The function may be called by the names: e01tmc, nag_interp_dim5_scat_shep or nag_5d_shep_interp.
3Description
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 $({\mathbf{x}}_{r},{f}_{r})$, 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 method, which is a generalization of the two-dimensional 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 $({\mathbf{x}}_{r},{f}_{r})$. 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 user-supplied 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 five-dimensional 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.
4References
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. Software25 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. Software14 139–148
Renka R J (1988b) Algorithm 661: QSHEP3D: Quadratic Shepard method for trivariate interpolation of scattered data ACM Trans. Math. Software14 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
5Arguments
1: $\mathbf{m}$ – 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 ${\mathbf{m}}\ge 4000$.
Note: the $(i,j)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[(j-1)\times 5+i-1\right]$.
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 four-dimensional hypersurface.
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: $\mathbf{nw}$ – 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 ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(32,{\mathbf{m}}-1)$ is used instead.
On entry: the number ${N}_{q}$ of data points to be used in the least squares fit for coefficients defining the quadratic functions ${q}_{r}\left(\mathbf{x}\right)$ (see Section 3). If ${\mathbf{nq}}\le 0$ the default value ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(50,{\mathbf{m}}-1)$ is used instead.
Constraint:
${\mathbf{nq}}\le 0$ or $20\le {\mathbf{nq}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}(70,{\mathbf{m}}-1)$.
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).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_DATA_HYPERSURFACE
On entry, all the data points lie on the same four-dimensional hypersurface.
No unique solution exists.
NE_DUPLICATE_NODE
There are duplicate nodes in the dataset. ${\mathbf{x}}\left[\left(k-1\right)\times 5+i-1\right]={\mathbf{x}}\left[\left(k-1\right)\times 5+j-1\right]$, for $i=\u27e8\mathit{\text{value}}\u27e9$, $j=\u27e8\mathit{\text{value}}\u27e9$ and $k=1,2,\dots ,5$. The interpolant cannot be derived.
NE_INT
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge 23$.
On entry, ${\mathbf{nq}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nq}}\le 0$ or
${\mathbf{nq}}\ge 20$.
NE_INT_2
On entry, ${\mathbf{nq}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nq}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}(70,{\mathbf{m}}-1)$.
On entry, ${\mathbf{nw}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nw}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}(50,{\mathbf{m}}-1)$.
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 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.
8Parallelism 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 implementation-specific information.
9Further Comments
9.1Timing
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.
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 e01tmc 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 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 ${\mathbf{nw}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(32,{\mathbf{m}}-1)$ and ${\mathbf{nq}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(50,{\mathbf{m}}-1)$ 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).
10Example
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.