findpt.c

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#include <stdio.h>
#include <stdlib.h>
#include <stdarg.h>
#include <math.h> /* for cos, fabs */
#include <float.h>
#include <string.h> /* for memcpy */
 
#include "moab/FindPtFuncs.h"
 
const unsigned opt_no_constraints_2 = 3 + 1;
const unsigned opt_no_constraints_3 = 9 + 3 + 1;
 
/*--------------------------------------------------------------------------
   Lobatto Polynomial Bounds
 
   Needed inputs are the Gauss-Lobatto quadrature nodes and weights:
     unsigned nr = ..., ns = ...;
     realType zr[nr], wr[nr];
     realType zs[ns], ws[ns];
 
     lobatto_nodes(zr,nr); lobatto_weights(zr,wr,nr);
     lobatto_nodes(zs,ns); lobatto_weights(zs,ws,ns);
 
   The number of points in the constructed piecewise (bi-)linear bounds
   is a parameter; more points give tighter bounds
 
     unsigned mr = 2*nr, ms = 2*ns;
 
   The necessary setup is accomplished via:
     lob_bnd_base b_data_r;
     lob_bnd_ext  e_data_s;
 
     lob_bnd_base_alloc(&b_data_r,nr,mr);
     lob_bnd_base_setup(&b_data_r,zr,wr);
     lob_bnd_ext_alloc(&e_data_s,ns,ms);
     lob_bnd_ext_setup(&e_data_s,zs,ws);
 
   Bounds may then be computed via:
     realType work1r[2*mr], work1s[2*ms], work2[2*mr + 2*mr*ns + 2*mr*ms];
     realType ur[nr], us[ns];    // 1-d polynomials on the zr[] and zs[] nodes
     realType u[ns][nr];         // 2-d polynomial on zr[] (x) zs[]
     realType bound[2];          // = { min, max } (to be computed)
 
     lob_bnd_1(&b_data_r  ,ur,bound,work1r); // compute bounds on ur
     lob_bnd_1(&e_data_s.b,us,bound,work1s); // compute bounds on us
     lob_bnd_2(&b_data_r, &e_data_s,
               (const double*)&u[0][0],bound,work2); // compute bounds on u
   The above routines access the zr,zs arrays passed to *_setup
     (so do not delete them between calls)
 
   Memory allocated in *_setup is freed with
     lob_bnd_base_free(&b_data_r);
     lob_bnd_ext_free(&e_data_s);
 
  --------------------------------------------------------------------------*/
 
typedef struct
{
    unsigned n;        /* number of Lobatto nodes in input */
    unsigned m;        /* number of Chebyshev nodes used to calculate bounds */
    realType *Q0, *Q1; /* Q0[n], Q1[n] -- first two rows of change of basis matrix
                      from Lobatto node Lagrangian to Legendre */
    const realType* z; /* z[n] -- external; Lobatto nodes */
    realType* h;       /* h[m] -- Chebyshev nodes */
    realType *uv, *ov; /* uv[n][m], ov[n][m] --
                              uv[j][:] is a piecewise linear function in the nodal
                                       basis with nodes h[m] that is everywhere less
                                       than or equal to the jth Lagrangian basis
                                       function (with the Lobatto nodes z[n])
                              ov[j][:] is everywhere greater than or equal */
} lob_bnd_base;
 
typedef struct
{
    lob_bnd_base b;
    realType *uvp, *uvn, *ovp, *ovn; /* [n][m] -- uv and ov split into
                                         positive and negative parts */
} lob_bnd_ext;
 
static void lob_bnd_base_alloc( lob_bnd_base* p, unsigned n, unsigned m )
{
    p->n = n, p->m = m;
    p->Q0 = tmalloc( realType, 2 * n + m + 2 * n * m );
    p->Q1 = p->Q0 + n;
    p->h  = p->Q1 + n;
    p->uv = p->h + m;
    p->ov = p->uv + n * m;
}
 
static void lob_bnd_base_free( lob_bnd_base* p )
{
    free( p->Q0 );
}
 
static void lob_bnd_ext_alloc( lob_bnd_ext* p, unsigned n, unsigned m )
{
    p->b.n = n, p->b.m = m;
    p->b.Q0 = tmalloc( realType, 2 * n + m + 6 * n * m );
    p->b.Q1 = p->b.Q0 + n;
    p->b.h  = p->b.Q1 + n;
    p->b.uv = p->b.h + m;
    p->b.ov = p->b.uv + n * m;
    p->uvp  = p->b.ov + n * m;
    p->uvn  = p->uvp + n * m;
    p->ovp  = p->uvn + n * m;
    p->ovn  = p->ovp + n * m;
}
 
static void lob_bnd_ext_free( lob_bnd_ext* p )
{
    free( p->b.Q0 );
}
 
static void lob_bnd_base_setup( lob_bnd_base* p, const realType* z, const realType* w )
{
    unsigned i, j, m = p->m, n = p->n, mm = 2 * m - 1;
    realType *q = tmalloc( realType, ( 2 * n + 1 ) * mm + 6 * n ), *J = q + mm, *D = J + n * mm, *work = D + n * mm;
    p->z = z;
    for( i = 0; i < n; ++i )
        p->Q0[i] = w[i] / 2, p->Q1[i] = 3 * p->Q0[i] * z[i];
    p->h[0] = -1, p->h[m - 1] = 1;
    for( j = 1; j < m - 1; ++j )
        p->h[j] = mbcos( ( m - j - 1 ) * MOAB_POLY_PI / ( m - 1 ) );
    for( j = 0; j < m - 1; ++j )
        q[2 * j] = p->h[j], q[2 * j + 1] = ( p->h[j] + p->h[j + 1] ) / 2;
    q[mm - 1] = p->h[m - 1];
    lagrange_weights_deriv( z, n, q, mm, J, D, work );
    for( i = 0; i < n; ++i )
    {
        realType *uv = p->uv + i * m, *ov = p->ov + i * m;
        ov[0] = uv[0] = J[i];
        ov[m - 1] = uv[m - 1] = J[( mm - 1 ) * n + i];
        for( j = 1; j < m - 1; ++j )
        {
            unsigned jj = 2 * j;
            realType c0 = J[( jj - 1 ) * n + i] + ( q[jj] - q[jj - 1] ) * D[( jj - 1 ) * n + i];
            realType c1 = J[( jj + 1 ) * n + i] + ( q[jj] - q[jj + 1] ) * D[( jj + 1 ) * n + i];
            realType c2 = J[jj * n + i];
            rminmax_3( &uv[j], &ov[j], c0, c1, c2 );
        }
    }
    free( q );
}
 
static void lob_bnd_ext_setup( lob_bnd_ext* p, const realType* z, const realType* w )
{
    unsigned i, mn = p->b.m * p->b.n;
    lob_bnd_base_setup( &p->b, z, w );
    for( i = 0; i < mn; ++i )
    {
        realType uvi = p->b.uv[i], ovi = p->b.ov[i];
        p->uvp[i] = p->uvn[i] = p->ovp[i] = p->ovn[i] = 0;
        if( uvi > 0 )
            p->uvp[i] = uvi;
        else
            p->uvn[i] = uvi;
        if( ovi > 0 )
            p->ovp[i] = ovi;
        else
            p->ovn[i] = ovi;
    }
}
 
static void lob_bnd_lines( const lob_bnd_base* p, const realType* u, realType* a, realType* b )
{
    unsigned i, j;
    realType a0 = 0, a1 = 0;
    const realType *uv = p->uv, *ov = p->ov;
    for( i = 0; i < p->n; ++i )
        a0 += p->Q0[i] * u[i], a1 += p->Q1[i] * u[i];
    for( j = 0; j < p->m; ++j )
        b[j] = a[j] = a0 + a1 * p->h[j];
    for( i = 0; i < p->n; ++i )
    {
        realType w = u[i] - ( a0 + a1 * p->z[i] );
        if( w >= 0 )
            for( j = 0; j < p->m; ++j )
                a[j] += w * ( *uv++ ), b[j] += w * ( *ov++ );
        else
            for( j = 0; j < p->m; ++j )
                a[j] += w * ( *ov++ ), b[j] += w * ( *uv++ );
    }
}
 
/* work holds p->m * 2 doubles */
static void lob_bnd_1( const lob_bnd_base* p, const realType* u, realType bnd[2], realType* work )
{
    unsigned j;
    realType *a = work, *b = work + p->m;
    lob_bnd_lines( p, u, a, b );
    bnd[0] = a[0], bnd[1] = b[0];
    for( j = 1; j < p->m; ++j )
    {
        if( a[j] < bnd[0] ) bnd[0] = a[j];
        if( b[j] > bnd[1] ) bnd[1] = b[j];
    }
}
 
/* work holds 2*mr + 2*mr*ns + 2*mr*ms doubles */
static void lob_bnd_2( const lob_bnd_base* pr,
                       const lob_bnd_ext* ps,
                       const realType* u,
                       realType bnd[2],
                       realType* work )
{
    unsigned nr = pr->n, mr = pr->m, ns = ps->b.n, ms = ps->b.m;
    realType *a0 = work, *a1 = a0 + mr, *ar_ = a1 + mr, *ar = ar_, *br_ = ar + mr * ns, *br = br_, *a_ = br + mr * ns,
             *a = a_, *b_ = a + mr * ms, *b = b_, *uvp, *ovp, *uvn, *ovn;
    realType b0, b1;
    unsigned i, j, k;
    for( i = 0; i < mr; ++i )
        a0[i] = a1[i] = 0;
    for( j = 0; j < ns; ++j, u += nr )
    {
        realType q0 = ps->b.Q0[j], q1 = ps->b.Q1[j];
        lob_bnd_lines( pr, u, ar, br );
        for( i = 0; i < mr; ++i )
        {
            realType t = ( *ar++ + *br++ ) / 2;
            a0[i] += q0 * t, a1[i] += q1 * t;
        }
    }
    for( i = 0; i < mr; ++i )
    {
        realType a0i = a0[i], a1i = a1[i];
        for( k = 0; k < ms; ++k )
            *b++ = *a++ = a0i + a1i * ps->b.h[k];
    }
    ar = ar_, br = br_;
    uvp = ps->uvp, ovp = ps->ovp, uvn = ps->uvn, ovn = ps->ovn;
    for( j = 0; j < ns; ++j, uvp += ms, uvn += ms, ovp += ms, ovn += ms )
    {
        realType zj = ps->b.z[j];
        a = a_, b = b_;
        for( i = 0; i < mr; ++i )
        {
            realType t  = a0[i] + a1[i] * zj;
            realType uw = *ar++ - t;
            realType ow = *br++ - t;
            if( uw >= 0 ) /* 0  <= uw <= ow */
                for( k = 0; k < ms; ++k )
                    *a++ += uw * uvp[k] + ow * uvn[k], *b++ += ow * ovp[k] + uw * ovn[k];
            else if( ow <= 0 ) /* uw <= ow <= 0  */
                for( k = 0; k < ms; ++k )
                    *a++ += uw * ovp[k] + ow * ovn[k], *b++ += ow * uvp[k] + uw * uvn[k];
            else /* uw <  0  <  ow */
                for( k = 0; k < ms; ++k )
                    *a++ += uw * ovp[k] + ow * uvn[k], *b++ += ow * ovp[k] + uw * uvn[k];
        }
    }
    b0 = a_[0], b1 = b_[0];
    for( i = 1; i < mr * ms; ++i )
    {
        if( a_[i] < b0 ) b0 = a_[i];
        if( b_[i] > b1 ) b1 = b_[i];
    }
    bnd[0] = b0, bnd[1] = b1;
}
 
/*--------------------------------------------------------------------------
   Small Matrix Inverse
  --------------------------------------------------------------------------*/
 
static void mat_inv_2( const realType A[4], realType inv[4] )
{
    const realType idet = 1 / ( A[0] * A[3] - A[1] * A[2] );
    inv[0]              = idet * A[3];
    inv[1]              = -( idet * A[1] );
    inv[2]              = -( idet * A[2] );
    inv[3]              = idet * A[0];
}
 
static void mat_inv_3( const realType A[9], realType inv[9] )
{
    const realType a = A[4] * A[8] - A[5] * A[7], b = A[5] * A[6] - A[3] * A[8], c = A[3] * A[7] - A[4] * A[6],
                   idet = 1 / ( A[0] * a + A[1] * b + A[2] * c );
    inv[0]              = idet * a;
    inv[1]              = idet * ( A[2] * A[7] - A[1] * A[8] );
    inv[2]              = idet * ( A[1] * A[5] - A[2] * A[4] );
    inv[3]              = idet * b;
    inv[4]              = idet * ( A[0] * A[8] - A[2] * A[6] );
    inv[5]              = idet * ( A[2] * A[3] - A[0] * A[5] );
    inv[6]              = idet * c;
    inv[7]              = idet * ( A[1] * A[6] - A[0] * A[7] );
    inv[8]              = idet * ( A[0] * A[4] - A[1] * A[3] );
}
 
static void mat_app_2r( realType y[2], const realType A[4], const realType x[2] )
{
    y[0] = A[0] * x[0] + A[1] * x[1];
    y[1] = A[2] * x[0] + A[3] * x[1];
}
 
static void mat_app_2c( realType y[2], const realType A[4], const realType x[2] )
{
    y[0] = A[0] * x[0] + A[2] * x[1];
    y[1] = A[1] * x[0] + A[3] * x[1];
}
 
static void mat_app_3r( realType y[3], const realType A[9], const realType x[3] )
{
    y[0] = A[0] * x[0] + A[1] * x[1] + A[2] * x[2];
    y[1] = A[3] * x[0] + A[4] * x[1] + A[5] * x[2];
    y[2] = A[6] * x[0] + A[7] * x[1] + A[8] * x[2];
}
 
static void mat_app_3c( realType y[3], const realType A[9], const realType x[3] )
{
    y[0] = A[0] * x[0] + A[3] * x[1] + A[6] * x[2];
    y[1] = A[1] * x[0] + A[4] * x[1] + A[7] * x[2];
    y[2] = A[2] * x[0] + A[5] * x[1] + A[8] * x[2];
}
 
static void tinyla_solve_2( realType x[2], const realType A[4], const realType b[2] )
{
    realType inv[4];
    mat_inv_2( A, inv );
    mat_app_2r( x, inv, b );
}
 
static void tinyla_solve_3( realType x[3], const realType A[9], const realType b[3] )
{
    realType inv[9];
    mat_inv_3( A, inv );
    mat_app_3r( x, inv, b );
}
 
/* solve
   A[0] x0 + A[2] x1 = b0,
   A[2] x0 + A[1] x1 = b1
*/
static void tinyla_solve_sym_2( realType* x0, realType* x1, const realType A[3], realType b0, realType b1 )
{
    const realType idet = 1 / ( A[0] * A[1] - A[2] * A[2] );
    *x0                 = idet * ( A[1] * b0 - A[2] * b1 );
    *x1                 = idet * ( A[0] * b1 - A[2] * b0 );
}
 
/*--------------------------------------------------------------------------
   Oriented Bounding Box
 
   Suffixes on names are _2 for 2-d and _3 for 3-d
 
   Needed inputs are the Gauss-Lobatto quadrature nodes and weights:
     unsigned nr = ..., ns = ...;
     realType zr[nr], wr[nr];
     realType zs[ns], ws[ns];
 
     lobatto_nodes(zr,nr); lobatto_weights(zr,wr,nr);
     lobatto_nodes(zs,ns); lobatto_weights(zs,ws,ns);
 
   The number of points in the constructed piecewise (bi-)linear bounds
   for the boundaries is a parameter; more points give tighter bounds
 
     unsigned mr = 2*nr, ms = 2*ns;
 
   Bounding boxes are increased by a relative amount as a parameter
 
     realType tol = 0.01;
 
   Setup is accomplished via:
 
     const realType *z[2] = {zr,zs}, *w[2] = {wr,ws};
     const unsigned n[2] = {nr,ns}, m[2] = {mr,ms};
     obbox_data_2 *data = obbox_setup_2(z,w,n,m);
 
   Bounding box data may then be computed:
 
     obbox_2 box;                 // will store bounding box information
     realType xm[ns][nr], ym[ns][nr]; // x, y coordinates of the element nodes
 
     obbox_calc_2(data, tol, (const realType *)&xm[0][0],
                             (const realType *)&ym[0][0], &box);
 
   A point may be tested:
 
     const realType x[2]; // point to test
     realType r[2];
 
     if( obbox_axis_test_2(&box, x) )
       ... // x failed axis-aligned bounding box test
 
     if( obbox_test_2(&box, x, r) )
       ... // x failed oriented bounding box test
     else
       ... // r suitable as initial guess for parametric coords
 
   Once all bounding box information has been computed
 
     obbox_free_2(data);
 
   to free the memory allocated with obbox_setup_2.
 
  --------------------------------------------------------------------------*/
 
typedef struct
{
    lob_bnd_base dr, ds;
    realType *Jr0, *Dr0, *Js0, *Ds0, *work;
} obbox_data_2;
 
typedef struct
{
    lob_bnd_base dr;
    lob_bnd_ext ds, dt;
    realType *Jr0, *Dr0, *Js0, *Ds0, *Jt0, *Dt0, *work;
} obbox_data_3;
 
static void obbox_data_alloc_2( obbox_data_2* p, const unsigned n[2], const unsigned m[2] )
{
    const unsigned max_npm = umax_2( n[0] + m[0], n[1] + m[1] );
    lob_bnd_base_alloc( &p->dr, n[0], m[0] );
    lob_bnd_base_alloc( &p->ds, n[1], m[1] );
    p->Jr0  = tmalloc( realType, 2 * n[0] + 2 * n[1] + 2 * max_npm );
    p->Dr0  = p->Jr0 + n[0];
    p->Js0  = p->Dr0 + n[0];
    p->Ds0  = p->Js0 + n[1];
    p->work = p->Ds0 + n[1];
}
 
static void obbox_data_free_2( obbox_data_2* p )
{
    lob_bnd_base_free( &p->dr );
    lob_bnd_base_free( &p->ds );
    free( p->Jr0 );
}
 
static void obbox_data_alloc_3( obbox_data_3* p, const unsigned n[3], const unsigned m[3] )
{
    const unsigned wk1    = 3 * n[0] * n[1] + 2 * m[0] + 2 * m[0] * n[1] + 2 * m[0] * m[1];
    const unsigned wk2    = 3 * n[0] * n[2] + 2 * m[0] + 2 * m[0] * n[2] + 2 * m[0] * m[2];
    const unsigned wk3    = 3 * n[1] * n[2] + 2 * m[1] + 2 * m[1] * n[2] + 2 * m[1] * m[2];
    const unsigned wk_max = umax_3( wk1, wk2, wk3 );
    lob_bnd_base_alloc( &p->dr, n[0], m[0] );
    lob_bnd_ext_alloc( &p->ds, n[1], m[1] );
    lob_bnd_ext_alloc( &p->dt, n[2], m[2] );
    p->Jr0  = tmalloc( realType, 2 * n[0] + 2 * n[1] + 2 * n[2] + wk_max );
    p->Dr0  = p->Jr0 + n[0];
    p->Js0  = p->Dr0 + n[0];
    p->Ds0  = p->Js0 + n[1];
    p->Jt0  = p->Ds0 + n[1];
    p->Dt0  = p->Jt0 + n[2];
    p->work = p->Dt0 + n[2];
}
 
static void obbox_data_free_3( obbox_data_3* p )
{
    lob_bnd_base_free( &p->dr );
    lob_bnd_ext_free( &p->ds );
    lob_bnd_ext_free( &p->dt );
    free( p->Jr0 );
}
 
static obbox_data_2* obbox_setup_2( const realType* const z[2],
                                    const realType* const w[2],
                                    const unsigned n[2],
                                    const unsigned m[2] )
{
    const realType zero = 0;
    realType* work;
    obbox_data_2* p = tmalloc( obbox_data_2, 1 );
    obbox_data_alloc_2( p, n, m );
    lob_bnd_base_setup( &p->dr, z[0], w[0] );
    lob_bnd_base_setup( &p->ds, z[1], w[1] );
    work = tmalloc( realType, 6 * umax_2( n[0], n[1] ) );
    lagrange_weights_deriv( z[0], n[0], &zero, 1, p->Jr0, p->Dr0, work );
    lagrange_weights_deriv( z[1], n[1], &zero, 1, p->Js0, p->Ds0, work );
    free( work );
    return p;
}
 
static obbox_data_3* obbox_setup_3( const realType* const z[3],
                                    const realType* const w[3],
                                    const unsigned n[3],
                                    const unsigned m[3] )
{
    const realType zero = 0;
    realType* work;
    obbox_data_3* p = tmalloc( obbox_data_3, 1 );
    obbox_data_alloc_3( p, n, m );
    lob_bnd_base_setup( &p->dr, z[0], w[0] );
    lob_bnd_ext_setup( &p->ds, z[1], w[1] );
    lob_bnd_ext_setup( &p->dt, z[2], w[2] );
    work = tmalloc( realType, 6 * umax_3( n[0], n[1], n[2] ) );
    lagrange_weights_deriv( z[0], n[0], &zero, 1, p->Jr0, p->Dr0, work );
    lagrange_weights_deriv( z[1], n[1], &zero, 1, p->Js0, p->Ds0, work );
    lagrange_weights_deriv( z[2], n[2], &zero, 1, p->Jt0, p->Dt0, work );
    free( work );
    return p;
}
 
static void obbox_free_2( obbox_data_2* p )
{
    obbox_data_free_2( p );
    free( p );
}
 
static void obbox_free_3( obbox_data_3* p )
{
    obbox_data_free_3( p );
    free( p );
}
 
int obbox_axis_test_2( const obbox_2* p, const realType x[2] )
{
    return ( x[0] < p->axis_bnd[0] || x[0] > p->axis_bnd[1] || x[1] < p->axis_bnd[2] || x[1] > p->axis_bnd[3] );
}
 
int obbox_axis_test_3( const obbox_3* p, const realType x[3] )
{
    return ( x[0] < p->axis_bnd[0] || x[0] > p->axis_bnd[1] || x[1] < p->axis_bnd[2] || x[1] > p->axis_bnd[3] ||
             x[2] < p->axis_bnd[4] || x[2] > p->axis_bnd[5] );
}
 
int obbox_test_2( const obbox_2* p, const realType x[2], realType r[2] )
{
    realType xt[2];
    xt[0] = x[0] - p->x[0];
    xt[1] = x[1] - p->x[1];
    r[0]  = p->A[0] * xt[0] + p->A[1] * xt[1];
    if( mbabs( r[0] ) > 1 ) return 1;
    r[1] = p->A[2] * xt[0] + p->A[3] * xt[1];
    return mbabs( r[1] ) > 1;
}
 
int obbox_test_3( const obbox_3* p, const realType x[3], realType r[3] )
{
    realType xt[3];
    xt[0] = x[0] - p->x[0];
    xt[1] = x[1] - p->x[1];
    xt[2] = x[2] - p->x[2];
    r[0]  = p->A[0] * xt[0] + p->A[1] * xt[1] + p->A[2] * xt[2];
    if( mbabs( r[0] ) > 1 ) return 1;
    r[1] = p->A[3] * xt[0] + p->A[4] * xt[1] + p->A[5] * xt[2];
    if( mbabs( r[1] ) > 1 ) return 1;
    r[2] = p->A[6] * xt[0] + p->A[7] * xt[1] + p->A[8] * xt[2];
    return mbabs( r[2] ) > 1;
}
 
void obbox_calc_tfm_2( const realType* x,
                       const realType* y,
                       unsigned n,
                       unsigned s,
                       const realType c0[2],
                       const realType A[4],
                       realType* u )
{
    unsigned i;
    realType* v = u + n;
    for( i = 0; i < n; ++i, x += s, y += s )
    {
        const realType xt = *x - c0[0], yt = *y - c0[1];
        u[i] = A[0] * xt + A[1] * yt;
        v[i] = A[2] * xt + A[3] * yt;
    }
}
 
void obbox_calc_tfm_3( const realType* x,
                       const realType* y,
                       const realType* z,
                       unsigned nr,
                       unsigned sr,
                       unsigned ns,
                       unsigned ss,
                       const realType c0[3],
                       const realType A[9],
                       realType* u )
{
    unsigned i, j;
    realType *v = u + nr * ns, *w = v + nr * ns;
    for( j = 0; j < ns; ++j, x += ss, y += ss, z += ss )
    {
        for( i = 0; i < nr; ++i, x += sr, y += sr, z += sr )
        {
            const realType xt = *x - c0[0], yt = *y - c0[1], zt = *z - c0[2];
            *u++ = A[0] * xt + A[1] * yt + A[2] * zt;
            *v++ = A[3] * xt + A[4] * yt + A[5] * zt;
            *w++ = A[6] * xt + A[7] * yt + A[8] * zt;
        }
    }
}
 
void obbox_merge_2( realType* b, const realType* ob )
{
    if( ob[0] < b[0] ) b[0] = ob[0];
    if( ob[1] > b[1] ) b[1] = ob[1];
    if( ob[2] < b[2] ) b[2] = ob[2];
    if( ob[3] > b[3] ) b[3] = ob[3];
}
 
void obbox_merge_3( realType* b, const realType* ob )
{
    if( ob[0] < b[0] ) b[0] = ob[0];
    if( ob[1] > b[1] ) b[1] = ob[1];
    if( ob[2] < b[2] ) b[2] = ob[2];
    if( ob[3] > b[3] ) b[3] = ob[3];
    if( ob[4] < b[4] ) b[4] = ob[4];
    if( ob[5] > b[5] ) b[5] = ob[5];
}
 
/* work holds 2*n + 2*m realTypes */
void obbox_side_2( const realType* x,
                   const realType* y,
                   unsigned n,
                   unsigned s,
                   const realType c0[2],
                   const realType A[4],
                   realType* work,
                   const lob_bnd_base* lbd,
                   realType bnd[4] )
{
    obbox_calc_tfm_2( x, y, n, s, c0, A, work );
    lob_bnd_1( lbd, work, bnd, work + 2 * n );
    lob_bnd_1( lbd, work + n, bnd + 2, work + 2 * n );
}
 
/* work holds 3*nr*ns + 2*mr + 2*mr*ns + 2*mr*ms realTypes */
void obbox_side_3( const realType* x,
                   const realType* y,
                   const realType* z,
                   unsigned nr,
                   unsigned sr,
                   unsigned ns,
                   unsigned ss,
                   const realType c0[3],
                   const realType A[9],
                   realType* work,
                   const lob_bnd_base* dr,
                   const lob_bnd_ext* ds,
                   realType bnd[6] )
{
    obbox_calc_tfm_3( x, y, z, nr, sr, ns, ss, c0, A, work );
    lob_bnd_2( dr, ds, work, bnd, work + 3 * nr * ns );
    lob_bnd_2( dr, ds, work + nr * ns, bnd + 2, work + 3 * nr * ns );
    lob_bnd_2( dr, ds, work + 2 * nr * ns, bnd + 4, work + 3 * nr * ns );
}
 
/* return bounds on u = A (x - c0)
   bnd[0] <= u_0 <= bnd[1]
   bnd[2] <= u_1 <= bnd[3] */
void obbox_bnd_2( const obbox_data_2* p,
                  const realType* x,
                  const realType* y,
                  const realType c0[2],
                  const realType A[4],
                  realType bnd[4] )
{
    unsigned i, nr = p->dr.n, ns = p->ds.n;
    realType obnd[4];
 
    i = nr * ( ns - 1 );
    obbox_side_2( x, y, nr, 1, c0, A, p->work, &p->dr, bnd );
    obbox_side_2( x + i, y + i, nr, 1, c0, A, p->work, &p->dr, obnd );
    obbox_merge_2( bnd, obnd );
 
    i = nr - 1;
    obbox_side_2( x, y, ns, nr, c0, A, p->work, &p->ds, obnd );
    obbox_merge_2( bnd, obnd );
    obbox_side_2( x + i, y + i, nr, nr, c0, A, p->work, &p->ds, obnd );
    obbox_merge_2( bnd, obnd );
}
 
/* return bounds on u = A (x - c0)
   bnd[0] <= u_0 <= bnd[1]
   bnd[2] <= u_1 <= bnd[3]
   bnd[4] <= u_2 <= bnd[5] */
void obbox_bnd_3( const obbox_data_3* p,
                  const realType* x,
                  const realType* y,
                  const realType* z,
                  const realType c0[3],
                  const realType A[9],
                  realType bnd[6] )
{
    unsigned i, nr = p->dr.n, ns = p->ds.b.n, nt = p->dt.b.n;
    realType obnd[6];
 
    i = nr * ns * ( nt - 1 );
    obbox_side_3( x, y, z, nr, 1, ns, 0, c0, A, p->work, &p->dr, &p->ds, bnd );
    obbox_side_3( x + i, y + i, z + i, nr, 1, ns, 0, c0, A, p->work, &p->dr, &p->ds, obnd );
    obbox_merge_3( bnd, obnd );
 
    i = nr * ( ns - 1 );
    obbox_side_3( x, y, z, nr, 1, nt, i, c0, A, p->work, &p->dr, &p->dt, obnd );
    obbox_merge_3( bnd, obnd );
    obbox_side_3( x + i, y + i, z + i, nr, 1, nt, i, c0, A, p->work, &p->dr, &p->dt, obnd );
    obbox_merge_3( bnd, obnd );
 
    i = nr - 1;
    obbox_side_3( x, y, z, ns, nr, nt, 0, c0, A, p->work, &p->ds.b, &p->dt, obnd );
    obbox_merge_3( bnd, obnd );
    obbox_side_3( x + i, y + i, z + i, ns, nr, nt, 0, c0, A, p->work, &p->ds.b, &p->dt, obnd );
    obbox_merge_3( bnd, obnd );
}
 
void obbox_calc_2( const obbox_data_2* p, realType tol, const realType* x, const realType* y, obbox_2* b )
{
    const realType zero[2] = { 0, 0 }, id[4] = { 1, 0, 0, 1 };
    realType c0[2], jac[4], inv[4], bnd[4], u0[2], d[2];
 
    obbox_bnd_2( p, x, y, zero, id, b->axis_bnd );
    d[0] = b->axis_bnd[1] - b->axis_bnd[0];
    d[1] = b->axis_bnd[3] - b->axis_bnd[2];
    b->axis_bnd[0] -= tol * d[0], b->axis_bnd[1] += tol * d[0];
    b->axis_bnd[2] -= tol * d[1], b->axis_bnd[3] += tol * d[1];
 
    c0[0] = tensor_ig2( p->Jr0, p->Dr0, p->dr.n, p->Js0, p->Ds0, p->ds.n, x, jac, p->work );
    c0[1] = tensor_ig2( p->Jr0, p->Dr0, p->dr.n, p->Js0, p->Ds0, p->ds.n, y, jac + 2, p->work );
    mat_inv_2( jac, inv );
 
    obbox_bnd_2( p, x, y, c0, inv, bnd );
 
    u0[0]   = ( bnd[0] + bnd[1] ) / 2;
    u0[1]   = ( bnd[2] + bnd[3] ) / 2;
    d[0]    = 2 / ( ( 1 + tol ) * ( bnd[1] - bnd[0] ) );
    d[1]    = 2 / ( ( 1 + tol ) * ( bnd[3] - bnd[2] ) );
    b->x[0] = c0[0] + jac[0] * u0[0] + jac[1] * u0[1];
    b->x[1] = c0[1] + jac[2] * u0[0] + jac[3] * u0[1];
    b->A[0] = d[0] * inv[0], b->A[1] = d[0] * inv[1];
    b->A[2] = d[1] * inv[2], b->A[3] = d[1] * inv[3];
}
 
void obbox_calc_3( const obbox_data_3* p,
                   realType tol,
                   const realType* x,
                   const realType* y,
                   const realType* z,
                   obbox_3* b )
{
    const realType zero[3] = { 0, 0 }, id[9] = { 1, 0, 0, 0, 1, 0, 0, 0, 1 };
    realType c0[3], jac[9], inv[9], bnd[6], u0[3], d[3];
 
    obbox_bnd_3( p, x, y, z, zero, id, b->axis_bnd );
    d[0] = b->axis_bnd[1] - b->axis_bnd[0];
    d[1] = b->axis_bnd[3] - b->axis_bnd[2];
    d[2] = b->axis_bnd[5] - b->axis_bnd[4];
    b->axis_bnd[0] -= tol * d[0], b->axis_bnd[1] += tol * d[0];
    b->axis_bnd[2] -= tol * d[1], b->axis_bnd[3] += tol * d[1];
    b->axis_bnd[4] -= tol * d[2], b->axis_bnd[5] += tol * d[2];
 
    c0[0] =
        tensor_ig3( p->Jr0, p->Dr0, p->dr.n, p->Js0, p->Ds0, p->ds.b.n, p->Jt0, p->Dt0, p->dt.b.n, x, jac, p->work );
    c0[1] = tensor_ig3( p->Jr0, p->Dr0, p->dr.n, p->Js0, p->Ds0, p->ds.b.n, p->Jt0, p->Dt0, p->dt.b.n, y, jac + 3,
                        p->work );
    c0[2] = tensor_ig3( p->Jr0, p->Dr0, p->dr.n, p->Js0, p->Ds0, p->ds.b.n, p->Jt0, p->Dt0, p->dt.b.n, z, jac + 6,
                        p->work );
    mat_inv_3( jac, inv );
 
    obbox_bnd_3( p, x, y, z, c0, inv, bnd );
 
    u0[0]   = ( bnd[0] + bnd[1] ) / 2;
    u0[1]   = ( bnd[2] + bnd[3] ) / 2;
    u0[2]   = ( bnd[4] + bnd[5] ) / 2;
    d[0]    = 2 / ( ( 1 + tol ) * ( bnd[1] - bnd[0] ) );
    d[1]    = 2 / ( ( 1 + tol ) * ( bnd[3] - bnd[2] ) );
    d[2]    = 2 / ( ( 1 + tol ) * ( bnd[5] - bnd[4] ) );
    b->x[0] = c0[0] + jac[0] * u0[0] + jac[1] * u0[1] + jac[2] * u0[2];
    b->x[1] = c0[1] + jac[3] * u0[0] + jac[4] * u0[1] + jac[5] * u0[2];
    b->x[2] = c0[2] + jac[6] * u0[0] + jac[7] * u0[1] + jac[8] * u0[2];
    b->A[0] = d[0] * inv[0], b->A[1] = d[0] * inv[1], b->A[2] = d[0] * inv[2];
    b->A[3] = d[1] * inv[3], b->A[4] = d[1] * inv[4], b->A[5] = d[1] * inv[5];
    b->A[6] = d[2] * inv[6], b->A[7] = d[2] * inv[7], b->A[8] = d[2] * inv[8];
}
 
/*--------------------------------------------------------------------------
   Point to Possible Elements Hashing
 
   Initializing the data:
     unsigned nel;        // number of elements
     const unsigned n[3]; // number of nodes in r, s, t directions
     const realType *xm[3];   // n[0]*n[1]*n[2]*nel x,y,z coordinates
     realType tol = 0.01;     // how far point is allowed to be outside element
                          //   relative to element size
     unsigned max_size = n[0]*n[1]*n[2]*nel; // maximum size of hash table
 
     hash_data_3 data;
     hash_build_3(&data, xm, n, nel, max_size, tol);
 
   Using the data:
     realType x[3];   // point to find
 
     unsigned index = hash_index_3(&data, x);
     unsigned i, b = data.offset[index], e = data.offset[index+1];
 
     // point may be in elements
     //   data.offset[b], data.offset[b+1], ... , data.offset[e-1]
     //
     // list has maximum size data.max (e.g., e-b <= data.max)
 
     for(i=b; i!=e; ++i) {
       unsigned el = data.offset[i];
       const obbox_3 *obb = &data.obb[el]; // bounding box data for element el
       ...
     }
 
   When done:
     hash_free_3(&data);
 
  --------------------------------------------------------------------------*/
 
static int ifloor( realType x )
{
    /*
    int y = x;
    return (double)y > x ? y-1 : y;
    */
    return mbfloor( x );
}
 
static int iceil( realType x )
{
    /*
    int y = x;
    return (double)y < x ? y+1 : y;
    */
    return mbceil( x );
}
 
static unsigned hash_index_helper( realType low, realType fac, unsigned n, realType x )
{
    const int i = ifloor( ( x - low ) * fac );
    if( i < 0 ) return 0;
    return umin_2( i, n - 1 );
}
 
static uint hash_index_2( const hash_data_2* p, const realType x[2] )
{
    const unsigned n = p->hash_n;
    return (uint)hash_index_helper( p->bnd[2], p->fac[1], n, x[1] ) * n +
           hash_index_helper( p->bnd[0], p->fac[0], n, x[0] );
}
 
static uint hash_index_3( const hash_data_3* p, const realType x[3] )
{
    const unsigned n = p->hash_n;
    return ( (uint)hash_index_helper( p->bnd[4], p->fac[2], n, x[2] ) * n +
             hash_index_helper( p->bnd[2], p->fac[1], n, x[1] ) ) *
               n +
           hash_index_helper( p->bnd[0], p->fac[0], n, x[0] );
}
 
static void hash_setfac_2( hash_data_2* p, unsigned n )
{
    p->hash_n = n;
    p->fac[0] = n / ( p->bnd[1] - p->bnd[0] );
    p->fac[1] = n / ( p->bnd[3] - p->bnd[2] );
}
 
static void hash_setfac_3( hash_data_3* p, unsigned n )
{
    p->hash_n = n;
    p->fac[0] = n / ( p->bnd[1] - p->bnd[0] );
    p->fac[1] = n / ( p->bnd[3] - p->bnd[2] );
    p->fac[2] = n / ( p->bnd[5] - p->bnd[4] );
}
 
static void hash_range_2( const hash_data_2* p, uint i, unsigned d, unsigned* ia, unsigned* ib )
{
    const realType a  = p->obb[i].axis_bnd[d * 2];
    const realType b  = p->obb[i].axis_bnd[d * 2 + 1];
    const int i0      = ifloor( ( a - p->bnd[d * 2] ) * p->fac[d] );
    const unsigned i1 = iceil( ( b - p->bnd[d * 2] ) * p->fac[d] );
    *ia               = imax_2( 0, i0 );
    *ib               = imin_2( i1, p->hash_n );
    if( *ib == *ia ) ++( *ib );
}
 
static void hash_range_3( const hash_data_3* p, uint i, unsigned d, unsigned* ia, unsigned* ib )
{
    const realType a  = p->obb[i].axis_bnd[d * 2];
    const realType b  = p->obb[i].axis_bnd[d * 2 + 1];
    const int i0      = ifloor( ( a - p->bnd[d * 2] ) * p->fac[d] );
    const unsigned i1 = iceil( ( b - p->bnd[d * 2] ) * p->fac[d] );
    *ia               = imax_2( 0, i0 );
    *ib               = imin_2( i1, p->hash_n );
    if( *ib == *ia ) ++( *ib );
}
 
static uint hash_count_2( hash_data_2* p, uint nel, unsigned n )
{
    uint i, count = 0;
    hash_setfac_2( p, n );
    for( i = 0; i < nel; ++i )
    {
        unsigned ia, ib;
        uint ci;
        hash_range_2( p, i, 0, &ia, &ib );
        ci = ib - ia;
        hash_range_2( p, i, 1, &ia, &ib );
        ci *= ib - ia;
        count += ci;
    }
    return count;
}
 
static uint hash_count_3( hash_data_3* p, uint nel, unsigned n )
{
    uint i, count = 0;
    hash_setfac_3( p, n );
    for( i = 0; i < nel; ++i )
    {
        unsigned ia, ib;
        uint ci;
        hash_range_3( p, i, 0, &ia, &ib );
        ci = ib - ia;
        hash_range_3( p, i, 1, &ia, &ib );
        ci *= ib - ia;
        hash_range_3( p, i, 2, &ia, &ib );
        ci *= ib - ia;
        count += ci;
    }
    return count;
}
 
static uint hash_opt_size_2( hash_data_2* p, uint nel, uint max_size )
{
    unsigned nl = 1, nu = ceil( sqrt( max_size - nel ) );
    uint size_low = 2 + nel;
    while( nu - nl > 1 )
    {
        unsigned nm = nl + ( nu - nl ) / 2;
        uint size   = (uint)nm * nm + 1 + hash_count_2( p, nel, nm );
        if( size <= max_size )
            nl = nm, size_low = size;
        else
            nu = nm;
    }
    hash_setfac_2( p, nl );
    return size_low;
}
 
static uint hash_opt_size_3( hash_data_3* p, uint nel, uint max_size )
{
    unsigned nl = 1, nu = ceil( pow( max_size - nel, 1.0 / 3 ) );
    uint size_low = 2 + nel;
    while( nu - nl > 1 )
    {
        unsigned nm = nl + ( nu - nl ) / 2;
        uint size   = (uint)nm * nm * nm + 1 + hash_count_3( p, nel, nm );
        if( size <= max_size )
            nl = nm, size_low = size;
        else
            nu = nm;
    }
    hash_setfac_3( p, nl );
    return size_low;
}
 
static void hash_getbb_2( hash_data_2* p, const realType* const elx[2], const unsigned n[2], uint nel, realType tol )
{
    obbox_data_2* data;
    realType *z[2], *w[2];
    uint i;
    unsigned d;
    const unsigned nn = n[0] * n[1];
    unsigned int m[2];
    const realType* x[2];
 
    for( i = 0; i < 2; ++i )
    {
        x[i] = elx[i];
        m[i] = 2 * n[i];
    }
 
    z[0] = tmalloc( realType, 2 * ( n[0] + n[1] ) );
    w[0] = z[0] + n[0];
    z[1] = w[0] + n[0], w[1] = z[1] + n[1];
    for( d = 0; d < 2; ++d )
        lobatto_nodes( z[d], n[d] ), lobatto_weights( z[d], w[d], n[d] );
    data = obbox_setup_2( (const realType* const*)z, (const realType* const*)w, n, m );
    obbox_calc_2( data, tol, x[0], x[1], &p->obb[0] );
    memcpy( &p->bnd[0], (const realType*)&p->obb[0].axis_bnd[0], 4 * sizeof( realType ) );
    for( i = 0; i < nel; ++i, x[0] += nn, x[1] += nn )
    {
        obbox_calc_2( data, tol, x[0], x[1], &p->obb[i] );
        obbox_merge_2( &p->bnd[0], (const realType*)&p->obb[i].axis_bnd[0] );
    }
    obbox_free_2( data );
    free( z[0] );
}
 
static void hash_getbb_3( hash_data_3* p, const realType* const elx[3], const unsigned n[3], uint nel, realType tol )
{
    obbox_data_3* data;
    const realType* x[3];
    realType *z[3], *w[3];
    uint i;
    unsigned d;
    const unsigned nn = n[0] * n[1] * n[2];
    unsigned int m[3];
 
    for( i = 0; i < 3; ++i )
    {
        x[i] = elx[i];
        m[i] = 2 * n[i];
    }
 
    z[0] = tmalloc( realType, 2 * ( n[0] + n[1] + n[2] ) );
    w[0] = z[0] + n[0];
    for( d = 1; d < 3; ++d )
        z[d] = w[d - 1] + n[d - 1], w[d] = z[d] + n[d];
    for( d = 0; d < 3; ++d )
        lobatto_nodes( z[d], n[d] ), lobatto_weights( z[d], w[d], n[d] );
    data = obbox_setup_3( (const realType* const*)z, (const realType* const*)w, n, m );
    obbox_calc_3( data, tol, x[0], x[1], x[2], &p->obb[0] );
    memcpy( &p->bnd[0], (const realType*)&p->obb[0].axis_bnd[0], 6 * sizeof( realType ) );
    for( i = 0; i < nel; ++i, x[0] += nn, x[1] += nn, x[2] += nn )
    {
        obbox_calc_3( data, tol, x[0], x[1], x[2], &p->obb[i] );
        obbox_merge_3( &p->bnd[0], (const realType*)&p->obb[i].axis_bnd[0] );
    }
    obbox_free_3( data );
    free( z[0] );
}
 
static void hash_build_2( hash_data_2* p,
                          const realType* const x[2],
                          const unsigned n[2],
                          uint nel,
                          uint max_hash_size,
                          realType tol )
{
    uint i, el, size, hn2, sum;
    unsigned hn;
    unsigned* count;
    p->obb = tmalloc( obbox_2, nel );
    hash_getbb_2( p, x, n, nel, tol );
    size      = hash_opt_size_2( p, nel, max_hash_size );
    p->offset = tmalloc( uint, size );
    hn        = p->hash_n;
    hn2       = (uint)hn * hn;
    count     = tcalloc( unsigned, hn2 );
    for( el = 0; el < nel; ++el )
    {
        unsigned ia, ib, j, ja, jb;
        hash_range_2( p, el, 0, &ia, &ib );
        hash_range_2( p, el, 1, &ja, &jb );
        for( j = ja; j < jb; ++j )
            for( i = ia; i < ib; ++i )
                ++count[(uint)j * hn + i];
    }
    sum = hn2 + 1, p->max = count[0];
    p->offset[0] = sum;
    for( i = 0; i < hn2; ++i )
    {
        if( count[i] > p->max ) p->max = count[i];
        sum += count[i];
        p->offset[i + 1] = sum;
    }
    for( el = 0; el < nel; ++el )
    {
        unsigned ia, ib, j, ja, jb;
        hash_range_2( p, el, 0, &ia, &ib );
        hash_range_2( p, el, 1, &ja, &jb );
        for( j = ja; j < jb; ++j )
            for( i = ia; i < ib; ++i )
            {
                uint arrindex                                        = (uint)j * hn + i;
                p->offset[p->offset[arrindex + 1] - count[arrindex]] = el;
                --count[arrindex];
            }
    }
    free( count );
}
 
static void hash_build_3( hash_data_3* p,
                          const realType* const x[3],
                          const unsigned n[3],
                          uint nel,
                          uint max_hash_size,
                          realType tol )
{
    uint i, el, size, hn3, sum;
    unsigned hn;
    unsigned* count;
    p->obb = tmalloc( obbox_3, nel );
    hash_getbb_3( p, x, n, nel, tol );
    size      = hash_opt_size_3( p, nel, max_hash_size );
    p->offset = tmalloc( uint, size );
    hn        = p->hash_n;
    hn3       = (uint)hn * hn * hn;
    count     = tcalloc( unsigned, hn3 );
    for( el = 0; el < nel; ++el )
    {
        unsigned ia, ib, j, ja, jb, k, ka, kb;
        hash_range_3( p, el, 0, &ia, &ib );
        hash_range_3( p, el, 1, &ja, &jb );
        hash_range_3( p, el, 2, &ka, &kb );
        for( k = ka; k < kb; ++k )
            for( j = ja; j < jb; ++j )
                for( i = ia; i < ib; ++i )
                    ++count[( (uint)k * hn + j ) * hn + i];
    }
    sum = hn3 + 1, p->max = count[0];
    p->offset[0] = sum;
    for( i = 0; i < hn3; ++i )
    {
        if( count[i] > p->max ) p->max = count[i];
        sum += count[i];
        p->offset[i + 1] = sum;
    }
    for( el = 0; el < nel; ++el )
    {
        unsigned ia, ib, j, ja, jb, k, ka, kb;
        hash_range_3( p, el, 0, &ia, &ib );
        hash_range_3( p, el, 1, &ja, &jb );
        hash_range_3( p, el, 2, &ka, &kb );
        for( k = ka; k < kb; ++k )
            for( j = ja; j < jb; ++j )
                for( i = ia; i < ib; ++i )
                {
                    uint arrindex                                        = ( (uint)k * hn + j ) * hn + i;
                    p->offset[p->offset[arrindex + 1] - count[arrindex]] = el;
                    --count[arrindex];
                }
    }
    free( count );
}
 
static void hash_free_2( hash_data_2* p )
{
    free( p->obb );
    free( p->offset );
}
 
static void hash_free_3( hash_data_3* p )
{
    free( p->obb );
    free( p->offset );
}
 
/*--------------------------------------------------------------------------
   Optimization algorithm to find a point within an element
 
   Given x(r)  (as values of x,y,z at all Lobatto nodes) and x_star,
   find the r that minimizes || x_star - x(r) ||_2
 
   As a minimization problem, the Newton step is
 
               __ 3
     [ J^T J - >_ d=1  resid_d H_d ] dr = J^t resid
 
   where resid = x_star - x(r), J = [ dx_i/dr_j ],
   and H_d = [ d^2 x_d/dr_i dr_j ].
 
   This is the appropriate step to take whenever constraints are active,
   and the current iterate is on a boundary of the element. When the current
   iterate is inside, J is square ( dim r = dim x ), resid will become small,
   and the step
 
     J dr = resid
 
   may be used instead, still giving quadratic convergence.
 
 
   Names use a _3 suffix for 3-d and _2 for 2-d.
   The routines require an initialized lagrange_data array as input:
     unsigned d, n[3] = { ... };
     realType *z[3] = { tmalloc(realType, n[0]), ... };
     for(d=0;d<3;++d) lobatto_nodes(z[d],n[d]);
 
     lagrange_data ld[3];
     for(d=0;d<3;++d) lagrange_setup(&ld[d],z[d],n[d]);
 
   Initialization:
     opt_data_3 data;
     opt_alloc_3(&data, ld);
 
   Use:
     const realType *xm[3];  // 3 pointers, each to n[0]*n[1]*n[2] realTypes
                         //   giving the nodal x, y, or z coordinates
 
     const realType x_star[3] = { ... };  // point to find
     realType r[3] = { 0,0,0 };           // initial guess with
     unsigned c = opt_no_constraints_3;   //   these constraints active
 
     realType dist = opt_findpt_3(&data,xm,x_star,r,&c);
     // minimizer is r with constraints c; 2-norm of resid is dist
 
   Clean-up:
     opt_free_3(&data);
 
     for(d=0;d<3;++d) lagrange_free(&ld[d]);
     for(d=0;d<3;++d) free(z[d]);
 
   The constraint number works as follows. Let cr be the constraints
   on the r variable:
      cr = 0       r fixed at -1
      cr = 1       r not fixed
      cr = 2       r fixed at 1
   Then the constraint number is (ct*3+cs)*3+cr
 
  --------------------------------------------------------------------------*/
 
/* how many directions are constrained? */
static const char opt_constr_num_2[9]  = { 2, 1, 2, 1, 0, 1, 2, 1, 2 };
static const char opt_constr_num_3[27] = { 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 1, 0,
                                           1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3 };
 
/* which direction is constrained? */
/* static const char opt_constr_dir_2[9] = {-1, 1,-1,  0,-1, 0, -1, 1,-1}; */
static const char opt_constr_dir_3[27] = { -1, -1, -1, -1, 2,  -1, -1, -1, -1, -1, 1,  -1, 0, -1,
                                           0,  -1, 1,  -1, -1, -1, -1, -1, 2,  -1, -1, -1, -1 };
 
/* which direction is not constrained? */
static const char opt_constr_not[27] = { -1, 0, -1, 1, -1, 1, -1, 0, -1, 2, -1, 2, -1, -1,
                                         -1, 2, -1, 2, -1, 0, -1, 1, -1, 1, -1, 0, -1 };
 
static const char opt_constr_wide[27] = { 0x00, 0x01, 0x02, 0x04, 0x05, 0x06, 0x08, 0x09, 0x0a,
                                          0x10, 0x11, 0x12, 0x14, 0x15, 0x16, 0x18, 0x19, 0x1a,
                                          0x20, 0x21, 0x22, 0x24, 0x25, 0x26, 0x28, 0x29, 0x2a };
 
static const unsigned opt_other1_3[3] = { 1, 0, 0 }, opt_other2_3[3] = { 2, 2, 1 };
 
static unsigned opt_constr( unsigned constraints, unsigned d )
{
    return ( opt_constr_wide[constraints] >> ( d * 2 ) ) & 3;
}
 
static void opt_constr_unpack_2( unsigned constraints, unsigned* c )
{
    const char cw = opt_constr_wide[constraints];
    c[0]          = cw & 3;
    c[1]          = cw >> 2;
}
 
static void opt_constr_unpack_3( unsigned constraints, unsigned* c )
{
    const char cw = opt_constr_wide[constraints];
    c[0]          = cw & 3;
    c[1]          = ( cw >> 2 ) & 3;
    c[2]          = cw >> 4;
}
 
static unsigned opt_constr_pack_2( const unsigned* c )
{
    return c[1] * 3 + c[0];
}
 
static unsigned opt_constr_pack_3( const unsigned* c )
{
    return ( c[2] * 3 + c[1] ) * 3 + c[0];
}
 
/*--------------------------------------------------------------------------
 
   3 - D
 
  --------------------------------------------------------------------------*/
 
void opt_alloc_3( opt_data_3* p, lagrange_data* ld )
{
    const unsigned nr = ld[0].n, ns = ld[1].n, nt = ld[2].n, nf = umax_3( nr * ns, nr * nt, ns * nt ),
                   ne = umax_3( nr, ns, nt ), nw = 2 * ns * nt + 3 * ns;
    p->size[0]   = 1;
    p->size[1]   = nr;
    p->size[2]   = nr * ns;
    p->size[3]   = p->size[2] * nt;
    p->ld        = ld;
    p->work      = tmalloc( realType, 6 * nf + 9 * ne + nw );
    p->fd.x[0]   = p->work + nw;
    p->fd.x[1]   = p->fd.x[0] + nf;
    p->fd.x[2]   = p->fd.x[1] + nf;
    p->fd.fdn[0] = p->fd.x[2] + nf;
    p->fd.fdn[1] = p->fd.fdn[0] + nf;
    p->fd.fdn[2] = p->fd.fdn[1] + nf;
    p->ed.x[0]   = p->fd.fdn[2] + nf;
    p->ed.x[1]   = p->ed.x[0] + ne;
    p->ed.x[2]   = p->ed.x[1] + ne;
    p->ed.fd1[0] = p->ed.x[2] + ne;
    p->ed.fd1[1] = p->ed.fd1[0] + ne;
    p->ed.fd1[2] = p->ed.fd1[1] + ne;
    p->ed.fd2[0] = p->ed.fd1[2] + ne;
    p->ed.fd2[1] = p->ed.fd2[0] + ne;
    p->ed.fd2[2] = p->ed.fd2[1] + ne;
}
 
void opt_free_3( opt_data_3* p )
{
    free( p->work );
}
 
static void opt_vol_set_3( opt_data_3* p, const realType r[3] )
{
    lagrange_1( &p->ld[0], r[0] );
    lagrange_1( &p->ld[1], r[1] );
    lagrange_1( &p->ld[2], r[2] );
}
 
/* work holds 2*ns*nt + 3*ns realTypes */
static void opt_vol_intp_3( opt_data_3* p )
{
    unsigned d;
    const lagrange_data* ld = p->ld;
 
    for( d = 0; d < 3; ++d )
        p->x[d] = tensor_ig3( ld[0].J, ld[0].D, ld[0].n, ld[1].J, ld[1].D, ld[1].n, ld[2].J, ld[2].D, ld[2].n,
                              p->elx[d], &p->jac[d * 3], p->work );
}
 
void opt_vol_set_intp_3( opt_data_3* p, const realType r[3] )
{
    opt_vol_set_3( p, r );
    opt_vol_intp_3( p );
}
 
static void opt_face_proj_3( opt_data_3* p )
{
    unsigned d, off = 0;
    const unsigned dn = p->fd.dn, d1 = p->fd.d1, d2 = p->fd.d2, so = p->size[d2] - p->size[d1 + 1], s1 = p->size[d1],
                   sn = p->size[dn], n1 = p->ld[d1].n, n2 = p->ld[d2].n, nn = p->ld[dn].n;
    const realType* D = p->ld[dn].D_z0;
    if( opt_constr( p->fd.constraints, dn ) == 2 ) off = p->size[dn + 1] - p->size[dn], D = p->ld[dn].D_zn;
    for( d = 0; d < 3; ++d )
    {
        unsigned i, j, k, arrindex = 0;
        const realType* in = p->elx[d] + off;
        for( j = n2; j; --j, in += so )
            for( i = n1; i; --i, ++arrindex, in += s1 )
            {
                const realType* ind  = in - off;
                realType* fdn        = &p->fd.fdn[d][arrindex];
                p->fd.x[d][arrindex] = *in;
                *fdn                 = 0;
                for( k = 0; k < nn; ++k, ind += sn )
                    *fdn += *ind * D[k];
            }
    }
}
 
static void opt_face_set_3( opt_data_3* p, const realType r[3], unsigned constr )
{
    if( p->fd.constraints != constr )
    {
        p->fd.constraints = constr;
        p->fd.dn          = opt_constr_dir_3[constr];
        p->fd.d1          = opt_other1_3[p->fd.dn];
        p->fd.d2          = opt_other2_3[p->fd.dn];
        opt_face_proj_3( p );
    }
    lagrange_1( &p->ld[p->fd.d1], r[p->fd.d1] );
    lagrange_1( &p->ld[p->fd.d2], r[p->fd.d2] );
}
 
/* work holds 2*ld[d2].n realTypes */
static void opt_face_intp_3( opt_data_3* p )
{
    unsigned d;
    const unsigned dn = p->fd.dn, d1 = p->fd.d1, d2 = p->fd.d2, n1 = p->ld[d1].n, n2 = p->ld[d2].n;
    const realType *J1 = p->ld[d1].J, *J2 = p->ld[d2].J, *D1 = p->ld[d1].D, *D2 = p->ld[d2].D;
 
    for( d = 0; d < 3; ++d )
    {
        realType g[2];
        p->x[d]            = tensor_ig2( J1, D1, n1, J2, D2, n2, p->fd.x[d], &g[0], p->work );
        p->jac[d * 3 + d1] = g[0];
        p->jac[d * 3 + d2] = g[1];
        p->jac[d * 3 + dn] = tensor_i2( J1, n1, J2, n2, p->fd.fdn[d], p->work );
    }
}
 
static void opt_face_set_intp_3( opt_data_3* p, const realType r[3], unsigned constr )
{
    opt_face_set_3( p, r, constr );
    opt_face_intp_3( p );
}
 
static void opt_face_hess_3( opt_data_3* p, realType hess[9] )
{
    unsigned d;
    const unsigned d1 = p->fd.d1, d2 = p->fd.d2, n1 = p->ld[d1].n, n2 = p->ld[d2].n;
    const realType *J1 = p->ld[d1].J, *J2 = p->ld[d2].J, *D1 = p->ld[d1].D, *D2 = p->ld[d2].D, *S1 = p->ld[d1].D2,
                   *S2 = p->ld[d2].D2;
 
    lagrange_2u( &p->ld[d1] );
    lagrange_2u( &p->ld[d2] );
 
    for( d = 0; d < 3; ++d )
    {
        (void)tensor_ig2( J1, S1, n1, J2, S2, n2, p->fd.x[d], hess + d * 3, p->work );
        hess[d * 3 + 0] = tensor_i2( S1, n1, J2, n2, p->fd.x[d], p->work );
        hess[d * 3 + 1] = tensor_i2( J1, n1, S2, n2, p->fd.x[d], p->work );
        hess[d * 3 + 2] = tensor_i2( D1, n1, D2, n2, p->fd.x[d], p->work );
    }
}
 
static void opt_edge_proj_3( opt_data_3* p )
{
    unsigned d, off, off1 = 0, off2 = 0;
    const unsigned de = p->ed.de, d1 = p->ed.d1, d2 = p->ed.d2, se = p->size[de], s1 = p->size[d1], s2 = p->size[d2],
                   ne = p->ld[de].n, n1 = p->ld[d1].n, n2 = p->ld[d2].n;
    const realType *fD1, *fD2;
    if( opt_constr( p->ed.constraints, d1 ) == 0 )
        fD1 = p->ld[d1].D_z0;
    else
        fD1 = p->ld[d1].D_zn, off1 = p->size[d1 + 1] - p->size[d1];
    if( opt_constr( p->ed.constraints, d2 ) == 0 )
        fD2 = p->ld[d2].D_z0;
    else
        fD2 = p->ld[d2].D_zn, off2 = p->size[d2 + 1] - p->size[d2];
    off = off1 + off2;
    for( d = 0; d < 3; ++d )
    {
        unsigned i, j;
        const realType* in = p->elx[d] + off;
        for( i = 0; i < ne; ++i, in += se )
        {
            const realType *in1 = in - off1, *in2 = in - off2;
            realType *fd1 = &p->ed.fd1[d][i], *fd2 = &p->ed.fd2[d][i];
            p->ed.x[d][i] = *in;
            *fd1 = *fd2 = 0;
            for( j = 0; j < n1; ++j, in1 += s1 )
                *fd1 += *in1 * fD1[j];
            for( j = 0; j < n2; ++j, in2 += s2 )
                *fd2 += *in2 * fD2[j];
        }
    }
}
 
static void opt_edge_set_3( opt_data_3* p, const realType r[3], unsigned constr )
{
    if( p->ed.constraints != constr )
    {
        p->ed.constraints = constr;
        p->ed.de          = opt_constr_not[constr];
        p->ed.d1          = opt_other1_3[p->ed.de];
        p->ed.d2          = opt_other2_3[p->ed.de];
        opt_edge_proj_3( p );
    }
    lagrange_1( &p->ld[p->ed.de], r[p->ed.de] );
}
 
static void opt_edge_intp_3( opt_data_3* p )
{
    unsigned d;
    const unsigned de = p->ed.de, d1 = p->ed.d1, d2 = p->ed.d2, n = p->ld[de].n;
    const realType *J = p->ld[de].J, *D = p->ld[de].D;
 
    for( d = 0; d < 3; ++d )
    {
        p->x[d]            = tensor_ig1( J, D, n, p->ed.x[d], &p->jac[d * 3 + de] );
        p->jac[d * 3 + d1] = tensor_i1( J, n, p->ed.fd1[d] );
        p->jac[d * 3 + d2] = tensor_i1( J, n, p->ed.fd2[d] );
    }
}
 
static void opt_edge_set_intp_3( opt_data_3* p, const realType r[3], unsigned constr )
{
    opt_edge_set_3( p, r, constr );
    opt_edge_intp_3( p );
}
 
static void opt_edge_hess_3( opt_data_3* p, realType hess[3] )
{
    unsigned d;
    const unsigned de = p->ed.de, n = p->ld[de].n;
    const realType* D2 = p->ld[de].D2;
    lagrange_2u( &p->ld[de] );
    for( d = 0; d < 3; ++d )
        hess[d] = tensor_i1( D2, n, p->ed.x[d] );
}
 
static void opt_point_proj_3( opt_data_3* p )
{
    unsigned off[3], offt, d, c[3];
    const realType* fD[3];
    opt_constr_unpack_3( p->pd.constraints, c );
    for( d = 0; d < 3; ++d )
        if( c[d] == 0 )
            fD[d] = p->ld[d].D_z0, off[d] = 0;
        else
            fD[d] = p->ld[d].D_zn, off[d] = p->size[d + 1] - p->size[d];
    offt = off[0] + off[1] + off[2];
    for( d = 0; d < 9; ++d )
        p->pd.jac[d] = 0;
    for( d = 0; d < 3; ++d )
    {
        unsigned i, j;
        p->pd.x[d] = p->elx[d][offt];
        for( i = 0; i < 3; ++i )
        {
            const realType* in = p->elx[d] + offt - off[i];
            for( j = 0; j < p->ld[i].n; ++j, in += p->size[i] )
                p->pd.jac[d * 3 + i] += *in * fD[i][j];
        }
    }
}
 
static void opt_point_set_3( opt_data_3* p, unsigned constr )
{
    if( p->pd.constraints != constr )
    {
        p->pd.constraints = constr;
        opt_point_proj_3( p );
    }
}
 
static void opt_point_intp_3( opt_data_3* p )
{
    memcpy( p->x, p->pd.x, 3 * sizeof( realType ) );
    memcpy( p->jac, p->pd.jac, 9 * sizeof( realType ) );
}
 
static void opt_point_set_intp_3( opt_data_3* p, unsigned constr )
{
    opt_point_set_3( p, constr );
    opt_point_intp_3( p );
}
 
#define DIAGNOSTICS 0
 
double opt_findpt_3( opt_data_3* p,
                     const realType* const elx[3],
                     const realType xstar[3],
                     realType r[3],
                     unsigned* constr )
{
    realType dr[3], resid[3], steep[3];
 
    unsigned c = *constr, ac, d, cc[3], step = 0;
 
    p->elx[0] = elx[0], p->elx[1] = elx[1], p->elx[2] = elx[2];
 
    p->fd.constraints = opt_no_constraints_3;
    p->ed.constraints = opt_no_constraints_3;
    p->pd.constraints = opt_no_constraints_3;
 
#if DIAGNOSTICS
    printf( "opt_findpt: xstar = %g, %g, %g\n", xstar[0], xstar[1], xstar[2] );
#endif
 
    do
    {
        ++step;
        if( step == 50 ) /*fail("%s: opt_findpt_3 did not converge\n",__FILE__);*/
            return 1.e+30;
#if DIAGNOSTICS
        printf( "  iteration %u\n", step );
        printf( "    %d constraint(s) active\n", (int)opt_constr_num_3[c] );
#endif
        /* update face/edge/point data if necessary,
           and evaluate x(r) as well as the jacobian */
        switch( opt_constr_num_3[c] )
        {
            case 0:
                opt_vol_set_intp_3( p, r );
                break;
            case 1:
                opt_face_set_intp_3( p, r, c );
                break;
            case 2:
                opt_edge_set_intp_3( p, r, c );
                break;
            case 3:
                opt_point_set_intp_3( p, c );
                break;
        }
#if DIAGNOSTICS
        printf( "    r = %g, %g, %g\n", r[0], r[1], r[2] );
        printf( "    x = %g, %g, %g\n", p->x[0], p->x[1], p->x[2] );
#endif
        /* compute residual */
        for( d = 0; d < 3; ++d )
            resid[d] = xstar[d] - p->x[d];
#if DIAGNOSTICS
        printf( "    resid = %g, %g, %g\n", resid[0], resid[1], resid[2] );
        printf( "    2-norm = %g\n", r2norm_3( resid[0], resid[1], resid[2] ) );
#endif
        /* check constraints against steepest descent direction */
        ac = c;
        if( opt_constr_num_3[c] )
        {
            opt_constr_unpack_3( c, cc );
            mat_app_3c( steep, p->jac, resid ); /* steepest descent = J^T r */
#if DIAGNOSTICS
            printf( "    steepest descent = %g, %g, %g\n", steep[0], steep[1], steep[2] );
#endif
            for( d = 0; d < 3; ++d )
                if( ( cc[d] == 0 && steep[d] > 0 ) || ( cc[d] == 2 && steep[d] < 0 ) ) cc[d] = 1;
            ac = opt_constr_pack_3( cc );
        }
        /* update face/edge/point data if necessary */
        if( ac != c )
        {
            c = ac;
#if DIAGNOSTICS
            printf( "    relaxed to %d constraints\n", (int)opt_constr_num_3[c] );
#endif
            switch( opt_constr_num_3[c] )
            {
                case 1:
                    opt_face_set_3( p, r, c );
                    break;
                case 2:
                    opt_edge_set_3( p, r, c );
                    break;
                case 3:
                    opt_point_set_3( p, c );
                    break;
            }
        }
        /* compute Newton step */
        switch( opt_constr_num_3[c] )
        {
            case 0:
                tinyla_solve_3( dr, p->jac, resid );
                break;
            case 1: {
                const unsigned dn = p->fd.dn, d1 = p->fd.d1, d2 = p->fd.d2;
                realType A[4], H[9];
                const realType* J = p->jac;
                opt_face_hess_3( p, H );
                A[0] = J[d1] * J[d1] + J[3 + d1] * J[3 + d1] + J[6 + d1] * J[6 + d1];
                A[1] = J[d2] * J[d2] + J[3 + d2] * J[3 + d2] + J[6 + d2] * J[6 + d2];
                A[2] = J[d1] * J[d2] + J[3 + d1] * J[3 + d2] + J[6 + d1] * J[6 + d2];
                A[0] -= resid[0] * H[0] + resid[1] * H[3] + resid[2] * H[6];
                A[1] -= resid[0] * H[1] + resid[1] * H[4] + resid[2] * H[7];
                A[2] -= resid[0] * H[2] + resid[1] * H[5] + resid[2] * H[8];
                tinyla_solve_sym_2( &dr[d1], &dr[d2], A, steep[d1], steep[d2] );
                dr[dn] = 0;
            }
            break;
            case 2: {
                const unsigned de = p->ed.de, d1 = p->ed.d1, d2 = p->ed.d2;
                realType fac, H[3];
                const realType* J = p->jac + de;
                opt_edge_hess_3( p, H );
                fac = J[0] * J[0] + J[3] * J[3] + J[6] * J[6] - ( resid[0] * H[0] + resid[1] * H[1] + resid[2] * H[2] );
                dr[de] = steep[de] / fac;
                dr[d1] = 0, dr[d2] = 0;
            }
            break;
            case 3:
                dr[0] = dr[1] = dr[2] = 0;
                break;
        }
#if DIAGNOSTICS
        printf( "    dr = %g, %g, %g\n", dr[0], dr[1], dr[2] );
#endif
        /* project new iteration onto [-1,1]^3 */
        opt_constr_unpack_3( c, cc );
        for( d = 0; d < 3; ++d )
        {
            if( cc[d] != 1 ) continue;
            r[d] += dr[d];
            if( r[d] <= -1 )
                dr[d] -= r[d] + 1, r[d] = -1, cc[d] = 0;
            else if( r[d] >= 1 )
                dr[d] -= r[d] - 1, r[d] = 1, cc[d] = 2;
        }
        c = opt_constr_pack_3( cc );
    } while( r1norm_3( dr[0], dr[1], dr[2] ) > 3 * MOAB_POLY_EPS );
    *constr = c;
#if 0
  printf("opt_findpt_3 converged in %u iterations\n", step);
#endif
    return r2norm_3( resid[0], resid[1], resid[2] );
}
 
#undef DIAGNOSTICS
 
void opt_alloc_2( opt_data_2* p, lagrange_data* ld )
{
    const unsigned nr = ld[0].n, ns = ld[1].n, ne = umax_2( nr, ns ), nw = 2 * ns;
    p->size[0]   = 1;
    p->size[1]   = nr;
    p->size[2]   = nr * ns;
    p->ld        = ld;
    p->work      = tmalloc( realType, 4 * ne + nw );
    p->ed.x[0]   = p->work + nw;
    p->ed.x[1]   = p->ed.x[0] + ne;
    p->ed.fd1[0] = p->ed.x[1] + ne;
    p->ed.fd1[1] = p->ed.fd1[0] + ne;
}
 
void opt_free_2( opt_data_2* p )
{
    free( p->work );
}
 
static void opt_area_set_2( opt_data_2* p, const realType r[2] )
{
    lagrange_1( &p->ld[0], r[0] );
    lagrange_1( &p->ld[1], r[1] );
}
 
/* work holds 2*ns realTypes */
static void opt_area_intp_2( opt_data_2* p )
{
    unsigned d;
    const lagrange_data* ld = p->ld;
 
    for( d = 0; d < 2; ++d )
        p->x[d] =
            tensor_ig2( ld[0].J, ld[0].D, ld[0].n, ld[1].J, ld[1].D, ld[1].n, p->elx[d], &p->jac[d * 2], p->work );
}
 
static void opt_area_set_intp_2( opt_data_2* p, const realType r[2] )
{
    opt_area_set_2( p, r );
    opt_area_intp_2( p );
}
 
static void opt_edge_proj_2( opt_data_2* p )
{
    unsigned d, off = 0;
    const unsigned de = p->ed.de, d1 = p->ed.d1, se = p->size[de], s1 = p->size[d1], ne = p->ld[de].n, n1 = p->ld[d1].n;
    const realType* fD1;
    if( opt_constr( p->ed.constraints, d1 ) == 0 )
        fD1 = p->ld[d1].D_z0;
    else
        fD1 = p->ld[d1].D_zn, off = p->size[d1 + 1] - p->size[d1];
    for( d = 0; d < 2; ++d )
    {
        unsigned i, j;
        const realType* in = p->elx[d] + off;
        for( i = 0; i < ne; ++i, in += se )
        {
            const realType* in1 = in - off;
            realType* fd1       = &p->ed.fd1[d][i];
            p->ed.x[d][i]       = *in;
            *fd1                = 0;
            for( j = 0; j < n1; ++j, in1 += s1 )
                *fd1 += *in1 * fD1[j];
        }
    }
}
 
static void opt_edge_set_2( opt_data_2* p, const realType r[2], unsigned constr )
{
    if( p->ed.constraints != constr )
    {
        p->ed.constraints = constr;
        p->ed.de          = opt_constr_not[constr];
        p->ed.d1          = 1 - p->ed.de;
        opt_edge_proj_2( p );
    }
    lagrange_1( &p->ld[p->ed.de], r[p->ed.de] );
}
 
static void opt_edge_intp_2( opt_data_2* p )
{
    unsigned d;
    const unsigned de = p->ed.de, d1 = p->ed.d1, n = p->ld[de].n;
    const realType *J = p->ld[de].J, *D = p->ld[de].D;
    for( d = 0; d < 2; ++d )
    {
        p->x[d]            = tensor_ig1( J, D, n, p->ed.x[d], &p->jac[d * 2 + de] );
        p->jac[d * 2 + d1] = tensor_i1( J, n, p->ed.fd1[d] );
    }
}
 
static void opt_edge_set_intp_2( opt_data_2* p, const realType r[2], unsigned constr )
{
    opt_edge_set_2( p, r, constr );
    opt_edge_intp_2( p );
}
 
static void opt_edge_hess_2( opt_data_2* p, realType hess[2] )
{
    unsigned d;
    const unsigned de = p->ed.de, n = p->ld[de].n;
    const realType* D2 = p->ld[de].D2;
    lagrange_2u( &p->ld[de] );
    for( d = 0; d < 2; ++d )
        hess[d] = tensor_i1( D2, n, p->ed.x[d] );
}
 
static void opt_point_proj_2( opt_data_2* p )
{
    unsigned off[2], offt, d, c[2];
    const realType* fD[2];
    opt_constr_unpack_2( p->pd.constraints, c );
    for( d = 0; d < 2; ++d )
        if( c[d] == 0 )
            fD[d] = p->ld[d].D_z0, off[d] = 0;
        else
            fD[d] = p->ld[d].D_zn, off[d] = p->size[d + 1] - p->size[d];
    offt = off[0] + off[1];
    for( d = 0; d < 4; ++d )
        p->pd.jac[d] = 0;
    for( d = 0; d < 2; ++d )
    {
        unsigned i, j;
        p->pd.x[d] = p->elx[d][offt];
        for( i = 0; i < 2; ++i )
        {
            const realType* in = p->elx[d] + offt - off[i];
            for( j = 0; j < p->ld[i].n; ++j, in += p->size[i] )
                p->pd.jac[d * 2 + i] += *in * fD[i][j];
        }
    }
}
 
static void opt_point_set_2( opt_data_2* p, unsigned constr )
{
    if( p->pd.constraints != constr )
    {
        p->pd.constraints = constr;
        opt_point_proj_2( p );
    }
}
 
static void opt_point_intp_2( opt_data_2* p )
{
    memcpy( p->x, p->pd.x, 2 * sizeof( realType ) );
    memcpy( p->jac, p->pd.jac, 4 * sizeof( realType ) );
}
 
static void opt_point_set_intp_2( opt_data_2* p, unsigned constr )
{
    opt_point_set_2( p, constr );
    opt_point_intp_2( p );
}
 
#define DIAGNOSTICS 0
 
double opt_findpt_2( opt_data_2* p,
                     const realType* const elx[2],
                     const realType xstar[2],
                     realType r[2],
                     unsigned* constr )
{
    realType dr[2], resid[2], steep[2];
 
    unsigned c = *constr, ac, d, cc[2], step = 0;
 
    p->elx[0] = elx[0], p->elx[1] = elx[1];
 
    p->ed.constraints = opt_no_constraints_2;
    p->pd.constraints = opt_no_constraints_2;
 
#if DIAGNOSTICS
    printf( "opt_findpt: xstar = %g, %g\n", xstar[0], xstar[1] );
#endif
 
    do
    {
        ++step;
        if( step == 50 ) /*fail("%s: opt_findpt_2 did not converge\n",__FILE__);*/
            return 1.e+30;
#if DIAGNOSTICS
        printf( "  iteration %u\n", step );
        printf( "    %d constraint(s) active\n", (int)opt_constr_num_2[c] );
#endif
        /* update face/edge/point data if necessary,
           and evaluate x(r) as well as the jacobian */
        switch( opt_constr_num_2[c] )
        {
            case 0:
                opt_area_set_intp_2( p, r );
                break;
            case 1:
                opt_edge_set_intp_2( p, r, c );
                break;
            case 2:
                opt_point_set_intp_2( p, c );
                break;
        }
#if DIAGNOSTICS
        printf( "    r = %g, %g\n", r[0], r[1] );
        printf( "    x = %g, %g\n", p->x[0], p->x[1] );
#endif
        /* compute residual */
        for( d = 0; d < 2; ++d )
            resid[d] = xstar[d] - p->x[d];
#if DIAGNOSTICS
        printf( "    resid = %g, %g\n", resid[0], resid[1] );
        printf( "    2-norm = %g\n", r2norm_2( resid[0], resid[1] ) );
#endif
        /* check constraints against steepest descent direction */
        ac = c;
        if( opt_constr_num_2[c] )
        {
            opt_constr_unpack_2( c, cc );
            mat_app_2c( steep, p->jac, resid ); /* steepest descent = J^T r */
#if DIAGNOSTICS
            printf( "    steepest descent = %g, %g\n", steep[0], steep[1] );
#endif
            for( d = 0; d < 2; ++d )
                if( ( cc[d] == 0 && steep[d] > 0 ) || ( cc[d] == 2 && steep[d] < 0 ) ) cc[d] = 1;
            ac = opt_constr_pack_2( cc );
        }
        /* update face/edge/point data if necessary */
        if( ac != c )
        {
            c = ac;
#if DIAGNOSTICS
            printf( "    relaxed to %d constraints\n", (int)opt_constr_num_2[c] );
#endif
            switch( opt_constr_num_2[c] )
            {
                case 1:
                    opt_edge_set_2( p, r, c );
                    break;
                case 2:
                    opt_point_set_2( p, c );
                    break;
            }
        }
        /* compute Newton step */
        switch( opt_constr_num_2[c] )
        {
            case 0:
                tinyla_solve_2( dr, p->jac, resid );
                break;
            case 1: {
                const unsigned de = p->ed.de, d1 = p->ed.d1;
                realType fac, H[2];
                const realType* J = p->jac + de;
                opt_edge_hess_2( p, H );
                fac    = J[0] * J[0] + J[2] * J[2] - ( resid[0] * H[0] + resid[1] * H[1] );
                dr[de] = steep[de] / fac;
                dr[d1] = 0;
            }
            break;
            case 2:
                dr[0] = dr[1] = 0;
                break;
        }
#if DIAGNOSTICS
        printf( "    dr = %g, %g\n", dr[0], dr[1] );
#endif
        /* project new iteration onto [-1,1]^2 */
        opt_constr_unpack_2( c, cc );
        for( d = 0; d < 2; ++d )
        {
            if( cc[d] != 1 ) continue;
            r[d] += dr[d];
            if( r[d] <= -1 )
                dr[d] -= r[d] + 1, r[d] = -1, cc[d] = 0;
            else if( r[d] >= 1 )
                dr[d] -= r[d] - 1, r[d] = 1, cc[d] = 2;
        }
        c = opt_constr_pack_2( cc );
    } while( r1norm_2( dr[0], dr[1] ) > 2 * MOAB_POLY_EPS );
    *constr = c;
    return r2norm_2( resid[0], resid[1] );
}
 
#undef DIAGNOSTICS
 
/*--------------------------------------------------------------------------
   Point Finding  (interface/top-level)
 
   Initializing the data:
     unsigned nel;        // number of elements
     const unsigned n[3]; // number of nodes in r, s, t directions
     const realType *xm[3];   // n[0]*n[1]*n[2]*nel x,y,z coordinates
     realType tol = 0.01;     // how far point is allowed to be outside element
                          //   relative to element size
     unsigned max_size = n[0]*n[1]*n[2]*nel; // maximum size of hash table
 
     findpt_data_3 *data = findpt_setup_3(xm,n,nel,max_size,tol);
 
   Using the data:
     realType x[3] = { ... };   // point to find
     int el; // element number
     realType r[3]; // parametric coordinates
     int guess = 0; // do we use (el,r,s,t) as an initial guess?
     int code; // 0 : normal, -1 : outside all elements,
               // 1 : border, or outside but within tolerance
     realType dist; // distance in xyz space from returned (el,r,s,t) to given
                // (x,y,z)
 
     code = findpt_3(data, x, guess, &el, r, &dist);
 
   When done:
     findpt_free_3(&data);
 
  --------------------------------------------------------------------------*/
 
/* heap sort on A[0:n-1] with key A[i]->dist
   precondition: n!=0 */
static void findpt_list_sort( findpt_listel** A, unsigned n )
{
    unsigned i;
    --A; /* make A have a base index of 1 */
    /* build heap */
    for( i = 2; i <= n; ++i )
    {
        findpt_listel* item = A[i];
        unsigned hole = i, parent = hole >> 1;
        if( A[parent]->dist >= item->dist ) continue;
        do
        {
            A[hole] = A[parent];
            hole    = parent;
            parent >>= 1;
        } while( parent && A[parent]->dist < item->dist );
        A[hole] = item;
    }
    /* extract */
    for( i = n - 1; i; --i )
    {
        findpt_listel* item = A[i + 1];
        unsigned hole       = 1;
        A[i + 1]            = A[1];
        for( ;; )
        {
            unsigned ch = hole << 1, r = ch + 1;
            if( r <= i && A[ch]->dist < A[r]->dist ) ch = r;
            if( ch > i || item->dist >= A[ch]->dist ) break;
            A[hole] = A[ch];
            hole    = ch;
        }
        A[hole] = item;
    }
}
 
findpt_data_2* findpt_setup_2( const realType* const xw[2],
                               const unsigned n[2],
                               uint nel,
                               uint max_hash_size,
                               realType bbox_tol )
{
    unsigned d;
    findpt_data_2* p = tmalloc( findpt_data_2, 1 );
 
    p->hash = tmalloc( hash_data_2, 1 );
    p->od   = tmalloc( opt_data_2, 1 );
 
    for( d = 0; d < 2; ++d )
        p->xw[d] = xw[d];
    p->nptel = n[0] * n[1];
 
    hash_build_2( p->hash, xw, n, nel, max_hash_size, bbox_tol );
 
    for( d = 0; d < 2; ++d )
    {
        p->z[d] = tmalloc( realType, n[d] );
        lobatto_nodes( p->z[d], n[d] );
        lagrange_setup( &p->ld[d], p->z[d], n[d] );
    }
 
    p->list   = tmalloc( findpt_listel, p->hash->max );
    p->sorted = tmalloc( findpt_listel*, p->hash->max );
 
    opt_alloc_2( p->od, p->ld );
    p->od_work = p->od->work;
 
    return p;
}
 
findpt_data_3* findpt_setup_3( const realType* const xw[3],
                               const unsigned n[3],
                               uint nel,
                               uint max_hash_size,
                               realType bbox_tol )
{
    unsigned d;
    findpt_data_3* p = tmalloc( findpt_data_3, 1 );
 
    p->hash = tmalloc( hash_data_3, 1 );
    p->od   = tmalloc( opt_data_3, 1 );
 
    for( d = 0; d < 3; ++d )
        p->xw[d] = xw[d];
    p->nptel = n[0] * n[1] * n[2];
 
    hash_build_3( p->hash, xw, n, nel, max_hash_size, bbox_tol );
 
    for( d = 0; d < 3; ++d )
    {
        p->z[d] = tmalloc( realType, n[d] );
        lobatto_nodes( p->z[d], n[d] );
        lagrange_setup( &p->ld[d], p->z[d], n[d] );
    }
 
    p->list   = tmalloc( findpt_listel, p->hash->max );
    p->sorted = tmalloc( findpt_listel*, p->hash->max );
 
    opt_alloc_3( p->od, p->ld );
    p->od_work = p->od->work;
 
    return p;
}
 
void findpt_free_2( findpt_data_2* p )
{
    unsigned d;
    opt_free_2( p->od );
    free( p->od );
    hash_free_2( p->hash );
    free( p->hash );
    free( p->list );
    free( p->sorted );
    for( d = 0; d < 2; ++d )
        free( p->z[d] );
    free( p );
}
 
void findpt_free_3( findpt_data_3* p )
{
    unsigned d;
    opt_free_3( p->od );
    free( p->od );
    hash_free_3( p->hash );
    free( p->hash );
    free( p->list );
    free( p->sorted );
    for( d = 0; d < 3; ++d )
        free( p->z[d] );
    free( p );
}
 
const realType* findpt_allbnd_2( const findpt_data_2* p )
{
    return p->hash->bnd;
}
 
const realType* findpt_allbnd_3( const findpt_data_3* p )
{
    return p->hash->bnd;
}
 
static void findpt_hash_2( findpt_data_2* p, const realType x[2] )
{
    findpt_listel *list = p->list, **sorted = p->sorted;
    const uint hi      = hash_index_2( p->hash, x );
    const uint* offset = p->hash->offset;
    uint i;
    const uint b = offset[hi], e = offset[hi + 1];
    for( i = b; i != e; ++i )
    {
        const uint el      = offset[i];
        realType* r        = &list->r[0];
        const obbox_2* obb = &p->hash->obb[el];
        if( obbox_axis_test_2( obb, x ) ) continue;
        if( obbox_test_2( obb, x, r ) ) continue;
        list->el   = el;
        list->dist = r1norm_2( r[0], r[1] );
        *sorted++  = list++;
    }
    p->end = sorted;
    if( p->end != p->sorted ) findpt_list_sort( p->sorted, p->end - p->sorted );
}
 
static void findpt_hash_3( findpt_data_3* p, const realType x[3] )
{
    findpt_listel *list = p->list, **sorted = p->sorted;
    const uint hi      = hash_index_3( p->hash, x );
    const uint* offset = p->hash->offset;
    uint i;
    const uint b = offset[hi], e = offset[hi + 1];
    for( i = b; i != e; ++i )
    {
        const uint el      = offset[i];
        realType* r        = &list->r[0];
        const obbox_3* obb = &p->hash->obb[el];
        if( obbox_axis_test_3( obb, x ) ) continue;
        if( obbox_test_3( obb, x, r ) ) continue;
        list->el   = el;
        list->dist = r1norm_3( r[0], r[1], r[2] );
        *sorted++  = list++;
    }
    p->end = sorted;
    if( p->end != p->sorted ) findpt_list_sort( p->sorted, p->end - p->sorted );
}
 
static int findpt_guess_2( findpt_data_2* p, const realType x[2], uint el, realType r[2], realType* dist )
{
    const uint arrindex = p->nptel * el;
    const realType* elx[2];
    realType g[2];
    unsigned c         = opt_no_constraints_2;
    const obbox_2* obb = &p->hash->obb[el];
    elx[0]             = p->xw[0] + arrindex;
    elx[1]             = p->xw[1] + arrindex;
    if( obbox_axis_test_2( obb, x ) || obbox_test_2( obb, x, g ) ) return 0;
    *dist = opt_findpt_2( p->od, elx, x, r, &c );
    return c == opt_no_constraints_2;
}
 
static int findpt_guess_3( findpt_data_3* p, const realType x[3], uint el, realType r[3], realType* dist )
{
    const uint arrindex = p->nptel * el;
    const realType* elx[3];
    realType g[3];
    unsigned c         = opt_no_constraints_3;
    const obbox_3* obb = &p->hash->obb[el];
    elx[0]             = p->xw[0] + arrindex;
    elx[1]             = p->xw[1] + arrindex;
    elx[2]             = p->xw[2] + arrindex;
    if( obbox_axis_test_3( obb, x ) || obbox_test_3( obb, x, g ) ) return 0;
    *dist = opt_findpt_3( p->od, elx, x, r, &c );
    return c == opt_no_constraints_3;
}
 
#define DIAGNOSTICS 0
 
static int findpt_pass_2( findpt_data_2* p, const realType x[2], uint* el, realType r[2], realType* dist_min )
{
    findpt_listel** qq = p->sorted;
    const realType* bnd;
    realType dist;
    do
    {
        findpt_listel* q    = *qq;
        const uint arrindex = p->nptel * q->el;
        const realType* elx[2];
        unsigned c = opt_no_constraints_2;
        elx[0]     = p->xw[0] + arrindex;
        elx[1]     = p->xw[1] + arrindex;
        dist       = opt_findpt_2( p->od, elx, x, q->r, &c );
        if( qq == p->sorted || dist < *dist_min || c == opt_no_constraints_2 )
        {
            *dist_min = dist;
            *el       = q->el;
            memcpy( r, q->r, 2 * sizeof( realType ) );
            if( c == opt_no_constraints_2 ) return 0;
        }
    } while( ++qq != p->end );
    bnd = p->hash->obb[*el].axis_bnd;
    return *dist_min > r2norm_2( bnd[1] - bnd[0], bnd[3] - bnd[2] ) ? -1 : 1;
}
 
static int findpt_pass_3( findpt_data_3* p, const realType x[3], uint* el, realType r[3], realType* dist_min )
{
    findpt_listel** qq = p->sorted;
    const realType* bnd;
    realType dist;
    do
    {
        findpt_listel* q    = *qq;
        const uint arrindex = p->nptel * q->el;
        const realType* elx[3];
        unsigned c = opt_no_constraints_3;
        elx[0]     = p->xw[0] + arrindex;
        elx[1]     = p->xw[1] + arrindex;
        elx[2]     = p->xw[2] + arrindex;
        dist       = opt_findpt_3( p->od, elx, x, q->r, &c );
        if( qq == p->sorted || dist < *dist_min || c == opt_no_constraints_3 )
        {
            *dist_min = dist;
            *el       = q->el;
            memcpy( r, q->r, 3 * sizeof( realType ) );
            if( c == opt_no_constraints_3 )
            {
#if DIAGNOSTICS
                printf( "point found in element #%d\n", qq - p->sorted );
#endif
                return 0;
            }
        }
    } while( ++qq != p->end );
    bnd = p->hash->obb[*el].axis_bnd;
    return *dist_min > r2norm_3( bnd[1] - bnd[0], bnd[3] - bnd[2], bnd[5] - bnd[4] ) ? -1 : 1;
}
 
int findpt_2( findpt_data_2* p, const realType x[2], int guess, uint* el, realType r[2], realType* dist )
{
    if( guess && findpt_guess_2( p, x, *el, r, dist ) ) return 0;
    findpt_hash_2( p, x );
    if( p->sorted == p->end ) return -1;
    return findpt_pass_2( p, x, el, r, dist );
}
 
int findpt_3( findpt_data_3* p, const realType x[3], int guess, uint* el, realType r[3], realType* dist )
{
    if( guess && findpt_guess_3( p, x, *el, r, dist ) ) return 0;
    findpt_hash_3( p, x );
#if DIAGNOSTICS
    printf( "hashing leaves %d elements to consider\n", p->end - p->sorted );
#endif
    if( p->sorted == p->end ) return -1;
    return findpt_pass_3( p, x, el, r, dist );
}
 
void findpt_weights_2( findpt_data_2* p, const realType r[2] )
{
    lagrange_0( &p->ld[0], r[0] );
    lagrange_0( &p->ld[1], r[1] );
}
 
void findpt_weights_3( findpt_data_3* p, const realType r[3] )
{
    lagrange_0( &p->ld[0], r[0] );
    lagrange_0( &p->ld[1], r[1] );
    lagrange_0( &p->ld[2], r[2] );
}
 
double findpt_eval_2( findpt_data_2* p, const realType* u )
{
    return tensor_i2( p->ld[0].J, p->ld[0].n, p->ld[1].J, p->ld[1].n, u, p->od_work );
}
 
double findpt_eval_3( findpt_data_3* p, const realType* u )
{
    return tensor_i3( p->ld[0].J, p->ld[0].n, p->ld[1].J, p->ld[1].n, p->ld[2].J, p->ld[2].n, u, p->od_work );
}