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// neville.cmd
// Neville's algorithm for computing Bspline values
// Programmer: Ken Brakke, brakke@susqu.edu, http://www.susqu.edu/brakke
// Contents:
// neville1 - interpolation and derivatives in one dimension
// neville2 - interpolation and derivatives in two dimensions.
// Neville algorithm for interpolating along one dimension.
// Usage of neville1:
// neville1_data should be set up by caller, redimensioning if necessary
// Indexed by point number along curve and data dimension.
define neville1_data real[0][0]; // not modified by neville1
// Returned from neville1. Index is data dimension.
define neville1_value real[0]; // interpolated values
define neville1_deriv real[0]; // interpolated deriv wrt u
procedure neville1 (
real neville1_order, // order of interpolation polynomial
real neville1_dim, // range dimension
real neville1_u // interpolation spot, between 0 and 1
)
{
local neville1_array; local neville1_darray;
define neville1_array real[neville1_order+1][neville1_dim];
define neville1_darray real[neville1_order+1][neville1_dim];
define neville1_value real[neville1_dim]; // interpolated values
define neville1_deriv real[neville1_dim]; // interpolated deriv wrt u
local inx;local dinx;
inx := 1;
while ( inx <= neville1_order+1 ) do
{
dinx := 1;
while ( dinx <= neville1_dim ) do
{ neville1_array[inx][dinx] := neville1_data[inx][dinx];
neville1_darray[inx][dinx] := 0;
dinx += 1;
};
inx += 1;
};
// scale npoint to integer based value
local nnpoint;
nnpoint := neville1_u*neville1_order;
local linx;
linx := 1;
while ( linx <= neville1_order ) do
{ inx := 1;
while ( inx + linx <= neville1_order + 1 ) do
{ dinx := 1;
while ( dinx <= neville1_dim ) do
{ // do derivatives first, since dependent on current value
neville1_darray[inx][dinx] :=
((inx-1+linx - nnpoint)*neville1_darray[inx][dinx]
+ (nnpoint - (inx-1))*neville1_darray[inx+1][dinx])/linx +
(-neville1_array[inx][dinx] + neville1_array[inx+1][dinx])/linx;
neville1_array[inx][dinx] :=
((inx-1+linx - nnpoint)*neville1_array[inx][dinx]
+ (nnpoint - (inx-1))*neville1_array[inx+1][dinx])/linx;
dinx += 1;
};
inx += 1;
};
linx += 1;
};
dinx := 1;
while ( dinx <= neville1_dim ) do
{ neville1_value[dinx] := neville1_array[1][dinx];
neville1_deriv[dinx] := neville1_darray[1][dinx]*neville1_order;
dinx += 1;
};
}
// Neville algorithm for interpolating in 2D
// Usage of neville2:
// Initialize neville2_data, neville2_u arrays.
// Call neville2.
// Input data, indexed by (i,j) node coordinate and data dimension.
define neville2_data real[0][0][0];
define neville2_u real[2]; // incoming; should sum to at most 1
neville2_u[1] := 1/3;
neville2_u[2] := 1/3;
// Return values, indexed by data dimension.
define neville2_value real[0] // return values of position
define neville2_deriv real[0][2] // return values of partials
procedure neville2 (
real neville2_order, // of polynomial
real neville2_dim // dimension of values
)
{
local neville2_array; local neville2_darray; local uupoint;
define neville2_array
real[neville2_order+1][neville2_order+1][neville2_dim];
define neville2_darray
real[neville2_order+1][neville2_order+1][neville2_dim][2];
define uupoint real[2]; // scaled target coordinates
define neville2_value real[neville2_dim]; // return values of position
define neville2_deriv real[neville2_dim][2]; // return values of partials
// initialize data; kludge due to fact that indexing on vertices
// only does the three corners.
local inx; local jnx; local dinx;
inx := 1;
while ( inx <= neville2_order+1 ) do
{ jnx := 1;
while ( inx + jnx <= neville2_order+2 ) do
{ dinx := 1;
while ( dinx <= neville2_dim ) do
{ neville2_array[inx][jnx][dinx] := neville2_data[inx][jnx][dinx];
neville2_darray[inx][jnx][dinx][1] := 0;
neville2_darray[inx][jnx][dinx][2] := 0;
dinx += 1;
};
jnx += 1;
};
inx += 1;
};
// scale npoint to integer based value
uupoint[1] := neville2_u[1]*neville2_order;
uupoint[2] := neville2_u[2]*neville2_order;
local linx;
linx := 1;
while ( linx <= neville2_order ) do
{ inx := 1;
while ( inx + linx <= neville2_order + 1 ) do
{ jnx := 1;
while ( inx + jnx + linx <= neville2_order + 2 ) do
{ dinx := 1;
while ( dinx <= neville2_dim ) do
{ // do derivatives first, since dependent on current value
neville2_darray[inx][jnx][dinx][1] :=
((inx+jnx-2+linx-uupoint[1]-uupoint[2])
*neville2_darray[inx][jnx][dinx][1]
+ (uupoint[1]-(inx-1))*neville2_darray[inx+1][jnx][dinx][1]
+ (uupoint[2]-(jnx-1))*neville2_darray[inx][jnx+1][dinx][1])/linx
+ (-neville2_array[inx][jnx][dinx]
+ neville2_array[inx+1][jnx][dinx] )/linx;
neville2_darray[inx][jnx][dinx][2] :=
((inx+jnx-2+linx-uupoint[1]-uupoint[2])
*neville2_darray[inx][jnx][dinx][2]
+ (uupoint[1]-(inx-1))*neville2_darray[inx+1][jnx][dinx][2]
+ (uupoint[2]-(jnx-1))*neville2_darray[inx][jnx+1][dinx][2])/linx
+ (-neville2_array[inx][jnx][dinx]
+ neville2_array[inx][jnx+1][dinx] )/linx;
neville2_array[inx][jnx][dinx] :=
((inx+jnx-2+linx-uupoint[1]-uupoint[2])*neville2_array[inx][jnx][dinx]
+ (uupoint[1] - (inx-1))*neville2_array[inx+1][jnx][dinx]
+ (uupoint[2] - (jnx-1))*neville2_array[inx][jnx+1][dinx])/linx;
dinx += 1;
};
jnx += 1;
};
inx += 1;
};
linx += 1;
};
dinx := 1;
while ( dinx <= neville2_dim ) do
{ neville2_value[dinx] := neville2_array[1][1][dinx];
neville2_deriv[dinx][1] := neville2_darray[1][1][dinx][1]*neville2_order;
neville2_deriv[dinx][2] := neville2_darray[1][1][dinx][2]*neville2_order;
dinx += 1;
};
}
// End neville.cmd
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