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NAG Toolbox: nag_interp_2d_scat_shep (e01sg)

Purpose

nag_interp_2d_scat_shep (e01sg) generates a two-dimensional interpolant to a set of scattered data points, using a modified Shepard method.

Syntax

[iq, rq, ifail] = e01sg(x, y, f, nw, nq, 'm', m)
[iq, rq, ifail] = nag_interp_2d_scat_shep(x, y, f, nw, nq, 'm', m)

Description

nag_interp_2d_scat_shep (e01sg) constructs a smooth function $Q\left(x,y\right)$ which interpolates a set of $m$ scattered data points $\left({x}_{r},{y}_{r},{f}_{r}\right)$, for $r=1,2,\dots ,m$, using a modification of Shepard's method. The surface is continuous and has continuous first partial derivatives.
The basic Shepard (1968) method interpolates the input data with the weighted mean
 $Qx,y=∑r=1mwrx,yqr ∑r=1mwrx,y ,$
where ${q}_{r}={f}_{r}$, ${w}_{r}\left(x,y\right)=\frac{1}{{d}_{r}^{2}}$ and ${d}_{r}^{2}={\left(x-{x}_{r}\right)}^{2}+{\left(y-{y}_{r}\right)}^{2}$.
The basic method is global in that the interpolated value at any point depends on all the data, but this function uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each ${w}_{r}\left(x,y\right)$ to be zero outside a circle with centre $\left({x}_{r},{y}_{r}\right)$ and some radius ${R}_{w}$. Also, to improve the performance of the basic method, each ${q}_{r}$ above is replaced by a function ${q}_{r}\left(x,y\right)$, which is a quadratic fitted by weighted least squares to data local to $\left({x}_{r},{y}_{r}\right)$ and forced to interpolate $\left({x}_{r},{y}_{r},{f}_{r}\right)$. In this context, a point $\left(x,y\right)$ is defined to be local to another point if it lies within some distance ${R}_{q}$ of it. Computation of these quadratics constitutes the main work done by this function.
The efficiency of the function is further enhanced by using a cell method for nearest neighbour searching due to Bentley and Friedman (1979).
The radii ${R}_{w}$ and ${R}_{q}$ are chosen to be just large enough to include ${N}_{w}$ and ${N}_{q}$ data points, respectively, for user-supplied constants ${N}_{w}$ and ${N}_{q}$. Default values of these arguments are provided by the function, and advice on alternatives is given in Choice of and .
This function is derived from the function QSHEP2 described by Renka (1988b).
Values of the interpolant $Q\left(x,y\right)$ generated by this function, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to nag_interp_2d_scat_shep_eval (e01sh).

References

Bentley J L and Friedman J H (1979) Data structures for range searching ACM Comput. Surv. 11 397–409
Franke R and Nielson G (1980) Smooth interpolation of large sets of scattered data Internat. J. Num. Methods Engrg. 15 1691–1704
Renka R J (1988a) Multivariate interpolation of large sets of scattered data ACM Trans. Math. Software 14 139–148
Renka R J (1988b) Algorithm 660: QSHEP2D: Quadratic Shepard method for bivariate interpolation of scattered data ACM Trans. Math. Software 14 149–150
Shepard D (1968) A two-dimensional interpolation function for irregularly spaced data Proc. 23rd Nat. Conf. ACM 517–523 Brandon/Systems Press Inc., Princeton

Parameters

Compulsory Input Parameters

1:     $\mathrm{x}\left({\mathbf{m}}\right)$ – double array
2:     $\mathrm{y}\left({\mathbf{m}}\right)$ – double array
The Cartesian coordinates of the data points $\left({x}_{\mathit{r}},{y}_{\mathit{r}}\right)$, for $\mathit{r}=1,2,\dots ,m$.
Constraint: these coordinates must be distinct, and must not all be collinear.
3:     $\mathrm{f}\left({\mathbf{m}}\right)$ – double array
${\mathbf{f}}\left(\mathit{r}\right)$ must be set to the data value ${f}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.
4:     $\mathrm{nw}$int64int32nag_int scalar
The number ${N}_{w}$ of data points that determines each radius of influence ${R}_{w}$, appearing in the definition of each of the weights ${w}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$ (see Description). Note that ${R}_{w}$ is different for each weight. If ${\mathbf{nw}}\le 0$ the default value ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(19,{\mathbf{m}}-1\right)$ is used instead.
Constraint: ${\mathbf{nw}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(40,{\mathbf{m}}-1\right)$.
5:     $\mathrm{nq}$int64int32nag_int scalar
The number ${N}_{q}$ of data points to be used in the least squares fit for coefficients defining the nodal functions ${q}_{r}\left(x,y\right)$ (see Description). If ${\mathbf{nq}}\le 0$ the default value ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(13,{\mathbf{m}}-1\right)$ is used instead.
Constraint: ${\mathbf{nq}}\le 0$ or $5\le {\mathbf{nq}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(40,{\mathbf{m}}-1\right)$.

Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the dimension of the arrays x, y, f. (An error is raised if these dimensions are not equal.)
$m$, the number of data points.
Constraint: ${\mathbf{m}}\ge 6$.

Output Parameters

1:     $\mathrm{iq}\left(\mathit{liq}\right)$int64int32nag_int array
$\mathit{liq}=2×{\mathbf{m}}+1$.
Integer data defining the interpolant $Q\left(x,y\right)$.
2:     $\mathrm{rq}\left(\mathit{lrq}\right)$ – double array
$\mathit{lrq}=6×{\mathbf{m}}+5$.
Real data defining the interpolant $Q\left(x,y\right)$.
3:     $\mathrm{ifail}$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:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{m}}<6$, or $0<{\mathbf{nq}}<5$, or ${\mathbf{nq}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(40,{\mathbf{m}}-1\right)$, or ${\mathbf{nw}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(40,{\mathbf{m}}-1\right)$, or $\mathit{liq}<2×{\mathbf{m}}+1$, or $\mathit{lrq}<6×{\mathbf{m}}+5$.
${\mathbf{ifail}}=2$
 On entry, $\left({\mathbf{x}}\left(i\right),{\mathbf{y}}\left(i\right)\right)=\left({\mathbf{x}}\left(j\right),{\mathbf{y}}\left(j\right)\right)$ for some $i\ne j$.
${\mathbf{ifail}}=3$
 On entry, all the data points are collinear. No unique solution exists.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Accuracy

On successful exit, the function generated interpolates the input data exactly and has quadratic accuracy.

Timing

The time taken for a call to nag_interp_2d_scat_shep (e01sg) will depend in general on the distribution of the data points. If x and y are uniformly randomly distributed, then the time taken should be $\mathit{O}\left({\mathbf{m}}\right)$. At worst $\mathit{O}\left({{\mathbf{m}}}^{2}\right)$ time will be required.

Choice of Nw and Nq

Default values of the arguments ${N}_{w}$ and ${N}_{q}$ may be selected by calling nag_interp_2d_scat_shep (e01sg) with ${\mathbf{nw}}\le 0$ and ${\mathbf{nq}}\le 0$. These default values may well be satisfactory for many applications.
If non-default values are required they must be supplied to nag_interp_2d_scat_shep (e01sg) through positive values of nw and nq. Increasing these arguments makes the method less local. This may increase the accuracy of the resulting interpolant at the expense of increased computational cost. The default values ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(19,{\mathbf{m}}-1\right)$ and ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(13,{\mathbf{m}}-1\right)$ have been chosen on the basis of experimental results reported in Renka (1988a). In these experiments the error norm was found to vary smoothly with ${N}_{w}$ and ${N}_{q}$, generally increasing monotonically and slowly with distance from the optimal pair. The method is not therefore thought to be particularly sensitive to the argument values. For further advice on the choice of these arguments see Renka (1988a).

Example

This program reads in a set of $30$ data points and calls nag_interp_2d_scat_shep (e01sg) to construct an interpolating function $Q\left(x,y\right)$. It then calls nag_interp_2d_scat_shep_eval (e01sh) to evaluate the interpolant at a set of points.
Note that this example is not typical of a realistic problem: the number of data points would normally be larger.
```function e01sg_example

fprintf('e01sg example results\n\n');

% Scattered Grid Data
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];

% Generate interpolant
nw = int64(0);
nq = int64(0);
[iq, rq, ifail] = e01sg(x, y, f, nw, nq);

% Interpolation points
u = [20.00;  6.41;  7.54;  9.91; 12.30];
v = [ 3.14; 15.44; 10.69; 18.27;  9.22];

% Interpolate at interpolation points
[q, qx, qy, ifail] = e01sh(x, y, f, iq, rq, u, v);

fprintf('Interpolated values Q and its derivatives at (u,v)\n');
fprintf('     u      v      q      qx     qy\n');
for i = 1:size(u,1)
fprintf('%7.2f%7.2f%7.2f%7.2f%7.2f\n', u(i), v(i), q(i), qx(i), qy(i));
end

```
```e01sg example results

Interpolated values Q and its derivatives at (u,v)
u      v      q      qx     qy
20.00   3.14  15.89  -1.28  -0.63
6.41  15.44  34.05  -3.62  -3.56
7.54  10.69  19.31  -2.84   0.81
9.91  18.27  13.68  -1.59  -4.71
12.30   9.22  14.56  -0.68  -0.78
```

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