The function may be called by the names: e01sgc, nag_interp_dim2_scat_shep or nag_2d_shep_interp.
3Description
e01sgc constructs a smooth function $Q(x,y)$ which interpolates a set of $m$ scattered data points $({x}_{r},{y}_{r},{f}_{r})$, 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 ${q}_{r}={f}_{r}$, ${w}_{r}(x,y)=\frac{1}{{d}_{r}^{2}}$ and ${d}_{r}^{2}={(x-{x}_{r})}^{2}+{(y-{y}_{r})}^{2}$.
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 ${w}_{r}(x,y)$ to be zero outside a circle with centre $({x}_{r},{y}_{r})$ and some radius ${R}_{w}$. Also, to improve the performance of the basic method, each ${q}_{r}$ above is replaced by a function ${q}_{r}(x,y)$, which is a quadratic fitted by weighted least squares to data local to $({x}_{r},{y}_{r})$ and forced to interpolate $({x}_{r},{y}_{r},{f}_{r})$. In this context, a point $(x,y)$ 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 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 ${R}_{w}$ 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 arguments are provided by the function, and advice on alternatives is given in Section 9.2.
This function is derived from the function QSHEP2 described by Renka (1988b).
Values of the interpolant $Q(x,y)$ 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 e01shc.
4References
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. Software14 139–148
Renka R J (1988b) Algorithm 660: QSHEP2D: Quadratic Shepard method for bivariate interpolation of scattered data ACM Trans. Math. Software14 149–150
Shepard D (1968) A two-dimensional interpolation function for irregularly spaced data Proc. 23rd Nat. Conf. ACM 517–523 Brandon/Systems Press Inc., Princeton
On entry: ${\mathbf{f}}\left[\mathit{r}-1\right]$ must be set to the data value ${f}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.
5: $\mathbf{nw}$ – 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 ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(19,{\mathbf{m}}-1)$ is used instead.
On entry: the number ${N}_{q}$ of data points to be used in the least squares fit for coefficients defining the nodal functions ${q}_{r}(x,y)$ (see Section 3). If ${\mathbf{nq}}\le 0$ the default value ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(13,{\mathbf{m}}-1)$ is used instead.
Constraint:
${\mathbf{nq}}\le 0$ or $5\le {\mathbf{nq}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}(40,{\mathbf{m}}-1)$.
On exit: real data defining the interpolant $Q(x,y)$.
9: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALL_DATA_COLLINEAR
All nodes are collinear. There is no unique solution.
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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_DATA_NOT_UNIQUE
There are duplicate nodes in the dataset. $({\mathbf{x}}\left[\mathit{I}-1\right],{\mathbf{y}}\left[\mathit{I}-1\right])=({\mathbf{x}}\left[\mathit{J}-1\right],{\mathbf{y}}\left[\mathit{J}-1\right])$, for $\mathit{I}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathit{J}=\u27e8\mathit{\text{value}}\u27e9$. The interpolant cannot be derived.
NE_INT
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge 6$.
On entry, ${\mathbf{nq}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nq}}\le 0$ or ${\mathbf{nq}}\ge 5$.
NE_INT_2
On entry, ${\mathbf{nq}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nq}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}(40,{\mathbf{m}}-1)$.
On entry, ${\mathbf{nw}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nw}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}(40,{\mathbf{m}}-1)$.
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.
7Accuracy
On successful exit, the function generated interpolates the input data exactly and has quadratic accuracy.
8Parallelism and Performance
e01sgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e01sgc 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 implementation-specific information.
9Further Comments
9.1Timing
The time taken for a call to e01sgc 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.
9.2Choice of ${\mathit{N}}_{\mathit{w}}$ and ${\mathit{N}}_{\mathit{q}}$
Default values of the arguments ${N}_{w}$ and ${N}_{q}$ may be selected by calling e01sgc with ${\mathbf{nw}}\le 0$ and ${\mathbf{nq}}\le 0$. These default values may well be satisfactory for many applications.
If non-default values are required they must be supplied to e01sgc 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 ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(19,{\mathbf{m}}-1)$ and ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(13,{\mathbf{m}}-1)$ 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).
10Example
This program reads in a set of $30$ data points and calls e01sgc to construct an interpolating function $Q(x,y)$. It then calls e01shc 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.