E01 Chapter Contents
E01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentE01TKF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

E01TKF generates a four-dimensional interpolant to a set of scattered data points, using a modified Shepard method.

## 2  Specification

 SUBROUTINE E01TKF ( M, X, F, NW, NQ, IQ, RQ, IFAIL)
 INTEGER M, NW, NQ, IQ(2*M+1), IFAIL REAL (KIND=nag_wp) X(4,M), F(M), RQ(15*M+9)

## 3  Description

E01TKF constructs a smooth function $Q\left(\mathbf{x}\right)$, $\mathbf{x}\in {ℝ}^{4}$ which interpolates a set of $m$ scattered data points $\left({\mathbf{x}}_{r},{f}_{r}\right)$, 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 method, which is a generalization of the two-dimensional method described in Shepard (1968), interpolates the input data with the weighted mean
 $Q x = ∑ r=1 m wr x qr ∑ r=1 m wr x ,$
where ${q}_{r}={f}_{r}$, ${w}_{r}\left(\mathbf{x}\right)=\frac{1}{{d}_{r}^{2}}$ and ${d}_{r}^{2}={{‖\mathbf{x}-{\mathbf{x}}_{r}‖}_{2}}^{2}$.
The basic method is global in that the interpolated value at any point depends on all the data, but E01TKF uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each ${w}_{r}\left(\mathbf{x}\right)$ to be zero outside a hypersphere with centre ${\mathbf{x}}_{r}$ and some radius ${R}_{w}$. Also, to improve the performance of the basic method, each ${q}_{r}$ above is replaced by a function ${q}_{r}\left(\mathbf{x}\right)$, which is a quadratic fitted by weighted least squares to data local to ${\mathbf{x}}_{r}$ and forced to interpolate $\left({\mathbf{x}}_{r},{f}_{r}\right)$. In this context, a point $\mathbf{x}$ is defined to be local to another point if it lies within some distance ${R}_{q}$ of it.
The efficiency of E01TKF is enhanced by using a cell method for nearest neighbour searching due to Bentley and Friedman (1979) with a cell density of $3$.
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 parameters are provided by the routine, and advice on alternatives is given in Section 8.2.
E01TKF is derived from the new implementation of QSHEP3 described by Renka (1988b). It uses the modification for high-dimensional interpolation described by Berry and Minser (1999).
Values of the interpolant $Q\left(\mathbf{x}\right)$ generated by E01TKF, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to E01TLF.

## 4  References

Bentley J L and Friedman J H (1979) Data structures for range searching ACM Comput. Surv. 11 397–409
Berry M W, Minser K S (1999) Algorithm 798: high-dimensional interpolation using the modified Shepard method ACM Trans. Math. Software 25 353–366
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

## 5  Parameters

1:     M – INTEGERInput
On entry: $m$, the number of data points.
Constraint: ${\mathbf{M}}\ge 16$.
2:     X($4$,M) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{X}}\left(1:4,\mathit{r}\right)$ must be set to the Cartesian coordinates of the data point ${\mathbf{x}}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.
Constraint: these coordinates must be distinct, and must not all lie on the same three-dimensional hypersurface.
3:     F(M) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{F}}\left(\mathit{r}\right)$ must be set to the data value ${f}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.
4:     NW – INTEGERInput
On entry: 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 Section 3). Note that ${R}_{w}$ is different for each weight. If ${\mathbf{NW}}\le 0$ the default value ${\mathbf{NW}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(32,{\mathbf{M}}-1\right)$ is used instead.
Constraint: ${\mathbf{NW}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(50,{\mathbf{M}}-1\right)$.
5:     NQ – INTEGERInput
On entry: the number ${N}_{q}$ of data points to be used in the least squares fit for coefficients defining the quadratic functions ${q}_{r}\left(\mathbf{x}\right)$ (see Section 3). If ${\mathbf{NQ}}\le 0$ the default value ${\mathbf{NQ}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(38,{\mathbf{M}}-1\right)$ is used instead.
Constraint: ${\mathbf{NQ}}\le 0$ or $14\le {\mathbf{NQ}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(50,{\mathbf{M}}-1\right)$.
6:     IQ($2×{\mathbf{M}}+1$) – INTEGER arrayOutput
On exit: integer data defining the interpolant $Q\left(\mathbf{x}\right)$.
7:     RQ($15×{\mathbf{M}}+9$) – REAL (KIND=nag_wp) arrayOutput
On exit: real data defining the interpolant $Q\left(\mathbf{x}\right)$.
8:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{M}}<16$, or $0<{\mathbf{NQ}}<14$, or ${\mathbf{NQ}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(50,{\mathbf{M}}-1\right)$, or ${\mathbf{NW}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(50,{\mathbf{M}}-1\right)$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{X}}\left(1:4,i\right)={\mathbf{X}}\left(1:4,j\right)$ for some $i\ne j$. The interpolant cannot be derived.
${\mathbf{IFAIL}}=3$
On entry, all the data points lie on the same three-dimensional hypersurface. No unique solution exists.

## 7  Accuracy

On successful exit, the routine generated interpolates the input data exactly and has quadratic precision. Overall accuracy of the interpolant is affected by the choice of parameters NW and NQ as well as the smoothness of the function represented by the input data.

## 8  Further Comments

### 8.1  Timing

The time taken for a call to E01TKF will depend in general on the distribution of the data points and on the choice of ${N}_{w}$ and ${N}_{q}$ parameters. If the data points are uniformly randomly distributed, then the time taken should be $\mathit{O}\left(m\right)$. At worst $\mathit{O}\left({m}^{2}\right)$ time will be required.

### 8.2  Choice of ${N}_{w}$ and ${N}_{q}$

Default values of the parameters ${N}_{w}$ and ${N}_{q}$ may be selected by calling E01TKF with ${\mathbf{NW}}\le 0$ and ${\mathbf{NQ}}\le 0$. These default values may well be satisfactory for many applications.
If nondefault values are required they must be supplied to E01TKF through positive values of NW and NQ. Increasing these parameter values makes the method less local. This may increase the accuracy of the resulting interpolant at the expense of increased computational cost.

## 9  Example

This program reads in a set of $30$ data points and calls E01TKF to construct an interpolating function $Q\left(\mathbf{x}\right)$. It then calls E01TLF 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.

### 9.1  Program Text

Program Text (e01tkfe.f90)

### 9.2  Program Data

Program Data (e01tkfe.d)

### 9.3  Program Results

Program Results (e01tkfe.r)