poly.c

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#include 
#include 
#include 
#include    /* for cos, fabs */
#include  /* for memcpy */
#include 

#include "moab/FindPtFuncs.h"

/*
  For brevity's sake, some names have been shortened
  Quadrature rules
    Gauss   -> Gauss-Legendre quadrature (open)
    Lobatto -> Gauss-Lobatto-Legendre quadrature (closed at both ends)
  Polynomial bases
    Legendre -> Legendre basis
    Gauss    -> Lagrangian basis using Gauss   quadrature nodes
    Lobatto  -> Lagrangian basis using Lobatto quadrature nodes
*/

/*--------------------------------------------------------------------------
   Legendre Polynomial Matrix Computation
   (compute P_i(x_j) for i = 0, ..., n and a given set of x)
  --------------------------------------------------------------------------*/

/* precondition: n >= 1
   inner index is x index (0 ... m-1);
   outer index is Legendre polynomial number (0 ... n)
 */
void legendre_matrix( const realType* x, int m, realType* P, int n )
{
    int i, j;
    realType *Pjm1 = P, *Pj = Pjm1 + m, *Pjp1 = Pj + m;
    for( i = 0; i < m; ++i )
        Pjm1[i] = 1;
    for( i = 0; i < m; ++i )
        Pj[i] = x[i];
    for( j = 1; j < n; ++j )
    {
        realType c = 1 / (realType)( j + 1 ), a = c * ( 2 * j + 1 ), b = c * j;
        for( i = 0; i < m; ++i )
            Pjp1[i] = a * x[i] * Pj[i] - b * Pjm1[i];
        Pjp1 += m, Pj += m, Pjm1 += m;
    }
}

/* precondition: n >= 1, n even */
static void legendre_row_even( realType x, realType* P, int n )
{
    int i;
    P[0] = 1, P[1] = x;
    for( i = 1; i <= n - 2; i += 2 )
    {
        P[i + 1] = ( ( 2 * i + 1 ) * x * P[i] - i * P[i - 1] ) / ( i + 1 );
        P[i + 2] = ( ( 2 * i + 3 ) * x * P[i - 1] - ( i + 1 ) * P[i] ) / ( i + 2 );
    }
    P[n] = ( ( 2 * n - 1 ) * x * P[n - 1] - ( n - 1 ) * P[n - 2] ) / n;
}

/* precondition: n >= 1, n odd */
static void legendre_row_odd( realType x, realType* P, int n )
{
    int i;
    P[0] = 1, P[1] = x;
    for( i = 1; i <= n - 2; i += 2 )
    {
        P[i + 1] = ( ( 2 * i + 1 ) * x * P[i] - i * P[i - 1] ) / ( i + 1 );
        P[i + 2] = ( ( 2 * i + 3 ) * x * P[i - 1] - ( i + 1 ) * P[i] ) / ( i + 2 );
    }
}

/* precondition: n >= 1
   compute P_i(x) with i = 0 ... n
 */
void legendre_row( realType x, realType* P, int n )
{
    if( n & 1 )
        legendre_row_odd( x, P, n );
    else
        legendre_row_even( x, P, n );
}

/* precondition: n >= 1
   inner index is Legendre polynomial number (0 ... n)
   outer index is x index (0 ... m-1);
 */
void legendre_matrix_t( const realType* x, int m, realType* P, int n )
{
    int i;
    if( n & 1 )
        for( i = 0; i < m; ++i, P += n + 1 )
            legendre_row_odd( x[i], P, n );
    else
        for( i = 0; i < m; ++i, P += n + 1 )
            legendre_row_even( x[i], P, n );
}

/*--------------------------------------------------------------------------
   Legendre Polynomial Computation
   compute P_n(x) or P_n'(x) or P_n''(x)
  --------------------------------------------------------------------------*/

/* precondition: n >= 0 */
static realType legendre( int n, realType x )
{
    realType p[2];
    int i;
    p[0] = 1;
    p[1] = x;
    for( i = 1; i < n; i += 2 )
    {
        p[0] = ( ( 2 * i + 1 ) * x * p[1] - i * p[0] ) / ( i + 1 );
        p[1] = ( ( 2 * i + 3 ) * x * p[0] - ( i + 1 ) * p[1] ) / ( i + 2 );
    }
    return p[n & 1];
}

/* precondition: n > 0 */
static realType legendre_d1( int n, realType x )
{
    realType p[2];
    int i;
    p[0] = 3 * x;
    p[1] = 1;
    for( i = 2; i < n; i += 2 )
    {
        p[1] = ( ( 2 * i + 1 ) * x * p[0] - ( i + 1 ) * p[1] ) / i;
        p[0] = ( ( 2 * i + 3 ) * x * p[1] - ( i + 2 ) * p[0] ) / ( i + 1 );
    }
    return p[n & 1];
}

/* precondition: n > 1 */
static realType legendre_d2( int n, realType x )
{
    realType p[2];
    int i;
    p[0] = 3;
    p[1] = 15 * x;
    for( i = 3; i < n; i += 2 )
    {
        p[0] = ( ( 2 * i + 1 ) * x * p[1] - ( i + 2 ) * p[0] ) / ( i - 1 );
        p[1] = ( ( 2 * i + 3 ) * x * p[0] - ( i + 3 ) * p[1] ) / i;
    }
    return p[n & 1];
}

/*--------------------------------------------------------------------------
   Quadrature Nodes and Weights Calculation
   compute the n Gauss-Legendre nodes and weights or
           the n Gauss-Lobatto-Legendre nodes and weights
  --------------------------------------------------------------------------*/

/* n nodes */
void gauss_nodes( realType* z, int n )
{
    int i, j;
    for( i = 0; i <= n / 2 - 1; ++i )
    {
        realType ox, x = mbcos( ( 2 * n - 2 * i - 1 ) * ( MOAB_POLY_PI / 2 ) / n );
        do
        {
            ox = x;
            x -= legendre( n, x ) / legendre_d1( n, x );
        } while( mbabs( x - ox ) > -x * MOAB_POLY_EPS );
        z[i] = x - legendre( n, x ) / legendre_d1( n, x );
    }
    if( n & 1 ) z[n / 2] = 0;
    for( j = ( n + 1 ) / 2, i = n / 2 - 1; j < n; ++j, --i )
        z[j] = -z[i];
}

/* n inner lobatto nodes (excluding -1,1) */
static void lobatto_nodes_aux( realType* z, int n )
{
    int i, j, np = n + 1;
    for( i = 0; i <= n / 2 - 1; ++i )
    {
        realType ox, x = mbcos( ( n - i ) * MOAB_POLY_PI / np );
        do
        {
            ox = x;
            x -= legendre_d1( np, x ) / legendre_d2( np, x );
        } while( mbabs( x - ox ) > -x * MOAB_POLY_EPS );
        z[i] = x - legendre_d1( np, x ) / legendre_d2( np, x );
    }
    if( n & 1 ) z[n / 2] = 0;
    for( j = ( n + 1 ) / 2, i = n / 2 - 1; j < n; ++j, --i )
        z[j] = -z[i];
}

/* n lobatto nodes */
void lobatto_nodes( realType* z, int n )
{
    z[0] = -1, z[n - 1] = 1;
    lobatto_nodes_aux( &z[1], n - 2 );
}

void gauss_weights( const realType* z, realType* w, int n )
{
    int i, j;
    for( i = 0; i <= ( n - 1 ) / 2; ++i )
    {
        realType d = ( n + 1 ) * legendre( n + 1, z[i] );
        w[i]       = 2 * ( 1 - z[i] * z[i] ) / ( d * d );
    }
    for( j = ( n + 1 ) / 2, i = n / 2 - 1; j < n; ++j, --i )
        w[j] = w[i];
}

void lobatto_weights( const realType* z, realType* w, int n )
{
    int i, j;
    for( i = 0; i <= ( n - 1 ) / 2; ++i )
    {
        realType d = legendre( n - 1, z[i] );
        w[i]       = 2 / ( ( n - 1 ) * n * d * d );
    }
    for( j = ( n + 1 ) / 2, i = n / 2 - 1; j < n; ++j, --i )
        w[j] = w[i];
}

/*--------------------------------------------------------------------------
   Lagrangian to Legendre Change-of-basis Matrix
   where the nodes of the Lagrangian basis are the GL or GLL quadrature nodes
  --------------------------------------------------------------------------*/

/* precondition: n >= 2
   given the Gauss quadrature rule (z,w,n), compute the square matrix J
   for transforming from the Gauss basis to the Legendre basis:

      u_legendre(i) = sum_j J(i,j) u_gauss(j)

   computes J   = .5 (2i+1) w  P (z )
             ij              j  i  j

   in column major format (inner index is i, the Legendre index)
 */
void gauss_to_legendre( const realType* z, const realType* w, int n, realType* J )
{
    int i, j;
    legendre_matrix_t( z, n, J, n - 1 );
    for( j = 0; j < n; ++j )
    {
        realType ww = w[j];
        for( i = 0; i < n; ++i )
            *J++ *= ( 2 * i + 1 ) * ww / 2;
    }
}

/* precondition: n >= 2
   same as above, but
   in row major format (inner index is j, the Gauss index)
 */
void gauss_to_legendre_t( const realType* z, const realType* w, int n, realType* J )
{
    int i, j;
    legendre_matrix( z, n, J, n - 1 );
    for( i = 0; i < n; ++i )
    {
        realType ii = (realType)( 2 * i + 1 ) / 2;
        for( j = 0; j < n; ++j )
            *J++ *= ii * w[j];
    }
}

/* precondition: n >= 3
   given the Lobatto quadrature rule (z,w,n), compute the square matrix J
   for transforming from the Gauss basis to the Legendre basis:

      u_legendre(i) = sum_j J(i,j) u_lobatto(j)

   in column major format (inner index is i, the Legendre index)
 */
void lobatto_to_legendre( const realType* z, const realType* w, int n, realType* J )
{
    int i, j, m = ( n + 1 ) / 2;
    realType *p = J, *q;
    realType ww, sum;
    if( n & 1 )
        for( j = 0; j < m; ++j, p += n )
            legendre_row_odd( z[j], p, n - 2 );
    else
        for( j = 0; j < m; ++j, p += n )
            legendre_row_even( z[j], p, n - 2 );
    p = J;
    for( j = 0; j < m; ++j )
    {
        ww = w[j], sum = 0;
        for( i = 0; i < n - 1; ++i )
            *p *= ( 2 * i + 1 ) * ww / 2, sum += *p++;
        *p++ = -sum;
    }
    q = J + ( n / 2 - 1 ) * n;
    if( n & 1 )
        for( ; j < n; ++j, p += n, q -= n )
        {
            for( i = 0; i < n - 1; i += 2 )
                p[i] = q[i], p[i + 1] = -q[i + 1];
            p[i] = q[i];
        }
    else
        for( ; j < n; ++j, p += n, q -= n )
        {
            for( i = 0; i < n - 1; i += 2 )
                p[i] = q[i], p[i + 1] = -q[i + 1];
        }
}

/*--------------------------------------------------------------------------
   Lagrangian to Lagrangian change-of-basis matrix, and derivative matrix
  --------------------------------------------------------------------------*/

/* given the Lagrangian nodes (z,n) and evaluation points (x,m)
   evaluate all Lagrangian basis functions at all points x

   inner index of output J is the basis function index (row-major format)
   provide work array with space for 4*n reals
 */
void lagrange_weights( const realType* z, unsigned n, const realType* x, unsigned m, realType* J, realType* work )
{
    unsigned i, j;
    realType *w = work, *d = w + n, *u = d + n, *v = u + n;
    for( i = 0; i < n; ++i )
    {
        realType ww = 1, zi = z[i];
        for( j = 0; j < i; ++j )
            ww *= zi - z[j];
        for( ++j; j < n; ++j )
            ww *= zi - z[j];
        w[i] = 1 / ww;
    }
    u[0] = v[n - 1] = 1;
    for( i = 0; i < m; ++i )
    {
        realType xi = x[i];
        for( j = 0; j < n; ++j )
            d[j] = xi - z[j];
        for( j = 0; j < n - 1; ++j )
            u[j + 1] = d[j] * u[j];
        for( j = n - 1; j; --j )
            v[j - 1] = d[j] * v[j];
        for( j = 0; j < n; ++j )
            *J++ = w[j] * u[j] * v[j];
    }
}

/* given the Lagrangian nodes (z,n) and evaluation points (x,m)
   evaluate all Lagrangian basis functions and their derivatives

   inner index of outputs J,D is the basis function index (row-major format)
   provide work array with space for 6*n reals
 */
void lagrange_weights_deriv( const realType* z,
                             unsigned n,
                             const realType* x,
                             unsigned m,
                             realType* J,
                             realType* D,
                             realType* work )
{
    unsigned i, j;
    realType *w = work, *d = w + n, *u = d + n, *v = u + n, *up = v + n, *vp = up + n;
    for( i = 0; i < n; ++i )
    {
        realType ww = 1, zi = z[i];
        for( j = 0; j < i; ++j )
            ww *= zi - z[j];
        for( ++j; j < n; ++j )
            ww *= zi - z[j];
        w[i] = 1 / ww;
    }
    u[0] = v[n - 1] = 1;
    up[0] = vp[n - 1] = 0;
    for( i = 0; i < m; ++i )
    {
        realType xi = x[i];
        for( j = 0; j < n; ++j )
            d[j] = xi - z[j];
        for( j = 0; j < n - 1; ++j )
            u[j + 1] = d[j] * u[j], up[j + 1] = d[j] * up[j] + u[j];
        for( j = n - 1; j; --j )
            v[j - 1] = d[j] * v[j], vp[j - 1] = d[j] * vp[j] + v[j];
        for( j = 0; j < n; ++j )
            *J++ = w[j] * u[j] * v[j], *D++ = w[j] * ( up[j] * v[j] + u[j] * vp[j] );
    }
}

/*--------------------------------------------------------------------------
   Speedy Lagrangian Interpolation

   Usage:

     lagrange_data p;
     lagrange_setup(&p,z,n);    // setup for nodes z[0 ... n-1]

     the weights
       p->J [0 ... n-1]     interpolation weights
       p->D [0 ... n-1]     1st derivative weights
       p->D2[0 ... n-1]     2nd derivative weights
     are computed for a given x with:
       lagrange_0(p,x);  // compute p->J
       lagrange_1(p,x);  // compute p->J, p->D
       lagrange_2(p,x);  // compute p->J, p->D, p->D2
       lagrange_2u(p);   // compute p->D2 after call of lagrange_1(p,x);
     These functions use the z array supplied to setup
       (that pointer should not be freed between calls)
     Weights for x=z[0] and x=z[n-1] are computed during setup; access as:
       p->J_z0, etc. and p->J_zn, etc.

     lagrange_free(&p);  // deallocate memory allocated by setup
  --------------------------------------------------------------------------*/

void lagrange_0( lagrange_data* p, realType x )
{
    unsigned i, n = p->n;
    for( i = 0; i < n; ++i )
        p->d[i] = x - p->z[i];
    for( i = 0; i < n - 1; ++i )
        p->u0[i + 1] = p->d[i] * p->u0[i];
    for( i = n - 1; i; --i )
        p->v0[i - 1] = p->d[i] * p->v0[i];
    for( i = 0; i < n; ++i )
        p->J[i] = p->w[i] * p->u0[i] * p->v0[i];
}

void lagrange_1( lagrange_data* p, realType x )
{
    unsigned i, n = p->n;
    for( i = 0; i < n; ++i )
        p->d[i] = x - p->z[i];
    for( i = 0; i < n - 1; ++i )
        p->u0[i + 1] = p->d[i] * p->u0[i], p->u1[i + 1] = p->d[i] * p->u1[i] + p->u0[i];
    for( i = n - 1; i; --i )
        p->v0[i - 1] = p->d[i] * p->v0[i], p->v1[i - 1] = p->d[i] * p->v1[i] + p->v0[i];
    for( i = 0; i < n; ++i )
        p->J[i] = p->w[i] * p->u0[i] * p->v0[i], p->D[i] = p->w[i] * ( p->u1[i] * p->v0[i] + p->u0[i] * p->v1[i] );
}

void lagrange_2( lagrange_data* p, realType x )
{
    unsigned i, n = p->n;
    for( i = 0; i < n; ++i )
        p->d[i] = x - p->z[i];
    for( i = 0; i < n - 1; ++i )
        p->u0[i + 1] = p->d[i] * p->u0[i], p->u1[i + 1] = p->d[i] * p->u1[i] + p->u0[i],
                  p->u2[i + 1] = p->d[i] * p->u2[i] + 2 * p->u1[i];
    for( i = n - 1; i; --i )
        p->v0[i - 1] = p->d[i] * p->v0[i], p->v1[i - 1] = p->d[i] * p->v1[i] + p->v0[i],
                  p->v2[i - 1] = p->d[i] * p->v2[i] + 2 * p->v1[i];
    for( i = 0; i < n; ++i )
        p->J[i] = p->w[i] * p->u0[i] * p->v0[i], p->D[i] = p->w[i] * ( p->u1[i] * p->v0[i] + p->u0[i] * p->v1[i] ),
        p->D2[i] = p->w[i] * ( p->u2[i] * p->v0[i] + 2 * p->u1[i] * p->v1[i] + p->u0[i] * p->v2[i] );
}

void lagrange_2u( lagrange_data* p )
{
    unsigned i, n = p->n;
    for( i = 0; i < n - 1; ++i )
        p->u2[i + 1] = p->d[i] * p->u2[i] + 2 * p->u1[i];
    for( i = n - 1; i; --i )
        p->v2[i - 1] = p->d[i] * p->v2[i] + 2 * p->v1[i];
    for( i = 0; i < n; ++i )
        p->D2[i] = p->w[i] * ( p->u2[i] * p->v0[i] + 2 * p->u1[i] * p->v1[i] + p->u0[i] * p->v2[i] );
}

void lagrange_setup( lagrange_data* p, const realType* z, unsigned n )
{
    unsigned i, j;
    p->n = n, p->z = z;
    p->w = tmalloc( realType, 17 * n );
    p->d = p->w + n;
    p->J = p->d + n, p->D = p->J + n, p->D2 = p->D + n;
    p->u0 = p->D2 + n, p->v0 = p->u0 + n;
    p->u1 = p->v0 + n, p->v1 = p->u1 + n;
    p->u2 = p->v1 + n, p->v2 = p->u2 + n;
    p->J_z0 = p->v2 + n, p->D_z0 = p->J_z0 + n, p->D2_z0 = p->D_z0 + n;
    p->J_zn = p->D2_z0 + n, p->D_zn = p->J_zn + n, p->D2_zn = p->D_zn + n;
    for( i = 0; i < n; ++i )
    {
        realType ww = 1, zi = z[i];
        for( j = 0; j < i; ++j )
            ww *= zi - z[j];
        for( ++j; j < n; ++j )
            ww *= zi - z[j];
        p->w[i] = 1 / ww;
    }
    p->u0[0] = p->v0[n - 1] = 1;
    p->u1[0] = p->v1[n - 1] = 0;
    p->u2[0] = p->v2[n - 1] = 0;
    lagrange_2( p, z[0] );
    memcpy( p->J_z0, p->J, 3 * n * sizeof( realType ) );
    lagrange_2( p, z[n - 1] );
    memcpy( p->J_zn, p->J, 3 * n * sizeof( realType ) );
}

void lagrange_free( lagrange_data* p )
{
    free( p->w );
}