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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_interp_3d_scat_shep (e01tg)

## Purpose

nag_interp_3d_scat_shep (e01tg) generates a three-dimensional interpolant to a set of scattered data points, using a modified Shepard method.

## Syntax

[iq, rq, ifail] = e01tg(x, y, z, f, nw, nq, 'm', m)
[iq, rq, ifail] = nag_interp_3d_scat_shep(x, y, z, f, nw, nq, 'm', m)

## Description

nag_interp_3d_scat_shep (e01tg) constructs a smooth function Q(x,y,z)$Q\left(x,y,z\right)$ which interpolates a set of m$m$ scattered data points (xr,yr,zr,fr)$\left({x}_{r},{y}_{r},{z}_{r},{f}_{r}\right)$, for r = 1,2,,m$r=1,2,\dots ,m$, using a modification of Shepard's method. The surface is continuous and has continuous first partial derivatives.
The basic Shepard method, which is a generalization of the two-dimensional method described in Shepard (1968), interpolates the input data with the weighted mean
 Q(x,y,z) = ( ∑ r = 1mwr(x,y,z)qr)/( ∑ r = 1mwr(x,y,z)), $Q(x,y,z)=∑r=1mwr(x,y,z)qr ∑r=1mwr(x,y,z) ,$
where
 qr = fr ​ and ​ wr(x,y,z) = 1/(dr2) ​ and ​ dr2 = (x − xr)2 + (y − yr)2 + (z − zr)2. $qr=fr ​ and ​ wr(x,y,z)= 1dr2 ​ and ​ dr2= (x-xr) 2+ (y-yr) 2+ (z-zr) 2.$
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 wr(x,y,z)${w}_{r}\left(x,y,z\right)$ to be zero outside a sphere with centre (xr,yr,zr)$\left({x}_{r},{y}_{r},{z}_{r}\right)$ and some radius Rw${R}_{w}$. Also, to improve the performance of the basic method, each qr${q}_{r}$ above is replaced by a function qr(x,y,z)${q}_{r}\left(x,y,z\right)$, which is a quadratic fitted by weighted least squares to data local to (xr,yr,zr)$\left({x}_{r},{y}_{r},{z}_{r}\right)$ and forced to interpolate (xr,yr,zr,fr)$\left({x}_{r},{y}_{r},{z}_{r},{f}_{r}\right)$. In this context, a point (x,y,z)$\left(x,y,z\right)$ is defined to be local to another point if it lies within some distance Rq${R}_{q}$ 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 Rw${R}_{w}$ and Rq${R}_{q}$ are chosen to be just large enough to include Nw${N}_{w}$ and Nq${N}_{q}$ data points, respectively, for user-supplied constants Nw${N}_{w}$ and Nq${N}_{q}$. Default values of these parameters are provided by the function, and advice on alternatives is given in Section [Choice of and ].
This function is derived from the function QSHEP3 described by Renka (1988b).
Values of the interpolant Q(x,y,z)$Q\left(x,y,z\right)$ 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_3d_scat_shep_eval (e01th).

## 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 661: QSHEP3D: Quadratic Shepard method for trivariate interpolation of scattered data ACM Trans. Math. Software 14 151–152
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:     x(m) – double array
2:     y(m) – double array
3:     z(m) – double array
m, the dimension of the array, must satisfy the constraint m10${\mathbf{m}}\ge 10$.
x(r)${\mathbf{x}}\left(\mathit{r}\right)$, y(r)${\mathbf{y}}\left(\mathit{r}\right)$, z(r)${\mathbf{z}}\left(\mathit{r}\right)$ must be set to the Cartesian coordinates of the data point (xr,yr,zr)$\left({x}_{\mathit{r}},{y}_{\mathit{r}},{z}_{\mathit{r}}\right)$, for r = 1,2,,m$\mathit{r}=1,2,\dots ,m$.
Constraint: these coordinates must be distinct, and must not all be coplanar.
4:     f(m) – double array
m, the dimension of the array, must satisfy the constraint m10${\mathbf{m}}\ge 10$.
f(r)${\mathbf{f}}\left(\mathit{r}\right)$ must be set to the data value fr${f}_{\mathit{r}}$, for r = 1,2,,m$\mathit{r}=1,2,\dots ,m$.
5:     nw – int64int32nag_int scalar
The number Nw${N}_{w}$ of data points that determines each radius of influence Rw${R}_{w}$, appearing in the definition of each of the weights wr${w}_{\mathit{r}}$, for r = 1,2,,m$\mathit{r}=1,2,\dots ,m$ (see Section [Description]). Note that Rw${R}_{w}$ is different for each weight. If nw0${\mathbf{nw}}\le 0$ the default value nw = min (32,m1)${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(32,{\mathbf{m}}-1\right)$ is used instead.
Constraint: nwmin (40,m1)${\mathbf{nw}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(40,{\mathbf{m}}-1\right)$.
6:     nq – int64int32nag_int scalar
The number Nq${N}_{q}$ of data points to be used in the least squares fit for coefficients defining the nodal functions qr(x,y,z)${q}_{r}\left(x,y,z\right)$ (see Section [Description]). If nq0${\mathbf{nq}}\le 0$ the default value nq = min (17,m1)${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(17,{\mathbf{m}}-1\right)$ is used instead.
Constraint: nq0${\mathbf{nq}}\le 0$ or 9nqmin (40,m1)$9\le {\mathbf{nq}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(40,{\mathbf{m}}-1\right)$.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the arrays x, y, z, f. (An error is raised if these dimensions are not equal.)
m$m$, the number of data points.
Constraint: m10${\mathbf{m}}\ge 10$.

liq lrq

### Output Parameters

1:     iq(liq) – int64int32nag_int array
liq2 × m + 1$\mathit{liq}\ge 2×{\mathbf{m}}+1$.
Integer data defining the interpolant Q(x,y,z)$Q\left(x,y,z\right)$.
2:     rq(lrq) – double array
lrq10 × m + 7$\mathit{lrq}\ge 10×{\mathbf{m}}+7$.
Real data defining the interpolant Q(x,y,z)$Q\left(x,y,z\right)$.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, m < 10${\mathbf{m}}<10$, or 0 < nq < 9$0<{\mathbf{nq}}<9$, or nq > min (40,m − 1) ${\mathbf{nq}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(40,{\mathbf{m}}-1\right)$, or nw > min (40,m − 1) ${\mathbf{nw}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(40,{\mathbf{m}}-1\right)$, or liq < 2 × m + 1$\mathit{liq}<2×{\mathbf{m}}+1$, or lrq < 10 × m + 7$\mathit{lrq}<10×{\mathbf{m}}+7$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, (x(i),y(i),z(i)) = (x(j),y(j),z(j))$\left({\mathbf{x}}\left(i\right),{\mathbf{y}}\left(i\right),{\mathbf{z}}\left(i\right)\right)=\left({\mathbf{x}}\left(j\right),{\mathbf{y}}\left(j\right),{\mathbf{z}}\left(j\right)\right)$ for some i ≠ j$i\ne j$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, all the data points are coplanar. No unique solution exists.

## 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_3d_scat_shep (e01tg) will depend in general on the distribution of the data points. If x, y and z are uniformly randomly distributed, then the time taken should be O(m). At worst O(m2)$\mathit{O}\left({{\mathbf{m}}}^{2}\right)$ time will be required.

### Choice of Nw and Nq

Default values of the parameters Nw${N}_{w}$ and Nq${N}_{q}$ may be selected by calling nag_interp_3d_scat_shep (e01tg) with nw0${\mathbf{nw}}\le 0$ and nq0${\mathbf{nq}}\le 0$. These default values may well be satisfactory for many applications.
If non-default values are required they must be supplied to nag_interp_3d_scat_shep (e01tg) through positive values of nw and nq. Increasing these parameters makes the method less local. This may increase the accuracy of the resulting interpolant at the expense of increased computational cost. The default values nw = min (32,m1) ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(32,{\mathbf{m}}-1\right)$ and nq = min (17,m1) ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(17,{\mathbf{m}}-1\right)$ 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 Nw${N}_{w}$ and Nq${N}_{q}$, generally increasing monotonically and slowly with distance from the optimal pair. The method is not therefore thought to be particularly sensitive to the parameter values. For further advice on the choice of these parameters see Renka (1988a).

## Example

```function nag_interp_3d_scat_shep_example
x = [0.8;
0.23;
0.18;
0.58;
0.64;
0.88;
0.3;
0.87;
0.04;
0.62;
0.87;
0.62;
0.86;
0.87;
0.49;
0.12;
0.02;
0.62;
0.49;
0.36;
0.62;
0.01;
0.41;
0.17;
0.51;
0.85;
0.2;
0.04;
0.31;
0.88];
y = [0.23;
0.88;
0.43;
0.95;
0.69;
0.35;
0.1;
0.09;
0.02;
0.9;
0.96;
0.64;
0.13;
0.6;
0.43;
0.61;
0.71;
0.93;
0.54;
0.56;
0.42;
0.72;
0.36;
0.99;
0.29;
0.05;
0.2;
0.67;
0.63;
0.27];
z = [0.37;
0.05;
0.04;
0.62;
0.2;
0.49;
0.78;
0.05;
0.4;
0.43;
0.24;
0.45;
0.47;
0.46;
0.13;
0;
0.82;
0.44;
0.04;
0.39;
0.97;
0.45;
0.52;
0.65;
0.59;
0.04;
0.87;
0.04;
0.18;
0.07];
f = [0.51;
1.8;
0.11;
2.65;
0.93;
0.72;
-0.11;
0.67;
0;
2.2;
3.17;
0.74;
0.64;
1.07;
0.22;
0.41;
0.58;
2.48;
0.37;
0.35;
-0.2;
0.78;
0.11;
2.82;
0.14;
0.61;
-0.25;
0.59;
0.5;
0.71];
nw = int64(0);
nq = int64(0);
[iq, rq, ifail] = nag_interp_3d_scat_shep(x, y, z, f, nw, nq);
ifail
```
```

ifail =

0

```
```function e01tg_example
x = [0.8;
0.23;
0.18;
0.58;
0.64;
0.88;
0.3;
0.87;
0.04;
0.62;
0.87;
0.62;
0.86;
0.87;
0.49;
0.12;
0.02;
0.62;
0.49;
0.36;
0.62;
0.01;
0.41;
0.17;
0.51;
0.85;
0.2;
0.04;
0.31;
0.88];
y = [0.23;
0.88;
0.43;
0.95;
0.69;
0.35;
0.1;
0.09;
0.02;
0.9;
0.96;
0.64;
0.13;
0.6;
0.43;
0.61;
0.71;
0.93;
0.54;
0.56;
0.42;
0.72;
0.36;
0.99;
0.29;
0.05;
0.2;
0.67;
0.63;
0.27];
z = [0.37;
0.05;
0.04;
0.62;
0.2;
0.49;
0.78;
0.05;
0.4;
0.43;
0.24;
0.45;
0.47;
0.46;
0.13;
0;
0.82;
0.44;
0.04;
0.39;
0.97;
0.45;
0.52;
0.65;
0.59;
0.04;
0.87;
0.04;
0.18;
0.07];
f = [0.51;
1.8;
0.11;
2.65;
0.93;
0.72;
-0.11;
0.67;
0;
2.2;
3.17;
0.74;
0.64;
1.07;
0.22;
0.41;
0.58;
2.48;
0.37;
0.35;
-0.2;
0.78;
0.11;
2.82;
0.14;
0.61;
-0.25;
0.59;
0.5;
0.71];
nw = int64(0);
nq = int64(0);
[iq, rq, ifail] = e01tg(x, y, z, f, nw, nq);
ifail
```
```

ifail =

0

```