NAG Library Routine Document
E01SGF
1 Purpose
E01SGF generates a twodimensional interpolant to a set of scattered data points, using a modified Shepard method.
2 Specification
SUBROUTINE E01SGF ( 
M, X, Y, F, NW, NQ, IQ, LIQ, RQ, LRQ, IFAIL) 
INTEGER 
M, NW, NQ, IQ(LIQ), LIQ, LRQ, IFAIL 
REAL (KIND=nag_wp) 
X(M), Y(M), F(M), RQ(LRQ) 

3 Description
E01SGF 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
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 routine 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 routine.
The efficiency of the routine 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 usersupplied constants
${N}_{w}$ and
${N}_{q}$. Default values of these arguments are provided by the routine, and advice on alternatives is given in
Section 9.2.
This routine is derived from the routine QSHEP2 described by
Renka (1988b).
Values of the interpolant
$Q\left(x,y\right)$ generated by this routine, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to
E01SHF.
4 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 twodimensional interpolation function for irregularly spaced data Proc. 23rd Nat. Conf. ACM 517–523 Brandon/Systems Press Inc., Princeton
5 Arguments
 1: $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of data points.
Constraint:
${\mathbf{M}}\ge 6$.
 2: $\mathrm{X}\left({\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayInput
 3: $\mathrm{Y}\left({\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: 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.
 4: $\mathrm{F}\left({\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: ${\mathbf{F}}\left(\mathit{r}\right)$ must be set to the data value ${f}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.
 5: $\mathrm{NW}$ – 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
${\mathbf{NW}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(19,{\mathbf{M}}1\right)$ is used instead.
Constraint:
${\mathbf{NW}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(40,{\mathbf{M}}1\right)$.
 6: $\mathrm{NQ}$ – INTEGERInput

On entry: 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
Section 3). 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 $5\le {\mathbf{NQ}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(40,{\mathbf{M}}1\right)$.
 7: $\mathrm{IQ}\left({\mathbf{LIQ}}\right)$ – INTEGER arrayOutput

On exit: integer data defining the interpolant $Q\left(x,y\right)$.
 8: $\mathrm{LIQ}$ – INTEGERInput

On entry: the dimension of the array
IQ as declared in the (sub)program from which E01SGF is called.
Constraint:
${\mathbf{LIQ}}\ge 2\times {\mathbf{M}}+1$.
 9: $\mathrm{RQ}\left({\mathbf{LRQ}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: real data defining the interpolant $Q\left(x,y\right)$.
 10: $\mathrm{LRQ}$ – INTEGERInput

On entry: the dimension of the array
RQ as declared in the (sub)program from which E01SGF is called.
Constraint:
${\mathbf{LRQ}}\ge 6\times {\mathbf{M}}+5$.
 11: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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
${\mathbf{IFAIL}}=0$ or
$1$, 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,  ${\mathbf{M}}<6$, 
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  ${\mathbf{LIQ}}<2\times {\mathbf{M}}+1$, 
or  ${\mathbf{LRQ}}<6\times {\mathbf{M}}+5$. 
 ${\mathbf{IFAIL}}=2$

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.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7 Accuracy
On successful exit, the function generated interpolates the input data exactly and has quadratic accuracy.
8 Parallelism and Performance
E01SGF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
E01SGF 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 routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time taken for a call to E01SGF 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.
Default values of the arguments ${N}_{w}$ and ${N}_{q}$ may be selected by calling E01SGF 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 E01SGF 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).
10 Example
This program reads in a set of
$30$ data points and calls E01SGF to construct an interpolating function
$Q\left(x,y\right)$. It then calls
E01SHF 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.
10.1 Program Text
Program Text (e01sgfe.f90)
10.2 Program Data
Program Data (e01sgfe.d)
10.3 Program Results
Program Results (e01sgfe.r)