    adj_m[8] = matr[8] * f + dg[8] * g;

    g_obj[0][0] = -adj_m[0] - adj_m[1];
    g_obj[1][0] = adj_m[0] - h * adj_m[2];
    g_obj[2][0] = adj_m[1] - h * adj_m[2];
    g_obj[3][0] = th * adj_m[2];

    g_obj[0][1] = -adj_m[3] - adj_m[4];
    g_obj[1][1] = adj_m[3] - h * adj_m[5];
    g_obj[2][1] = adj_m[4] - h * adj_m[5];
    g_obj[3][1] = th * adj_m[5];

    g_obj[0][2] = -adj_m[6] - adj_m[7];
    g_obj[1][2] = adj_m[6] - h * adj_m[8];
    g_obj[2][2] = adj_m[7] - h * adj_m[8];
    g_obj[3][2] = th * adj_m[8];

    /* Calculate the hessian of the objective.                   */
    loc0  = f;                            /* Constant on nabla^2 f       */
    loc1  = g;                            /* Constant on nabla^2 g       */
    cross = f * c.value() / loc4;         /* Constant on nabla g nabla f */
    f     = f * ( b.value() - 1 ) / loc3; /* Constant on nabla f nabla f */
    g     = g * ( c.value() - 1 ) / loc4; /* Constant on nabla g nabla g */
    f *= 2.0;                             /* Modification for nabla f    */

    /* First block of rows */
    loc3 = matr[0] * f + dg[0] * cross;
    loc4 = dg[0] * g + matr[0] * cross;

    J_A[0] = loc0 + loc3 * matr[0] + loc4 * dg[0];
    J_A[1] = loc3 * matr[1] + loc4 * dg[1];
    J_A[2] = loc3 * matr[2] + loc4 * dg[2];
    J_B[0] = loc3 * matr[3] + loc4 * dg[3];
    J_B[1] = loc3 * matr[4] + loc4 * dg[4];
    J_B[2] = loc3 * matr[5] + loc4 * dg[5];
    J_C[0] = loc3 * matr[6] + loc4 * dg[6];
    J_C[1] = loc3 * matr[7] + loc4 * dg[7];
    J_C[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[1] * f + dg[1] * cross;
    loc4 = dg[1] * g + matr[1] * cross;

    J_A[3] = loc0 + loc3 * matr[1] + loc4 * dg[1];
    J_A[4] = loc3 * matr[2] + loc4 * dg[2];
    J_B[3] = loc3 * matr[3] + loc4 * dg[3];
    J_B[4] = loc3 * matr[4] + loc4 * dg[4];
    J_B[5] = loc3 * matr[5] + loc4 * dg[5];
    J_C[3] = loc3 * matr[6] + loc4 * dg[6];
    J_C[4] = loc3 * matr[7] + loc4 * dg[7];
    J_C[5] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[2] * f + dg[2] * cross;
    loc4 = dg[2] * g + matr[2] * cross;

    J_A[5] = loc0 + loc3 * matr[2] + loc4 * dg[2];
    J_B[6] = loc3 * matr[3] + loc4 * dg[3];
    J_B[7] = loc3 * matr[4] + loc4 * dg[4];
    J_B[8] = loc3 * matr[5] + loc4 * dg[5];
    J_C[6] = loc3 * matr[6] + loc4 * dg[6];
    J_C[7] = loc3 * matr[7] + loc4 * dg[7];
    J_C[8] = loc3 * matr[8] + loc4 * dg[8];

    /* First diagonal block */
    A[0] = -J_A[0] - J_A[1];
    A[1] = J_A[0] - h * J_A[2];
    A[2] = J_A[1] - h * J_A[2];
    A[3] = th * J_A[2];

    A[4] = -J_A[1] - J_A[3];
    A[5] = J_A[1] - h * J_A[4];
    A[6] = J_A[3] - h * J_A[4];
    A[7] = th * J_A[4];

    A[8]  = -J_A[2] - J_A[4];
    A[9]  = J_A[2] - h * J_A[5];
    A[10] = J_A[4] - h * J_A[5];
    A[11] = th * J_A[5];

    h_obj[0][0][0] = -A[0] - A[4];
    h_obj[1][0][0] = A[0] - h * A[8];
    h_obj[2][0][0] = A[4] - h * A[8];
    h_obj[3][0][0] = th * A[8];

    h_obj[4][0][0] = A[1] - h * A[9];
    h_obj[5][0][0] = A[5] - h * A[9];
    h_obj[6][0][0] = th * A[9];

    h_obj[7][0][0] = A[6] - h * A[10];
    h_obj[8][0][0] = th * A[10];

    h_obj[9][0][0] = th * A[11];

    /* First off-diagonal block */
    loc2 = matr[8] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    loc2 = matr[7] * loc1;
    J_B[2] -= loc2;
    J_B[6] += loc2;

    loc2 = matr[6] * loc1;
    J_B[5] += loc2;
    J_B[7] -= loc2;

    A[0] = -J_B[0] - J_B[1];
    A[1] = J_B[0] - h * J_B[2];
    A[2] = J_B[1] - h * J_B[2];
    A[3] = th * J_B[2];

    A[4] = -J_B[3] - J_B[4];
    A[5] = J_B[3] - h * J_B[5];
    A[6] = J_B[4] - h * J_B[5];
    A[7] = th * J_B[5];

    A[8]  = -J_B[6] - J_B[7];
    A[9]  = J_B[6] - h * J_B[8];
    A[10] = J_B[7] - h * J_B[8];
    A[11] = th * J_B[8];

    h_obj[0][0][1] = -A[0] - A[4];
    h_obj[1][1][0] = A[0] - h * A[8];
    h_obj[2][1][0] = A[4] - h * A[8];
    h_obj[3][1][0] = th * A[8];

    h_obj[1][0][1] = -A[1] - A[5];
    h_obj[4][0][1] = A[1] - h * A[9];
    h_obj[5][1][0] = A[5] - h * A[9];
    h_obj[6][1][0] = th * A[9];

    h_obj[2][0][1] = -A[2] - A[6];
    h_obj[5][0][1] = A[2] - h * A[10];
    h_obj[7][0][1] = A[6] - h * A[10];
    h_obj[8][1][0] = th * A[10];

    h_obj[3][0][1] = -A[3] - A[7];
    h_obj[6][0][1] = A[3] - h * A[11];
    h_obj[8][0][1] = A[7] - h * A[11];
    h_obj[9][0][1] = th * A[11];

    /* Second off-diagonal block */
    loc2 = matr[5] * loc1;
    J_C[1] -= loc2;
    J_C[3] += loc2;

    loc2 = matr[4] * loc1;
    J_C[2] += loc2;
    J_C[6] -= loc2;

    loc2 = matr[3] * loc1;
    J_C[5] -= loc2;
    J_C[7] += loc2;

    A[0] = -J_C[0] - J_C[1];
    A[1] = J_C[0] - h * J_C[2];
    A[2] = J_C[1] - h * J_C[2];
    A[3] = th * J_C[2];

    A[4] = -J_C[3] - J_C[4];
    A[5] = J_C[3] - h * J_C[5];
    A[6] = J_C[4] - h * J_C[5];
    A[7] = th * J_C[5];

    A[8]  = -J_C[6] - J_C[7];
    A[9]  = J_C[6] - h * J_C[8];
    A[10] = J_C[7] - h * J_C[8];
    A[11] = th * J_C[8];

    h_obj[0][0][2] = -A[0] - A[4];
    h_obj[1][2][0] = A[0] - h * A[8];
    h_obj[2][2][0] = A[4] - h * A[8];
    h_obj[3][2][0] = th * A[8];

    h_obj[1][0][2] = -A[1] - A[5];
    h_obj[4][0][2] = A[1] - h * A[9];
    h_obj[5][2][0] = A[5] - h * A[9];
    h_obj[6][2][0] = th * A[9];

    h_obj[2][0][2] = -A[2] - A[6];
    h_obj[5][0][2] = A[2] - h * A[10];
    h_obj[7][0][2] = A[6] - h * A[10];
    h_obj[8][2][0] = th * A[10];

    h_obj[3][0][2] = -A[3] - A[7];
    h_obj[6][0][2] = A[3] - h * A[11];
    h_obj[8][0][2] = A[7] - h * A[11];
    h_obj[9][0][2] = th * A[11];

    /* Second block of rows */
    loc3 = matr[3] * f + dg[3] * cross;
    loc4 = dg[3] * g + matr[3] * cross;

    J_A[0] = loc0 + loc3 * matr[3] + loc4 * dg[3];
    J_A[1] = loc3 * matr[4] + loc4 * dg[4];
    J_A[2] = loc3 * matr[5] + loc4 * dg[5];
    J_B[0] = loc3 * matr[6] + loc4 * dg[6];
    J_B[1] = loc3 * matr[7] + loc4 * dg[7];
    J_B[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[4] * f + dg[4] * cross;
    loc4 = dg[4] * g + matr[4] * cross;

    J_A[3] = loc0 + loc3 * matr[4] + loc4 * dg[4];
    J_A[4] = loc3 * matr[5] + loc4 * dg[5];
    J_B[3] = loc3 * matr[6] + loc4 * dg[6];
    J_B[4] = loc3 * matr[7] + loc4 * dg[7];
    J_B[5] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[5] * f + dg[5] * cross;
    loc4 = dg[5] * g + matr[5] * cross;

    J_A[5] = loc0 + loc3 * matr[5] + loc4 * dg[5];
    J_B[6] = loc3 * matr[6] + loc4 * dg[6];
    J_B[7] = loc3 * matr[7] + loc4 * dg[7];
    J_B[8] = loc3 * matr[8] + loc4 * dg[8];

    /* Second diagonal block */
    A[0] = -J_A[0] - J_A[1];
    A[1] = J_A[0] - h * J_A[2];
    A[2] = J_A[1] - h * J_A[2];
    A[3] = th * J_A[2];

    A[4] = -J_A[1] - J_A[3];
    A[5] = J_A[1] - h * J_A[4];
    A[6] = J_A[3] - h * J_A[4];
    A[7] = th * J_A[4];

    A[8]  = -J_A[2] - J_A[4];
    A[9]  = J_A[2] - h * J_A[5];
    A[10] = J_A[4] - h * J_A[5];
    A[11] = th * J_A[5];

    h_obj[0][1][1] = -A[0] - A[4];
    h_obj[1][1][1] = A[0] - h * A[8];
    h_obj[2][1][1] = A[4] - h * A[8];
    h_obj[3][1][1] = th * A[8];

    h_obj[4][1][1] = A[1] - h * A[9];
    h_obj[5][1][1] = A[5] - h * A[9];
    h_obj[6][1][1] = th * A[9];

    h_obj[7][1][1] = A[6] - h * A[10];
    h_obj[8][1][1] = th * A[10];

    h_obj[9][1][1] = th * A[11];

    /* Third off-diagonal block */
    loc2 = matr[2] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    loc2 = matr[1] * loc1;
    J_B[2] -= loc2;
    J_B[6] += loc2;

    loc2 = matr[0] * loc1;
    J_B[5] += loc2;
    J_B[7] -= loc2;

    A[0] = -J_B[0] - J_B[1];
    A[1] = J_B[0] - h * J_B[2];
    A[2] = J_B[1] - h * J_B[2];
    A[3] = th * J_B[2];

    A[4] = -J_B[3] - J_B[4];
    A[5] = J_B[3] - h * J_B[5];
    A[6] = J_B[4] - h * J_B[5];
    A[7] = th * J_B[5];

    A[8]  = -J_B[6] - J_B[7];
    A[9]  = J_B[6] - h * J_B[8];
    A[10] = J_B[7] - h * J_B[8];
    A[11] = th * J_B[8];

    h_obj[0][1][2] = -A[0] - A[4];
    h_obj[1][2][1] = A[0] - h * A[8];
    h_obj[2][2][1] = A[4] - h * A[8];
    h_obj[3][2][1] = th * A[8];

    h_obj[1][1][2] = -A[1] - A[5];
    h_obj[4][1][2] = A[1] - h * A[9];
    h_obj[5][2][1] = A[5] - h * A[9];
    h_obj[6][2][1] = th * A[9];

    h_obj[2][1][2] = -A[2] - A[6];
    h_obj[5][1][2] = A[2] - h * A[10];
    h_obj[7][1][2] = A[6] - h * A[10];
    h_obj[8][2][1] = th * A[10];

    h_obj[3][1][2] = -A[3] - A[7];
    h_obj[6][1][2] = A[3] - h * A[11];
    h_obj[8][1][2] = A[7] - h * A[11];
    h_obj[9][1][2] = th * A[11];

    /* Third block of rows */
    loc3 = matr[6] * f + dg[6] * cross;
    loc4 = dg[6] * g + matr[6] * cross;

    J_A[0] = loc0 + loc3 * matr[6] + loc4 * dg[6];
    J_A[1] = loc3 * matr[7] + loc4 * dg[7];
    J_A[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[7] * f + dg[7] * cross;
    loc4 = dg[7] * g + matr[7] * cross;

    J_A[3] = loc0 + loc3 * matr[7] + loc4 * dg[7];
    J_A[4] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[8] * f + dg[8] * cross;
    loc4 = dg[8] * g + matr[8] * cross;

    J_A[5] = loc0 + loc3 * matr[8] + loc4 * dg[8];

    /* Third diagonal block */
    A[0] = -J_A[0] - J_A[1];
    A[1] = J_A[0] - h * J_A[2];
    A[2] = J_A[1] - h * J_A[2];
    A[3] = th * J_A[2];

    A[4] = -J_A[1] - J_A[3];
    A[5] = J_A[1] - h * J_A[4];
    A[6] = J_A[3] - h * J_A[4];
    A[7] = th * J_A[4];

    A[8]  = -J_A[2] - J_A[4];
    A[9]  = J_A[2] - h * J_A[5];
    A[10] = J_A[4] - h * J_A[5];
    A[11] = th * J_A[5];

    h_obj[0][2][2] = -A[0] - A[4];
    h_obj[1][2][2] = A[0] - h * A[8];
    h_obj[2][2][2] = A[4] - h * A[8];
    h_obj[3][2][2] = th * A[8];

    h_obj[4][2][2] = A[1] - h * A[9];
    h_obj[5][2][2] = A[5] - h * A[9];
    h_obj[6][2][2] = th * A[9];

    h_obj[7][2][2] = A[6] - h * A[10];
    h_obj[8][2][2] = th * A[10];

    h_obj[9][2][2] = th * A[11];

#if 0
  int i, j, k, l;
  double   nobj;
  Vector3D nx[4];
  Vector3D ng_obj[4];

  for (i = 0; i < 4; ++i) {
    nx[i] = x[i];
    ng_obj[i] = 0.0;
  }

  m_fcn_3p(obj, x, a, b, c);
  for (i = 0; i < 4; ++i) {
    for (j = 0; j < 3; ++j) {
      nx[i][j] += 1e-6;
      m_fcn_3p(nobj, nx, a, b, c);
      nx[i][j] = x[i][j];

      ng_obj[i][j] = (nobj - obj) / 1e-6;
      std::cout << i << " " << j << " " << g_obj[i][j] << " " << ng_obj[i][j] << std::endl;
    }
  }
  std::cout << std::endl;

  for (i = 0; i < 4; ++i) {
    for (j = 0; j < 3; ++j) {
      nx[i][j] += 1e-6;
      g_fcn_3p(nobj, ng_obj, nx, a, b, c);
      nx[i][j] = x[i][j];

      printf("%d %d", i, j);
      for (k = 0; k < 4; ++k) {
    for (l = 0; l < 3; ++l) {
      printf(" % 5.4e", (ng_obj[k][l] - g_obj[k][l]) / 1e-6);
    }
      }
      printf("\n");
    }
  }

  for (i = 0; i < 10; ++i) {
    std::cout << i << std::endl << h_obj[i] << std::endl << std::endl;
  }
#endif

    // completes diagonal blocks.
    h_obj[0].fill_lower_triangle();
    h_obj[4].fill_lower_triangle();
    h_obj[7].fill_lower_triangle();
    h_obj[9].fill_lower_triangle();
    return true;
}

/*********************************************************************
 * Reference tetrahedral elements to corners of an ideal wedge element.
 * Vertices should be ordered such that the first three vertices form
 * the ideally-equaliteral face of the tetrahedron (the end of the
 * wedge) and the first and fourth vertices form the tetrahedral edge
 * orthogonal to the ideally-equaliteral face (the lateral edge of the
 * edge.)
 *      1  1/2        0                     1  -1/sqrt(3)  0
 * W =  0  sqrt(3)/2  0           inv(W) =  0   2/sqrt(3)  0
 *      0  0          1                     0   0          1
 *
 *********************************************************************/
inline bool m_fcn_3w( double& obj, const Vector3D x[4], const double a, const Exponent& b, const Exponent& c )
{
    double matr[9], f, g;
    double loc1, loc2, loc3;

    /* Calculate M = A*inv(W). */
    matr[0] = x[1][0] - x[0][0];
    matr[1] = isqrt3 * ( 2 * x[2][0] - x[1][0] - x[0][0] );
    matr[2] = x[3][0] - x[0][0];

    matr[3] = x[1][1] - x[0][1];
    matr[4] = isqrt3 * ( 2 * x[2][1] - x[1][1] - x[0][1] );
    matr[5] = x[3][1] - x[0][1];

    matr[6] = x[1][2] - x[0][2];
    matr[7] = isqrt3 * ( 2 * x[2][2] - x[1][2] - x[0][2] );
    matr[8] = x[3][2] - x[0][2];

    /* Calculate det(M). */
    loc1 = matr[4] * matr[8] - matr[5] * matr[7];
    loc2 = matr[5] * matr[6] - matr[3] * matr[8];
    loc3 = matr[3] * matr[7] - matr[4] * matr[6];
    g    = matr[0] * loc1 + matr[1] * loc2 + matr[2] * loc3;
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );
    return true;
}

inline bool g_fcn_3w( double& obj,
                      Vector3D g_obj[4],
                      const Vector3D x[4],
                      const double a,
                      const Exponent& b,
                      const Exponent& c )
{
    double matr[9], f;
    double adj_m[9], g;
    double loc1, loc2, loc3;

    /* Calculate M = A*inv(W). */
    matr[0] = x[1][0] - x[0][0];
    matr[1] = isqrt3 * ( 2 * x[2][0] - x[1][0] - x[0][0] );
    matr[2] = x[3][0] - x[0][0];

    matr[3] = x[1][1] - x[0][1];
    matr[4] = isqrt3 * ( 2 * x[2][1] - x[1][1] - x[0][1] );
    matr[5] = x[3][1] - x[0][1];

    matr[6] = x[1][2] - x[0][2];
    matr[7] = isqrt3 * ( 2 * x[2][2] - x[1][2] - x[0][2] );
    matr[8] = x[3][2] - x[0][2];

    /* Calculate det(M). */
    loc1 = matr[4] * matr[8] - matr[5] * matr[7];
    loc2 = matr[5] * matr[6] - matr[3] * matr[8];
    loc3 = matr[3] * matr[7] - matr[4] * matr[6];
    g    = matr[0] * loc1 + matr[1] * loc2 + matr[2] * loc3;
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    adj_m[0] = matr[0] * f + loc1 * g;
    adj_m[1] = matr[1] * f + loc2 * g;
    adj_m[2] = matr[2] * f + loc3 * g;

    loc1 = matr[0] * g;
    loc2 = matr[1] * g;
    loc3 = matr[2] * g;

    adj_m[3] = matr[3] * f + loc3 * matr[7] - loc2 * matr[8];
    adj_m[4] = matr[4] * f + loc1 * matr[8] - loc3 * matr[6];
    adj_m[5] = matr[5] * f + loc2 * matr[6] - loc1 * matr[7];

    adj_m[6] = matr[6] * f + loc2 * matr[5] - loc3 * matr[4];
    adj_m[7] = matr[7] * f + loc3 * matr[3] - loc1 * matr[5];
    adj_m[8] = matr[8] * f + loc1 * matr[4] - loc2 * matr[3];

    g_obj[0][0] = -adj_m[0] - isqrt3 * adj_m[1] - adj_m[2];
    g_obj[1][0] = adj_m[0] - isqrt3 * adj_m[1];
    g_obj[2][0] = tisqrt3 * adj_m[1];
    g_obj[3][0] = adj_m[2];

    g_obj[0][1] = -adj_m[3] - isqrt3 * adj_m[4] - adj_m[5];
    g_obj[1][1] = adj_m[3] - isqrt3 * adj_m[4];
    g_obj[2][1] = tisqrt3 * adj_m[4];
    g_obj[3][1] = adj_m[5];

    g_obj[0][2] = -adj_m[6] - isqrt3 * adj_m[7] - adj_m[8];
    g_obj[1][2] = adj_m[6] - isqrt3 * adj_m[7];
    g_obj[2][2] = tisqrt3 * adj_m[7];
    g_obj[3][2] = adj_m[8];

    return true;
}

inline bool h_fcn_3w( double& obj,
                      Vector3D g_obj[4],
                      Matrix3D h_obj[10],
                      const Vector3D x[4],
                      const double a,
                      const Exponent& b,
                      const Exponent& c )
{
    double matr[9], f;
    double adj_m[9], g;
    double dg[9], loc0, loc1, loc2, loc3, loc4;
    double A[12], J_A[6], J_B[9], J_C[9], cross;

    /* Calculate M = A*inv(W). */
    matr[0] = x[1][0] - x[0][0];
    matr[1] = isqrt3 * ( 2 * x[2][0] - x[1][0] - x[0][0] );
    matr[2] = x[3][0] - x[0][0];

    matr[3] = x[1][1] - x[0][1];
    matr[4] = isqrt3 * ( 2 * x[2][1] - x[1][1] - x[0][1] );
    matr[5] = x[3][1] - x[0][1];

    matr[6] = x[1][2] - x[0][2];
    matr[7] = isqrt3 * ( 2 * x[2][2] - x[1][2] - x[0][2] );
    matr[8] = x[3][2] - x[0][2];

    /* Calculate det(M). */
    dg[0] = matr[4] * matr[8] - matr[5] * matr[7];
    dg[1] = matr[5] * matr[6] - matr[3] * matr[8];
    dg[2] = matr[3] * matr[7] - matr[4] * matr[6];
    g     = matr[0] * dg[0] + matr[1] * dg[1] + matr[2] * dg[2];
    if( g < MSQ_MIN )
    {
        obj = g;
        return false;
    }

    dg[3] = matr[2] * matr[7] - matr[1] * matr[8];
    dg[4] = matr[0] * matr[8] - matr[2] * matr[6];
    dg[5] = matr[1] * matr[6] - matr[0] * matr[7];
    dg[6] = matr[1] * matr[5] - matr[2] * matr[4];
    dg[7] = matr[2] * matr[3] - matr[0] * matr[5];
    dg[8] = matr[0] * matr[4] - matr[1] * matr[3];

    /* Calculate norm(M). */
    f = matr[0] * matr[0] + matr[1] * matr[1] + matr[2] * matr[2] + matr[3] * matr[3] + matr[4] * matr[4] +
        matr[5] * matr[5] + matr[6] * matr[6] + matr[7] * matr[7] + matr[8] * matr[8];

    loc3 = f;
    loc4 = g;

    /* Calculate objective function. */
    obj = a * b.raise( f ) * c.raise( g );

    /* Calculate the derivative of the objective function.    */
    f = b.value() * obj / f; /* Constant on nabla f      */
    g = c.value() * obj / g; /* Constant on nable g      */
    f *= 2.0;                /* Modification for nabla f */

    adj_m[0] = matr[0] * f + dg[0] * g;
    adj_m[1] = matr[1] * f + dg[1] * g;
    adj_m[2] = matr[2] * f + dg[2] * g;
    adj_m[3] = matr[3] * f + dg[3] * g;
    adj_m[4] = matr[4] * f + dg[4] * g;
    adj_m[5] = matr[5] * f + dg[5] * g;
    adj_m[6] = matr[6] * f + dg[6] * g;
    adj_m[7] = matr[7] * f + dg[7] * g;
    adj_m[8] = matr[8] * f + dg[8] * g;

    g_obj[0][0] = -adj_m[0] - isqrt3 * adj_m[1] - adj_m[2];
    g_obj[1][0] = adj_m[0] - isqrt3 * adj_m[1];
    g_obj[2][0] = tisqrt3 * adj_m[1];
    g_obj[3][0] = adj_m[2];

    g_obj[0][1] = -adj_m[3] - isqrt3 * adj_m[4] - adj_m[5];
    g_obj[1][1] = adj_m[3] - isqrt3 * adj_m[4];
    g_obj[2][1] = tisqrt3 * adj_m[4];
    g_obj[3][1] = adj_m[5];

    g_obj[0][2] = -adj_m[6] - isqrt3 * adj_m[7] - adj_m[8];
    g_obj[1][2] = adj_m[6] - isqrt3 * adj_m[7];
    g_obj[2][2] = tisqrt3 * adj_m[7];
    g_obj[3][2] = adj_m[8];

    /* Calculate the hessian of the objective.                   */
    loc0  = f;                            /* Constant on nabla^2 f       */
    loc1  = g;                            /* Constant on nabla^2 g       */
    cross = f * c.value() / loc4;         /* Constant on nabla g nabla f */
    f     = f * ( b.value() - 1 ) / loc3; /* Constant on nabla f nabla f */
    g     = g * ( c.value() - 1 ) / loc4; /* Constant on nabla g nabla g */
    f *= 2.0;                             /* Modification for nabla f    */

    /* First block of rows */
    loc3 = matr[0] * f + dg[0] * cross;
    loc4 = dg[0] * g + matr[0] * cross;

    J_A[0] = loc0 + loc3 * matr[0] + loc4 * dg[0];
    J_A[1] = loc3 * matr[1] + loc4 * dg[1];
    J_A[2] = loc3 * matr[2] + loc4 * dg[2];
    J_B[0] = loc3 * matr[3] + loc4 * dg[3];
    J_B[1] = loc3 * matr[4] + loc4 * dg[4];
    J_B[2] = loc3 * matr[5] + loc4 * dg[5];
    J_C[0] = loc3 * matr[6] + loc4 * dg[6];
    J_C[1] = loc3 * matr[7] + loc4 * dg[7];
    J_C[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[1] * f + dg[1] * cross;
    loc4 = dg[1] * g + matr[1] * cross;

    J_A[3] = loc0 + loc3 * matr[1] + loc4 * dg[1];
    J_A[4] = loc3 * matr[2] + loc4 * dg[2];
    J_B[3] = loc3 * matr[3] + loc4 * dg[3];
    J_B[4] = loc3 * matr[4] + loc4 * dg[4];
    J_B[5] = loc3 * matr[5] + loc4 * dg[5];
    J_C[3] = loc3 * matr[6] + loc4 * dg[6];
    J_C[4] = loc3 * matr[7] + loc4 * dg[7];
    J_C[5] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[2] * f + dg[2] * cross;
    loc4 = dg[2] * g + matr[2] * cross;

    J_A[5] = loc0 + loc3 * matr[2] + loc4 * dg[2];
    J_B[6] = loc3 * matr[3] + loc4 * dg[3];
    J_B[7] = loc3 * matr[4] + loc4 * dg[4];
    J_B[8] = loc3 * matr[5] + loc4 * dg[5];
    J_C[6] = loc3 * matr[6] + loc4 * dg[6];
    J_C[7] = loc3 * matr[7] + loc4 * dg[7];
    J_C[8] = loc3 * matr[8] + loc4 * dg[8];

    /* First diagonal block */
    A[0] = -J_A[0] - isqrt3 * J_A[1] - J_A[2];
    A[1] = J_A[0] - isqrt3 * J_A[1];
    // A[2]  =          tisqrt3*J_A[1];
    // A[3]  =                           J_A[2];

    A[4] = -J_A[1] - isqrt3 * J_A[3] - J_A[4];
    A[5] = J_A[1] - isqrt3 * J_A[3];
    A[6] = tisqrt3 * J_A[3];
    // A[7]  =                           J_A[4];

    A[8]  = -J_A[2] - isqrt3 * J_A[4] - J_A[5];
    A[9]  = J_A[2] - isqrt3 * J_A[4];
    A[10] = tisqrt3 * J_A[4];
    A[11] = J_A[5];

    h_obj[0][0][0] = -A[0] - isqrt3 * A[4] - A[8];
    h_obj[1][0][0] = A[0] - isqrt3 * A[4];
    h_obj[2][0][0] = tisqrt3 * A[4];
    h_obj[3][0][0] = A[8];

    h_obj[4][0][0] = A[1] - isqrt3 * A[5];
    h_obj[5][0][0] = tisqrt3 * A[5];
    h_obj[6][0][0] = A[9];

    h_obj[7][0][0] = tisqrt3 * A[6];
    h_obj[8][0][0] = A[10];

    h_obj[9][0][0] = A[11];

    /* First off-diagonal block */
    loc2 = matr[8] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    loc2 = matr[7] * loc1;
    J_B[2] -= loc2;
    J_B[6] += loc2;

    loc2 = matr[6] * loc1;
    J_B[5] += loc2;
    J_B[7] -= loc2;

    A[0] = -J_B[0] - isqrt3 * J_B[1] - J_B[2];
    A[1] = J_B[0] - isqrt3 * J_B[1];
    A[2] = tisqrt3 * J_B[1];
    A[3] = J_B[2];

    A[4] = -J_B[3] - isqrt3 * J_B[4] - J_B[5];
    A[5] = J_B[3] - isqrt3 * J_B[4];
    A[6] = tisqrt3 * J_B[4];
    A[7] = J_B[5];

    A[8]  = -J_B[6] - isqrt3 * J_B[7] - J_B[8];
    A[9]  = J_B[6] - isqrt3 * J_B[7];
    A[10] = tisqrt3 * J_B[7];
    A[11] = J_B[8];

    h_obj[0][0][1] = -A[0] - isqrt3 * A[4] - A[8];
    h_obj[1][1][0] = A[0] - isqrt3 * A[4];
    h_obj[2][1][0] = tisqrt3 * A[4];
    h_obj[3][1][0] = A[8];

    h_obj[1][0][1] = -A[1] - isqrt3 * A[5] - A[9];
    h_obj[4][0][1] = A[1] - isqrt3 * A[5];
    h_obj[5][1][0] = tisqrt3 * A[5];
    h_obj[6][1][0] = A[9];

    h_obj[2][0][1] = -A[2] - isqrt3 * A[6] - A[10];
    h_obj[5][0][1] = A[2] - isqrt3 * A[6];
    h_obj[7][0][1] = tisqrt3 * A[6];
    h_obj[8][1][0] = A[10];

    h_obj[3][0][1] = -A[3] - isqrt3 * A[7] - A[11];
    h_obj[6][0][1] = A[3] - isqrt3 * A[7];
    h_obj[8][0][1] = tisqrt3 * A[7];
    h_obj[9][0][1] = A[11];

    /* Second off-diagonal block */
    loc2 = matr[5] * loc1;
    J_C[1] -= loc2;
    J_C[3] += loc2;

    loc2 = matr[4] * loc1;
    J_C[2] += loc2;
    J_C[6] -= loc2;

    loc2 = matr[3] * loc1;
    J_C[5] -= loc2;
    J_C[7] += loc2;

    A[0] = -J_C[0] - isqrt3 * J_C[1] - J_C[2];
    A[1] = J_C[0] - isqrt3 * J_C[1];
    A[2] = tisqrt3 * J_C[1];
    A[3] = J_C[2];

    A[4] = -J_C[3] - isqrt3 * J_C[4] - J_C[5];
    A[5] = J_C[3] - isqrt3 * J_C[4];
    A[6] = tisqrt3 * J_C[4];
    A[7] = J_C[5];

    A[8]  = -J_C[6] - isqrt3 * J_C[7] - J_C[8];
    A[9]  = J_C[6] - isqrt3 * J_C[7];
    A[10] = tisqrt3 * J_C[7];
    A[11] = J_C[8];

    h_obj[0][0][2] = -A[0] - isqrt3 * A[4] - A[8];
    h_obj[1][2][0] = A[0] - isqrt3 * A[4];
    h_obj[2][2][0] = tisqrt3 * A[4];
    h_obj[3][2][0] = A[8];

    h_obj[1][0][2] = -A[1] - isqrt3 * A[5] - A[9];
    h_obj[4][0][2] = A[1] - isqrt3 * A[5];
    h_obj[5][2][0] = tisqrt3 * A[5];
    h_obj[6][2][0] = A[9];

    h_obj[2][0][2] = -A[2] - isqrt3 * A[6] - A[10];
    h_obj[5][0][2] = A[2] - isqrt3 * A[6];
    h_obj[7][0][2] = tisqrt3 * A[6];
    h_obj[8][2][0] = A[10];

    h_obj[3][0][2] = -A[3] - isqrt3 * A[7] - A[11];
    h_obj[6][0][2] = A[3] - isqrt3 * A[7];
    h_obj[8][0][2] = tisqrt3 * A[7];
    h_obj[9][0][2] = A[11];

    /* Second block of rows */
    loc3 = matr[3] * f + dg[3] * cross;
    loc4 = dg[3] * g + matr[3] * cross;

    J_A[0] = loc0 + loc3 * matr[3] + loc4 * dg[3];
    J_A[1] = loc3 * matr[4] + loc4 * dg[4];
    J_A[2] = loc3 * matr[5] + loc4 * dg[5];
    J_B[0] = loc3 * matr[6] + loc4 * dg[6];
    J_B[1] = loc3 * matr[7] + loc4 * dg[7];
    J_B[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[4] * f + dg[4] * cross;
    loc4 = dg[4] * g + matr[4] * cross;

    J_A[3] = loc0 + loc3 * matr[4] + loc4 * dg[4];
    J_A[4] = loc3 * matr[5] + loc4 * dg[5];
    J_B[3] = loc3 * matr[6] + loc4 * dg[6];
    J_B[4] = loc3 * matr[7] + loc4 * dg[7];
    J_B[5] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[5] * f + dg[5] * cross;
    loc4 = dg[5] * g + matr[5] * cross;

    J_A[5] = loc0 + loc3 * matr[5] + loc4 * dg[5];
    J_B[6] = loc3 * matr[6] + loc4 * dg[6];
    J_B[7] = loc3 * matr[7] + loc4 * dg[7];
    J_B[8] = loc3 * matr[8] + loc4 * dg[8];

    /* Second diagonal block */
    A[0] = -J_A[0] - isqrt3 * J_A[1] - J_A[2];
    A[1] = J_A[0] - isqrt3 * J_A[1];
    // A[2]  =          tisqrt3*J_A[1];
    // A[3]  =                           J_A[2];

    A[4] = -J_A[1] - isqrt3 * J_A[3] - J_A[4];
    A[5] = J_A[1] - isqrt3 * J_A[3];
    A[6] = tisqrt3 * J_A[3];
    // A[7]  =                           J_A[4];

    A[8]  = -J_A[2] - isqrt3 * J_A[4] - J_A[5];
    A[9]  = J_A[2] - isqrt3 * J_A[4];
    A[10] = tisqrt3 * J_A[4];
    A[11] = J_A[5];

    h_obj[0][1][1] = -A[0] - isqrt3 * A[4] - A[8];
    h_obj[1][1][1] = A[0] - isqrt3 * A[4];
    h_obj[2][1][1] = tisqrt3 * A[4];
    h_obj[3][1][1] = A[8];

    h_obj[4][1][1] = A[1] - isqrt3 * A[5];
    h_obj[5][1][1] = tisqrt3 * A[5];
    h_obj[6][1][1] = A[9];

    h_obj[7][1][1] = tisqrt3 * A[6];
    h_obj[8][1][1] = A[10];

    h_obj[9][1][1] = A[11];

    /* Third off-diagonal block */
    loc2 = matr[2] * loc1;
    J_B[1] += loc2;
    J_B[3] -= loc2;

    loc2 = matr[1] * loc1;
    J_B[2] -= loc2;
    J_B[6] += loc2;

    loc2 = matr[0] * loc1;
    J_B[5] += loc2;
    J_B[7] -= loc2;

    A[0] = -J_B[0] - isqrt3 * J_B[1] - J_B[2];
    A[1] = J_B[0] - isqrt3 * J_B[1];
    A[2] = tisqrt3 * J_B[1];
    A[3] = J_B[2];

    A[4] = -J_B[3] - isqrt3 * J_B[4] - J_B[5];
    A[5] = J_B[3] - isqrt3 * J_B[4];
    A[6] = tisqrt3 * J_B[4];
    A[7] = J_B[5];

    A[8]  = -J_B[6] - isqrt3 * J_B[7] - J_B[8];
    A[9]  = J_B[6] - isqrt3 * J_B[7];
    A[10] = tisqrt3 * J_B[7];
    A[11] = J_B[8];

    h_obj[0][1][2] = -A[0] - isqrt3 * A[4] - A[8];
    h_obj[1][2][1] = A[0] - isqrt3 * A[4];
    h_obj[2][2][1] = tisqrt3 * A[4];
    h_obj[3][2][1] = A[8];

    h_obj[1][1][2] = -A[1] - isqrt3 * A[5] - A[9];
    h_obj[4][1][2] = A[1] - isqrt3 * A[5];
    h_obj[5][2][1] = tisqrt3 * A[5];
    h_obj[6][2][1] = A[9];

    h_obj[2][1][2] = -A[2] - isqrt3 * A[6] - A[10];
    h_obj[5][1][2] = A[2] - isqrt3 * A[6];
    h_obj[7][1][2] = tisqrt3 * A[6];
    h_obj[8][2][1] = A[10];

    h_obj[3][1][2] = -A[3] - isqrt3 * A[7] - A[11];
    h_obj[6][1][2] = A[3] - isqrt3 * A[7];
    h_obj[8][1][2] = tisqrt3 * A[7];
    h_obj[9][1][2] = A[11];

    /* Third block of rows */
    loc3 = matr[6] * f + dg[6] * cross;
    loc4 = dg[6] * g + matr[6] * cross;

    J_A[0] = loc0 + loc3 * matr[6] + loc4 * dg[6];
    J_A[1] = loc3 * matr[7] + loc4 * dg[7];
    J_A[2] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[7] * f + dg[7] * cross;
    loc4 = dg[7] * g + matr[7] * cross;

    J_A[3] = loc0 + loc3 * matr[7] + loc4 * dg[7];
    J_A[4] = loc3 * matr[8] + loc4 * dg[8];

    loc3 = matr[8] * f + dg[8] * cross;
    loc4 = dg[8] * g + matr[8] * cross;

    J_A[5] = loc0 + loc3 * matr[8] + loc4 * dg[8];

    /* Third diagonal block */
    A[0] = -J_A[0] - isqrt3 * J_A[1] - J_A[2];
    A[1] = J_A[0] - isqrt3 * J_A[1];
    // A[2]  =          tisqrt3*J_A[1];
    // A[3]  =                           J_A[2];

    A[4] = -J_A[1] - isqrt3 * J_A[3] - J_A[4];
    A[5] = J_A[1] - isqrt3 * J_A[3];
    A[6] = tisqrt3 * J_A[3];
    // A[7]  =                           J_A[4];

    A[8]  = -J_A[2] - isqrt3 * J_A[4] - J_A[5];
    A[9]  = J_A[2] - isqrt3 * J_A[4];
    A[10] = tisqrt3 * J_A[4];
    A[11] = J_A[5];

    h_obj[0][2][2] = -A[0] - isqrt3 * A[4] - A[8];
    h_obj[1][2][2] = A[0] - isqrt3 * A[4];
    h_obj[2][2][2] = tisqrt3 * A[4];
    h_obj[3][2][2] = A[8];

    h_obj[4][2][2] = A[1] - isqrt3 * A[5];
    h_obj[5][2][2] = tisqrt3 * A[5];
    h_obj[6][2][2] = A[9];

    h_obj[7][2][2] = tisqrt3 * A[6];
    h_obj[8][2][2] = A[10];

    h_obj[9][2][2] = A[11];

    // completes diagonal blocks.
    h_obj[0].fill_lower_triangle();
    h_obj[4].fill_lower_triangle();
    h_obj[7].fill_lower_triangle();
    h_obj[9].fill_lower_triangle();

    return true;
}

}  // namespace MBMesquite

#endif