NAG CL Interface
e01tmc (dim5_scat_shep)
1
Purpose
e01tmc generates a fivedimensional interpolant to a set of scattered data points, using a modified Shepard method.
2
Specification
void 
e01tmc (Integer m,
const double x[],
const double f[],
Integer nw,
Integer nq,
Integer iq[],
double rq[],
NagError *fail) 

The function may be called by the names: e01tmc, nag_interp_dim5_scat_shep or nag_5d_shep_interp.
3
Description
e01tmc 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
e01tmc 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
e01tmc 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 arguments are provided, and advice on alternatives is given in
Section 9.2.
e01tmc 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
e01tmc, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to
e01tnc.
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
Arguments

1:
$\mathbf{m}$ – Integer
Input

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:
$\mathbf{x}\left[5\times {\mathbf{m}}\right]$ – const double
Input

Note: the $\left(i,j\right)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[\left(j1\right)\times 5+i1\right]$.
On entry: ${\mathbf{x}}\left[\left(\mathit{r}1\right)\times 5\right],\dots ,{\mathbf{x}}\left[\left(\mathit{r}1\right)\times 5+4\right]$ 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:
$\mathbf{f}\left[{\mathbf{m}}\right]$ – const double
Input

On entry: ${\mathbf{f}}\left[\mathit{r}1\right]$ must be set to the data value ${f}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.

4:
$\mathbf{nw}$ – Integer
Input

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:
$\mathbf{nq}$ – Integer
Input

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:
$\mathbf{iq}\left[2\times {\mathbf{m}}+1\right]$ – Integer
Output

On exit: integer data defining the interpolant $Q\left(\mathbf{x}\right)$.

7:
$\mathbf{rq}\left[21\times {\mathbf{m}}+11\right]$ – double
Output

On exit: real data defining the interpolant $Q\left(\mathbf{x}\right)$.

8:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_DATA_HYPERSURFACE

On entry, all the data points lie on the same fourdimensional hypersurface.
No unique solution exists.
 NE_DUPLICATE_NODE

There are duplicate nodes in the dataset. ${\mathbf{x}}\left[\left(k1\right)\times 5+i1\right]={\mathbf{x}}\left[\left(k1\right)\times 5+j1\right]$, for $i=\u2329\mathit{\text{value}}\u232a$, $j=\u2329\mathit{\text{value}}\u232a$ and $k=1,2,\dots ,5$. The interpolant cannot be derived.
 NE_INT

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge 23$.
On entry, ${\mathbf{nq}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nq}}\le 0$ or
${\mathbf{nq}}\ge 20$.
 NE_INT_2

On entry, ${\mathbf{nq}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nq}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(70,{\mathbf{m}}1\right)$.
On entry, ${\mathbf{nw}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nw}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(50,{\mathbf{m}}1\right)$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
On successful exit, the function generated interpolates the input data exactly and has quadratic precision. Overall accuracy of the interpolant is affected by the choice of arguments
nw and
nq as well as the smoothness of the function represented by the input data.
Berry and Minser (1999) report on the results obtained for a set of test functions.
8
Parallelism and Performance
e01tmc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e01tmc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time taken for a call to e01tmc 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 arguments ${N}_{w}$ and ${N}_{q}$ may be selected by calling e01tmc 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
e01tmc through positive values of
nw and
nq. Increasing these argument 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 function. For further advice on the choice of these arguments see
Berry and Minser (1999).
10
Example
This program reads in a set of
$30$ data points and calls
e01tmc to construct an interpolating function
$Q\left(\mathbf{x}\right)$. It then calls
e01tnc 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 10 in
e01tnc.
10.1
Program Text
10.2
Program Data
10.3
Program Results