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/* interpolate.c -- interpolation functions */
/* $Header$ */
/*
* interpolate_local - ... (local)
* interpolate_local_order1 - ... (order 1) (linear) (2-point)
* interpolate_local_order2 - ... (order 2) (3-point)
* interpolate_local_order3 - ... (order 3) (4-point)
* interpolate_local_order4 - ... (order 4) (5-point)
* interpolate_local_order5 - ... (order 5) (6-point)
*/
/* To make porting easy, only the interpolate_local* functions
* have been retained in this file. JT's original code (kept in
* the archive directory) also has functions for derivatives and
* interpolation on non-uniform grids. ---Sai.
*/
#include <math.h>
#include <stdio.h>
#include "cctk.h"
CCTK_REAL interpolate_local(int order, CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x,
CCTK_INT excised_points[]);
CCTK_REAL interpolate_eno(CCTK_INT order, CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x,
CCTK_INT excised_points[]);
/* prototypes for private functions defined in this file */
static CCTK_REAL interpolate_local_order1(CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x);
static CCTK_REAL interpolate_local_order2(CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x);
static CCTK_REAL interpolate_local_order3(CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x);
static CCTK_REAL interpolate_local_order4(CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x);
static CCTK_REAL interpolate_local_order5(CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x);
/*****************************************************************************/
/*
* This function does local Lagrange polynomial interpolation on a
* uniform grid. That is, given order+1 uniformly spaced data points
* (x0 + i*dx, y[i]) , i = 0...order, it interpolates a degree-(order)
* (Lagrange) polynomial through them and evaluates this polynomial at
* a specified x coordinate. In general the error in this computed
* function value is O(h^(order+1)) .
*
* Except for possible end effects (see end_action (below)), the local
* interpolation is internally centered to improve its conditioning. Even
* so, the interpolation is highly ill-conditioned for small dx and/or
* large order and/or x outside the domain of the data points.
*
* The interpolation formulas were (are) all derived via Maple, see
* "./interpolate.in" and "./interpolate.out".
*
* Arguments:
* order = (in) The order of the local interpolation, i.e. the degree
* of the interpolated polynomial.
* x0 = (in) The x coordinate corresponding to y[0].
* dx = (in) The x spacing between the data points.
* y[0...order] = (in) The y data array.
* x = (in) The x coordinate for the interpolation.
*/
CCTK_REAL interpolate_local(int order,
CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x,
CCTK_INT excised_points[])
{
/*
We assume that the excised region has one of the following 4 forms:
o o o o o (No excision)
x o o o o (left excision)
o o o o x (right excision)
x o o o x (left and right excision)
Anything else (e.g., x o x o x) will go horribly wrong.
*/
int start, end, available_order;
start = 0;
end = order;
while ((start < order + 1)&&(excised_points[start]))
{
++start;
}
while ((end > -1)&&(excised_points[end]))
{
--end;
}
if (end < start)
{
/* The whole block is excised */
return y[0];
}
/* The possible interpolation order that can be used */
available_order = end - start;
if (available_order == 0)
{
/* There is only one non-excised point */
return y[start];
}
if (dx * start + x0 > x)
{
/* We would have to extrapolate to the left */
return y[start];
}
if (dx * end + x0 < x)
{
/* We would have to extrapolate to the right */
return y[end];
}
switch (available_order)
{
case 1: return interpolate_local_order1(x0 + start * dx, dx, &y[start], x);
case 2: return interpolate_local_order2(x0 + start * dx, dx, &y[start], x);
case 3: return interpolate_local_order3(x0 + start * dx, dx, &y[start], x);
case 4: return interpolate_local_order4(x0 + start * dx, dx, &y[start], x);
case 5: return interpolate_local_order5(x0 + start * dx, dx, &y[start], x);
default: CCTK_WARN(0, "Interpolation order not supported");
}
}
/*****************************************************************************/
/*
* This function interpolates a 1st-order (linear) Lagrange polynomial
* in a 2-point [0...1] data array centered about the [0.5] grid position.
*/
static CCTK_REAL interpolate_local_order1(CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x)
{
CCTK_REAL c0 = (+1.0/2.0)*y[0] + (+1.0/2.0)*y[1];
CCTK_REAL c1 = - y[0] + + y[1];
CCTK_REAL xc = x0 + 0.5*dx;
CCTK_REAL xr = (x - xc) / dx;
return( c0 + xr*c1 );
}
/*****************************************************************************/
/*
* This function interpolates a 2nd-order (quadratic) Lagrange polynomial
* in a 3-point [0...2] data array centered about the [1] grid position.
*/
static CCTK_REAL interpolate_local_order2(CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x)
{
CCTK_REAL c0 = y[1];
CCTK_REAL c1 = (-1.0/2.0)*y[0] + (+1.0/2.0)*y[2];
CCTK_REAL c2 = (+1.0/2.0)*y[0] - y[1] + (+1.0/2.0)*y[2];
CCTK_REAL xc = x0 + 1.0*dx;
CCTK_REAL xr = (x - xc) / dx;
return( c0 + xr*(c1 + xr*c2) );
}
/*****************************************************************************/
/*
* This function interpolates a 3rd-order (cubic) Lagrange polynomial
* in a 4-point [0...3] data array centered about the [1.5] grid position.
*/
static CCTK_REAL interpolate_local_order3(CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x)
{
CCTK_REAL c0 = (-1.0/16.0)*y[0] + (+9.0/16.0)*y[1]
+ (+9.0/16.0)*y[2] + (-1.0/16.0)*y[3];
CCTK_REAL c1 = (+1.0/24.0)*y[0] + (-9.0/ 8.0)*y[1]
+ (+9.0/ 8.0)*y[2] + (-1.0/24.0)*y[3];
CCTK_REAL c2 = (+1.0/ 4.0)*y[0] + (-1.0/ 4.0)*y[1]
+ (-1.0/ 4.0)*y[2] + (+1.0/ 4.0)*y[3];
CCTK_REAL c3 = (-1.0/ 6.0)*y[0] + (+1.0/ 2.0)*y[1]
+ (-1.0/ 2.0)*y[2] + ( 1.0/ 6.0)*y[3];
CCTK_REAL xc = x0 + 1.5*dx;
CCTK_REAL xr = (x - xc) / dx;
return( c0 + xr*(c1 + xr*(c2 + xr*c3)) );
}
/*****************************************************************************/
/*
* This function interpolates a 4th-order (quartic) Lagrange polynomial
* in a 5-point [0...4] data array centered about the [2] grid position.
*/
static CCTK_REAL interpolate_local_order4(CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x)
{
CCTK_REAL c0 = y[2];
CCTK_REAL c1 = (+1.0/12.0)*y[0] + (-2.0/ 3.0)*y[1]
+ (+2.0/ 3.0)*y[3] + (-1.0/12.0)*y[4];
CCTK_REAL c2 = + (-1.0/24.0)*y[0] + (+2.0/ 3.0)*y[1] + (-5.0/ 4.0)*y[2]
+ (+2.0/ 3.0)*y[3] + (-1.0/24.0)*y[4];
CCTK_REAL c3 = + (-1.0/12.0)*y[0] + (+1.0/ 6.0)*y[1]
+ (-1.0/ 6.0)*y[3] + (+1.0/12.0)*y[4];
CCTK_REAL c4 = + (+1.0/24.0)*y[0] + (-1.0/ 6.0)*y[1] + (+1.0/ 4.0)*y[2]
+ (-1.0/ 6.0)*y[3] + (+1.0/24.0)*y[4];
CCTK_REAL xc = x0 + 2.0*dx;
CCTK_REAL xr = (x - xc) / dx;
return( c0 + xr*(c1 + xr*(c2 + xr*(c3 + xr*c4))) );
}
/*****************************************************************************/
/*
* This function interpolates a 5th-order (quintic) Lagrange polynomial
* in a 6-point [0...5] data array centered about the [2.5] grid position.
*/
static CCTK_REAL interpolate_local_order5(CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x)
{
CCTK_REAL c0 = (+ 3.0/256.0)*y[0] + (-25.0/256.0)*y[1]
+ (+75.0/128.0)*y[2] + (+75.0/128.0)*y[3]
+ (-25.0/256.0)*y[4] + (+ 3.0/256.0)*y[5];
CCTK_REAL c1 = (- 3.0/640.0)*y[0] + (+25.0/384.0)*y[1]
+ (-75.0/ 64.0)*y[2] + (+75.0/ 64.0)*y[3]
+ (-25.0/384.0)*y[4] + (+ 3.0/640.0)*y[5];
CCTK_REAL c2 = (- 5.0/ 96.0)*y[0] + (+13.0/ 32.0)*y[1]
+ (-17.0/ 48.0)*y[2] + (-17.0/ 48.0)*y[3]
+ (+13.0/ 32.0)*y[4] + (- 5.0/ 96.0)*y[5];
CCTK_REAL c3 = (+ 1.0/ 48.0)*y[0] + (-13.0/ 48.0)*y[1]
+ (+17.0/ 24.0)*y[2] + (-17.0/ 24.0)*y[3]
+ (+13.0/ 48.0)*y[4] + (- 1.0/ 48.0)*y[5];
CCTK_REAL c4 = (+ 1.0/ 48.0)*y[0] + (- 1.0/ 16.0)*y[1]
+ (+ 1.0/ 24.0)*y[2] + (+ 1.0/ 24.0)*y[3]
+ (- 1.0/ 16.0)*y[4] + (+ 1.0/ 48.0)*y[5];
CCTK_REAL c5 = (- 1.0/120.0)*y[0] + (+ 1.0/ 24.0)*y[1]
+ (- 1.0/ 12.0)*y[2] + (+ 1.0/ 12.0)*y[3]
+ (- 1.0/ 24.0)*y[4] + (+ 1.0/120.0)*y[5];
CCTK_REAL xc = x0 + 2.5*dx;
CCTK_REAL xr = (x - xc) / dx;
return( c0 + xr*(c1 + xr*(c2 + xr*(c3 + xr*(c4 + xr*c5)))) );
}
/*****************************************************************************/
CCTK_REAL interpolate_eno(CCTK_INT order,
CCTK_REAL x0, CCTK_REAL dx,
CCTK_REAL y[],
CCTK_REAL x,
CCTK_INT excised_points[])
{
/* CCTK_REAL diff[4]; */
/* CCTK_INT seed = 0; */
/* CCTK_INT j,r; */
/* CCTK_REAL c0; */
/* CCTK_REAL c1; */
/* CCTK_REAL c2; */
/* CCTK_REAL xc; */
/* CCTK_REAL xr; */
CCTK_REAL result;
CCTK_REAL undiv_diff [13][6];
CCTK_INT diff, i, ii, r;
CCTK_REAL fp_i, fp_ii, x_rel;
/* Find seed index */
/* while (x > x0 + dx * ((CCTK_REAL)seed+0.5)) seed++; */
/* if (seed!=2) { */
/* Not enough stencil, only perform linear interpolation */
/* seed = 0; */
/* while (x > x0 + dx * ((CCTK_REAL)(seed+1))) seed++; */
/* if (seed==4) seed=3; */
/* return y[seed] + (y[seed+1]-y[seed])/dx * (x-x0-(CCTK_REAL)seed*dx); */
/* } */
/* Calculate the undivided differences */
/* for (j=0; j<=3; j++) */
/* diff[j] = y[j+1] - y[j]; */
/* Find the stencil */
/* if ( fabs(diff[1]) < fabs(diff[2]) ) { */
/* if ( fabs(diff[1]-diff[0]) < fabs(diff[2]-diff[1]) ) */
/* r = 0; */
/* else */
/* r = 1; */
/* } else { */
/* if ( fabs(diff[2]-diff[1]) < fabs(diff[3]-diff[2]) ) */
/* r = 1; */
/* else */
/* r = 2; */
/* } */
/* Interpolate second order */
/* c0 = y[r+1]; */
/* c1 = (-1.0/2.0)*y[r] + (+1.0/2.0)*y[r+2]; */
/* c2 = (+1.0/2.0)*y[r] - y[r+1] + (+1.0/2.0)*y[r+2]; */
/* xc = x0 + dx * (CCTK_REAL)(r+1); */
/* xr = (x - xc) / dx; */
/* result = ( c0 + xr*(c1 + xr*c2) ); */
for (i = 1; i < 2 * order + 2; ++i)
{
undiv_diff[i][0] = y[i-1];
}
for (i = 0; i < 6; ++i)
{
undiv_diff[0][i] = 1e10;
undiv_diff[2 * order + 2][i] = 1e10;
}
for (diff = 1; diff < order + 1; ++diff)
{
for (i = 1; i < 2 * order + 2; ++i)
{
undiv_diff[i][diff] = undiv_diff[i][diff-1] - undiv_diff[i-1][diff-1];
}
undiv_diff[0][diff] = -10 * undiv_diff[1][diff];
undiv_diff[2 * order + 2][diff]= -10 * undiv_diff[2 * order + 1][diff];
}
fp_i = (x - x0) / dx;
fp_ii = floor(fp_i);
x_rel = fp_i - fp_ii;
ii = (CCTK_INT)(fp_ii) + 1;
if (ii < 1)
{
ii = 1;
x_rel = fp_i - (CCTK_REAL)(ii) + 1.0;
}
else if (ii > 2 * order + 1)
{
ii = 2 * order + 1;
x_rel = fp_i - (CCTK_REAL)(ii) + 1.0;
}
r = 0;
for (diff = 1; diff < order + 1; ++diff)
{
if (fabs(undiv_diff[ii-r][diff]) < fabs(undiv_diff[ii-r+1][diff]) )
{
++r;
}
if (ii - r < diff + 1)
{
--r;
}
else if (ii - r + diff > 2 * order + 1)
{
++r;
}
}
result = undiv_diff[ii-r][0];
switch (order)
{
case 1:
result += (x_rel + r - 0.0) / 1.0 * undiv_diff[ii-r+1][1];
break;
case 2:
result += (x_rel + r - 0.0) / 1.0 * (undiv_diff[ii-r+1][1] +
(x_rel + r - 1.0) / 2.0 * (undiv_diff[ii-r+2][2]));
break;
case 3:
result += (x_rel + r - 0.0) / 1.0 * (undiv_diff[ii-r+1][1] +
(x_rel + r - 1.0) / 2.0 * (undiv_diff[ii-r+2][2] +
(x_rel + r - 2.0) / 3.0 * (undiv_diff[ii-r+3][3])));
break;
case 4:
result += (x_rel + r - 0.0) / 1.0 * (undiv_diff[ii-r+1][1] +
(x_rel + r - 1.0) / 2.0 * (undiv_diff[ii-r+2][2] +
(x_rel + r - 2.0) / 3.0 * (undiv_diff[ii-r+3][3] +
(x_rel + r - 3.0) / 4.0 * (undiv_diff[ii-r+4][4]))));
break;
case 5:
result += (x_rel + r - 0.0) / 1.0 * (undiv_diff[ii-r+1][1] +
(x_rel + r - 1.0) / 2.0 * (undiv_diff[ii-r+2][2] +
(x_rel + r - 2.0) / 3.0 * (undiv_diff[ii-r+3][3] +
(x_rel + r - 3.0) / 4.0 * (undiv_diff[ii-r+4][4] +
(x_rel + r - 4.0) / 5.0 * (undiv_diff[ii-r+5][5])))));
break;
}
/* #define CARTOON_ENO_DBG 1 */
#ifdef CARTOON_ENO_DBG
printf("Cartoon ENO interpolation.\n"
"Input: order %d x0 %g dx %g x %g.\n",
order, x0, dx, x);
printf("Data is\n");
for (i = 0; i < 2 * order + 1; ++i)
{
printf(" %g", y[i]);
}
printf("\nEntries used were\n");
printf("0: %g 1: %g 2: %g\n",
undiv_diff[ii-r][0], undiv_diff[ii-r+1][1], undiv_diff[ii-r+2][2]);
printf("Parameters are fp_i %g fp_ii %g x_rel %g ii %d r %d\n",
fp_i, fp_ii, x_rel, ii, r);
printf("Result is %g\n", result);
#endif
return result;
}
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