poly.c File Reference

#include <stdio.h>
#include <stdlib.h>
#include <stdarg.h>
#include <math.h>
#include <string.h>
#include <float.h>
#include "moab/FindPtFuncs.h"

Include dependency graph for poly.c:

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Functions
void	legendre_matrix (const realType *x, int m, realType *P, int n)
static void	legendre_row_even (realType x, realType *P, int n)
static void	legendre_row_odd (realType x, realType *P, int n)
void	legendre_row (realType x, realType *P, int n)
void	legendre_matrix_t (const realType *x, int m, realType *P, int n)
static realType	legendre (int n, realType x)
static realType	legendre_d1 (int n, realType x)
static realType	legendre_d2 (int n, realType x)
void	gauss_nodes (realType *z, int n)
static void	lobatto_nodes_aux (realType *z, int n)
void	lobatto_nodes (realType *z, int n)
void	gauss_weights (const realType *z, realType *w, int n)
void	lobatto_weights (const realType *z, realType *w, int n)
void	gauss_to_legendre (const realType *z, const realType *w, int n, realType *J)
void	gauss_to_legendre_t (const realType *z, const realType *w, int n, realType *J)
void	lobatto_to_legendre (const realType *z, const realType *w, int n, realType *J)
void	lagrange_weights (const realType *z, unsigned n, const realType *x, unsigned m, realType *J, realType *work)
void	lagrange_weights_deriv (const realType *z, unsigned n, const realType *x, unsigned m, realType *J, realType *D, realType *work)
void	lagrange_0 (lagrange_data *p, realType x)
void	lagrange_1 (lagrange_data *p, realType x)
void	lagrange_2 (lagrange_data *p, realType x)
void	lagrange_2u (lagrange_data *p)
void	lagrange_setup (lagrange_data *p, const realType *z, unsigned n)
void	lagrange_free (lagrange_data *p)

Function Documentation

gauss_nodes()

void gauss_nodes	(	realType *	z,
		int	n
	)

Definition at line 155 of file poly.c.

{
 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];
}

References legendre(), legendre_d1(), mbabs, mbcos, MOAB_POLY_EPS, and MOAB_POLY_PI.

gauss_to_legendre()

void gauss_to_legendre	(	const realType *	z,
		const realType *	w,
		int	n,
		realType *	J
	)

Definition at line 239 of file poly.c.

{
 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;
 }
}

References legendre_matrix_t().

gauss_to_legendre_t()

void gauss_to_legendre_t	(	const realType *	z,
		const realType *	w,
		int	n,
		realType *	J
	)

Definition at line 255 of file poly.c.

{
 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];
 }
}

References legendre_matrix().

gauss_weights()

void gauss_weights	(	const realType *	z,
		realType *	w,
		int	n
	)

Definition at line 199 of file poly.c.

{
 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];
}

References legendre().

lagrange_0()

void lagrange_0	(	lagrange_data *	p,
		realType	x
	)

Definition at line 414 of file poly.c.

{
 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];
}

References lagrange_data::d, lagrange_data::J, lagrange_data::n, lagrange_data::u0, lagrange_data::v0, lagrange_data::w, and lagrange_data::z.

Referenced by moab::SpectralQuad::evalFcn(), moab::Element::SpectralHex::evaluate(), moab::Element::SpectralQuad::evaluate(), moab::element_utility::Spectral_hex_map< _Matrix >::evaluate(), moab::Element::SpectralHex::evaluate_scalar_field(), moab::Element::SpectralQuad::evaluate_scalar_field(), moab::element_utility::Spectral_hex_map< _Matrix >::evaluate_scalar_field(), findpt_weights_2(), findpt_weights_3(), and moab::ElemUtil::hex_eval().

lagrange_1()

void lagrange_1	(	lagrange_data *	p,
		realType	x
	)

Definition at line 427 of file poly.c.

{
 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] );
}

References lagrange_data::D, lagrange_data::d, lagrange_data::J, lagrange_data::n, lagrange_data::u0, lagrange_data::u1, lagrange_data::v0, lagrange_data::v1, lagrange_data::w, and lagrange_data::z.

Referenced by opt_area_set_2(), opt_edge_set_2(), opt_edge_set_3(), opt_face_set_3(), and opt_vol_set_3().

lagrange_2()

void lagrange_2	(	lagrange_data *	p,
		realType	x
	)

Definition at line 440 of file poly.c.

{
 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] );
}

References lagrange_data::D, lagrange_data::d, lagrange_data::D2, lagrange_data::J, lagrange_data::n, lagrange_data::u0, lagrange_data::u1, lagrange_data::u2, lagrange_data::v0, lagrange_data::v1, lagrange_data::v2, lagrange_data::w, and lagrange_data::z.

Referenced by lagrange_setup().

lagrange_2u()

void lagrange_2u

(

lagrange_data *

p

)

Definition at line 456 of file poly.c.

{
 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] );
}

References lagrange_data::d, lagrange_data::D2, lagrange_data::n, lagrange_data::u0, lagrange_data::u1, lagrange_data::u2, lagrange_data::v0, lagrange_data::v1, lagrange_data::v2, and lagrange_data::w.

Referenced by opt_edge_hess_2(), opt_edge_hess_3(), and opt_face_hess_3().

lagrange_free()

void lagrange_free

(

lagrange_data *

p

)

Definition at line 497 of file poly.c.

{
 free( p->w );
}

References lagrange_data::w.

Referenced by moab::element_utility::Spectral_hex_map< _Matrix >::free_data(), moab::Element::SpectralHex::freedata(), moab::Element::SpectralQuad::freedata(), moab::ElemUtil::hex_eval(), and moab::ElemUtil::hex_findpt().

lagrange_setup()

void lagrange_setup	(	lagrange_data *	p,
		const realType *	z,
		unsigned	n
	)

Definition at line 467 of file poly.c.

{
 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 ) );
}

References lagrange_data::D, lagrange_data::d, lagrange_data::D2, lagrange_data::D2_z0, lagrange_data::D2_zn, lagrange_data::D_z0, lagrange_data::D_zn, lagrange_data::J, lagrange_data::J_z0, lagrange_data::J_zn, lagrange_2(), lagrange_data::n, tmalloc, lagrange_data::u0, lagrange_data::u1, lagrange_data::u2, lagrange_data::v0, lagrange_data::v1, lagrange_data::v2, lagrange_data::w, and lagrange_data::z.

Referenced by findpt_setup_2(), findpt_setup_3(), moab::ElemUtil::hex_eval(), moab::ElemUtil::hex_findpt(), moab::Element::SpectralHex::Init(), moab::Element::SpectralQuad::Init(), moab::SpectralHex::initFcn(), and moab::element_utility::Spectral_hex_map< _Matrix >::initialize_spectral_hex().

lagrange_weights()

void lagrange_weights	(	const realType *	z,
		unsigned	n,
		const realType *	x,
		unsigned	m,
		realType *	J,
		realType *	work
	)

Definition at line 320 of file poly.c.

{
 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];
 }
}

lagrange_weights_deriv()

void lagrange_weights_deriv	(	const realType *	z,
		unsigned	n,
		const realType *	x,
		unsigned	m,
		realType *	J,
		realType *	D,
		realType *	work
	)

Definition at line 354 of file poly.c.

{
 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] );
 }
}

Referenced by lob_bnd_base_setup(), obbox_setup_2(), and obbox_setup_3().

legendre()

static realType legendre ( int  n, realType  x  )

static

Definition at line 104 of file poly.c.

{
 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];
}

Referenced by gauss_nodes(), gauss_weights(), and lobatto_weights().

legendre_d1()

static realType legendre_d1 ( int  n, realType  x  )

static

Definition at line 119 of file poly.c.

{
 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];
}

Referenced by gauss_nodes(), and lobatto_nodes_aux().

legendre_d2()

static realType legendre_d2 ( int  n, realType  x  )

static

Definition at line 134 of file poly.c.

{
 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];
}

Referenced by lobatto_nodes_aux().

legendre_matrix()

void legendre_matrix	(	const realType *	x,
		int	m,
		realType *	P,
		int	n
	)

Definition at line 30 of file poly.c.

{
 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;
 }
}

Referenced by gauss_to_legendre_t().

legendre_matrix_t()

void legendre_matrix_t	(	const realType *	x,
		int	m,
		realType *	P,
		int	n
	)

Definition at line 87 of file poly.c.

{
 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 );
}

References legendre_row_even(), and legendre_row_odd().

Referenced by gauss_to_legendre().

legendre_row()

void legendre_row	(	realType	x,
		realType *	P,
		int	n
	)

Definition at line 75 of file poly.c.

{
 if( n & 1 )
 legendre_row_odd( x, P, n );
 else
 legendre_row_even( x, P, n );
}

References legendre_row_even(), and legendre_row_odd().

legendre_row_even()

static void legendre_row_even ( realType  x, realType *  P, int  n  )

static

Definition at line 48 of file poly.c.

{
 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;
}

Referenced by legendre_matrix_t(), legendre_row(), and lobatto_to_legendre().

legendre_row_odd()

static void legendre_row_odd ( realType  x, realType *  P, int  n  )

static

Definition at line 61 of file poly.c.

{
 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 );
 }
}

Referenced by legendre_matrix_t(), legendre_row(), and lobatto_to_legendre().

lobatto_nodes()

void lobatto_nodes	(	realType *	z,
		int	n
	)

Definition at line 193 of file poly.c.

{
 z[0] = -1, z[n - 1] = 1;
 lobatto_nodes_aux( &z[1], n - 2 );
}

References lobatto_nodes_aux().

Referenced by findpt_setup_2(), findpt_setup_3(), hash_getbb_2(), hash_getbb_3(), moab::ElemUtil::hex_eval(), moab::ElemUtil::hex_findpt(), moab::Element::SpectralHex::Init(), moab::Element::SpectralQuad::Init(), moab::SpectralHex::initFcn(), and moab::element_utility::Spectral_hex_map< _Matrix >::initialize_spectral_hex().

lobatto_nodes_aux()

static void lobatto_nodes_aux ( realType *  z, int  n  )

static

Definition at line 174 of file poly.c.

{
 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];
}

References legendre_d1(), legendre_d2(), mbabs, mbcos, MOAB_POLY_EPS, and MOAB_POLY_PI.

Referenced by lobatto_nodes().

lobatto_to_legendre()

void lobatto_to_legendre	(	const realType *	z,
		const realType *	w,
		int	n,
		realType *	J
	)

Definition at line 275 of file poly.c.

{
 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];
 }
}

References legendre_row_even(), legendre_row_odd(), and moab::sum().

lobatto_weights()

void lobatto_weights	(	const realType *	z,
		realType *	w,
		int	n
	)

Definition at line 211 of file poly.c.

{
 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];
}

References legendre().

Referenced by hash_getbb_2(), and hash_getbb_3().