NAG Library Routine Document
E01TMF
1 Purpose
E01TMF generates a fivedimensional interpolant to a set of scattered data points, using a modified Shepard method.
2 Specification
INTEGER 
M, NW, NQ, IQ(2*M+1), IFAIL 
REAL (KIND=nag_wp) 
X(5,M), F(M), RQ(21*M+11) 

3 Description
E01TMF constructs a smooth function $Q\left(\mathbf{x}\right)$, $\mathbf{x}\in {\mathbb{R}}^{5}$ 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 twodimensional method described in
Shepard (1968), interpolates the input data with the weighted mean
where
${q}_{r}={f}_{r}$,
${w}_{r}\left(\mathbf{x}\right)=\frac{1}{{d}_{r}^{2}}$ and
${d}_{r}^{2}={{\Vert \mathbf{x}{\mathbf{x}}_{r}\Vert}_{2}}^{2}$.
The basic method is global in that the interpolated value at any point depends on all the data, but E01TMF 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 E01TMF 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 usersupplied constants
${N}_{w}$ and
${N}_{q}$. Default values of these parameters are provided, and advice on alternatives is given in
Section 8.2.
E01TMF is derived from the new implementation of QSHEP3 described by
Renka (1988b). It uses the modification for fivedimensional interpolation described by
Berry and Minser (1999).
Values of the interpolant
$Q\left(\mathbf{x}\right)$ generated by E01TMF, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to
E01TNF.
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: highdimensional 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 twodimensional 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.
Note: on the basis of experimental results reported in
Berry and Minser (1999), it is recommended to use
${\mathbf{M}}\ge 4000$.
Constraint:
${\mathbf{M}}\ge 23$.
 2: X($5$,M) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{X}}\left(1:5,\mathit{r}\right)$ 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 fourdimensional 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
${\mathbf{NQ}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(50,{\mathbf{M}}1\right)$ is used instead.
Constraint:
${\mathbf{NQ}}\le 0$ or $20\le {\mathbf{NQ}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(70,{\mathbf{M}}1\right)$.
 6: IQ($2\times {\mathbf{M}}+1$) – INTEGER arrayOutput
On exit: integer data defining the interpolant $Q\left(\mathbf{x}\right)$.
 7: RQ($21\times {\mathbf{M}}+11$) – 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,  ${\mathbf{M}}<23$, 
or  $0<{\mathbf{NQ}}<20$, 
or  ${\mathbf{NQ}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(70,{\mathbf{M}}1\right)$, 
or  ${\mathbf{NW}}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(50,{\mathbf{M}}1\right)$. 
 ${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{X}}\left(1:5,i\right)={\mathbf{X}}\left(1:5,j\right)$ for some $i\ne j$. The interpolant cannot be derived.
 ${\mathbf{IFAIL}}=3$
On entry, all the data points lie on the same fourdimensional 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 routine represented by the input data.
Berry and Minser (1999) report on the results obtained for a set of test routines.
The time taken for a call to E01TMF 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.
Default values of the parameters ${N}_{w}$ and ${N}_{q}$ may be selected by calling E01TMF 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 E01TMF 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. The default values
${\mathbf{NW}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(32,{\mathbf{M}}1\right)$ and
${\mathbf{NQ}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(50,{\mathbf{M}}1\right)$ have been chosen on the basis of experimental results reported in
Berry and Minser (1999). In these experiments the error norm was found to increase with the decrease of
${N}_{q}$, but to be little affected by the choice of
${N}_{w}$. The choice of both, directly affected the time taken by the routine. For further advice on the choice of these parameters see
Berry and Minser (1999).
9 Example
This program reads in a set of
$30$ data points and calls E01TMF to construct an interpolating function
$Q\left(\mathbf{x}\right)$. It then calls
E01TNF 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.
See also
Section 9 in E01TNF.
9.1 Program Text
Program Text (e01tmfe.f90)
9.2 Program Data
Program Data (e01tmfe.d)
9.3 Program Results
Program Results (e01tmfe.r)