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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_interp_2d_scat_eval (e01sb)

## Purpose

nag_interp_2d_scat_eval (e01sb) evaluates at a given point the two-dimensional interpolant function computed by nag_interp_2d_scat (e01sa).

## Syntax

[pf, ifail] = e01sb(x, y, f, triang, grads, px, py, 'm', m)
[pf, ifail] = nag_interp_2d_scat_eval(x, y, f, triang, grads, px, py, 'm', m)

## Description

nag_interp_2d_scat_eval (e01sb) takes as input the arguments defining the interpolant $F\left(x,y\right)$ of a set of scattered data points $\left({x}_{r},{y}_{r},{f}_{r}\right)$, for $\mathit{r}=1,2,\dots ,m$, as computed by nag_interp_2d_scat (e01sa), and evaluates the interpolant at the point $\left(px,py\right)$.
If $\left(px,py\right)$ is equal to $\left({x}_{r},{y}_{r}\right)$ for some value of $r$, the returned value will be equal to ${f}_{r}$.
If $\left(px,py\right)$ is not equal to $\left({x}_{r},{y}_{r}\right)$ for any $r$, the derivatives in grads will be used to compute the interpolant. A triangle is sought which contains the point $\left(px,py\right)$, and the vertices of the triangle along with the partial derivatives and ${f}_{r}$ values at the vertices are used to compute the value $F\left(px,py\right)$. If the point $\left(px,py\right)$ lies outside the triangulation defined by the input arguments, the returned value is obtained by extrapolation. In this case, the interpolating function ${\mathbf{f}}$ is extended linearly beyond the triangulation boundary. The method is described in more detail in Renka and Cline (1984) and the code is derived from Renka (1984).
nag_interp_2d_scat_eval (e01sb) must only be called after a call to nag_interp_2d_scat (e01sa).

## References

Renka R L (1984) Algorithm 624: triangulation and interpolation of arbitrarily distributed points in the plane ACM Trans. Math. Software 10 440–442
Renka R L and Cline A K (1984) A triangle-based ${C}^{1}$ interpolation method Rocky Mountain J. Math. 14 223–237

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}\left({\mathbf{m}}\right)$ – double array
2:     $\mathrm{y}\left({\mathbf{m}}\right)$ – double array
3:     $\mathrm{f}\left({\mathbf{m}}\right)$ – double array
4:     $\mathrm{triang}\left(7×{\mathbf{m}}\right)$int64int32nag_int array
5:     $\mathrm{grads}\left(2,{\mathbf{m}}\right)$ – double array
m, x, y, f, triang and grads must be unchanged from the previous call of nag_interp_2d_scat (e01sa).
6:     $\mathrm{px}$ – double scalar
7:     $\mathrm{py}$ – double scalar
The point $\left(px,py\right)$ at which the interpolant is to be evaluated.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the dimension of the arrays x, y, f, grads. (An error is raised if these dimensions are not equal.)
m, x, y, f, triang and grads must be unchanged from the previous call of nag_interp_2d_scat (e01sa).

### Output Parameters

1:     $\mathrm{pf}$ – double scalar
The value of the interpolant evaluated at the point $\left(px,py\right)$.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
 On entry, ${\mathbf{m}}<3$.
${\mathbf{ifail}}=2$
On entry, the triangulation information held in the array triang does not specify a valid triangulation of the data points. triang may have been corrupted since the call to nag_interp_2d_scat (e01sa).
W  ${\mathbf{ifail}}=3$
The evaluation point (px,py) lies outside the nodal triangulation, and the value returned in pf is computed by extrapolation.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Computational errors should be negligible in most practical situations.

The time taken for a call of nag_interp_2d_scat_eval (e01sb) is approximately proportional to the number of data points, $m$.
The results returned by this function are particularly suitable for applications such as graph plotting, producing a smooth surface from a number of scattered points.

## Example

See Example in nag_interp_2d_scat (e01sa).
```function e01sb_example

fprintf('e01sb example results\n\n');

x = [11.16; 12.85; 19.85; 19.72; 15.91;  0.00; 20.87;  3.45; 14.26; ...
17.43; 22.80;  7.58; 25.00;  0.00;  9.66;  5.22; 17.25; 25.00; ...
12.13; 22.23; 11.52; 15.20;  7.54; 17.32;  2.14;  0.51; 22.69; ...
5.47; 21.67;  3.31];
y = [ 1.24;  3.06; 10.72;  1.39;  7.74; 20.00; 20.00; 12.78; 17.87; ...
3.46; 12.39;  1.98; 11.87;  0.00; 20.00; 14.66; 19.57;  3.87; ...
10.79;  6.21;  8.53;  0.00; 10.69; 13.78; 15.03;  8.37; 19.63; ...
17.13; 14.36; 0.33];
f = [22.15; 22.11;  7.97; 16.83; 15.30; 34.60;  5.74; 41.24; 10.74; ...
18.60;  5.47; 29.87;  4.40; 58.20;  4.73; 40.36;  6.43;  8.74; ...
13.71; 10.25; 15.74; 21.60; 19.31; 12.11; 53.10; 49.43;  3.25; ...
28.63;  5.52; 44.08];

% Triangulate and obtain details of interpolant
x,y,f);

px = [3:3:21];
py = [2:3:17];
% Evaluate interpolant at on regular mesh (px,py)
for i = 1:6
for j = 1:7
[pf(i,j), ifail] = e01sb( ...
x, y, f, triang, grads, px(j), py(i));
end
end

% Display interpolated values
matrix = 'General';
diag = 'Non-unit';
format = 'F7.2';
title  = 'Spline evaluated on a regular mesh (x across, y down):';
chlab  = 'Character';
rlabs  = cellstr(num2str(py'));
clabs  = cellstr(num2str(px'));
ncols  = int64(80);
indent = int64(0);
[ifail] =  x04cb( ...
matrix, diag, pf, format, title, chlab, ...
rlabs, chlab, clabs, ncols, indent);

```
```e01sb example results

Spline evaluated on a regular mesh (x across, y down):
3      6      9     12     15     18     21
2   43.52  33.91  26.59  22.23  21.15  18.67  14.88
5   40.49  29.26  22.51  20.72  19.30  16.72  12.87
8   37.90  23.97  16.79  16.43  15.46  13.02   9.30
11   38.55  25.25  16.72  13.83  13.08  10.71   6.88
14   47.61  36.66  22.87  14.02  13.44  11.20   6.46
17   41.25  27.62  18.03  12.29  11.68   9.09   5.37
```