NAG CL Interface
e01tmc (dim5_​scat_​shep)

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1 Purpose

e01tmc generates a five-dimensional interpolant to a set of scattered data points, using a modified Shepard method.

2 Specification

#include <nag.h>
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 (x) , x5 which interpolates a set of m scattered data points (xr,fr) , for r=1,2,,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
Q (x) = r=1 m wr (x) qr r=1 m wr (x) ,  
where qr = fr , wr (x) = 1dr2 and dr2 = x-xr2 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 wr (x) to be zero outside a hypersphere with centre xr and some radius Rw. Also, to improve the performance of the basic method, each qr above is replaced by a function qr (x) , which is a quadratic fitted by weighted least squares to data local to xr and forced to interpolate (xr,fr) . In this context, a point x is defined to be local to another point if it lies within some distance Rq 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 Rw and Rq are chosen to be just large enough to include Nw and Nq data points, respectively, for user-supplied constants Nw and Nq. 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 (x) 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: high-dimensional 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 two-dimensional interpolation function for irregularly spaced data Proc. 23rd Nat. Conf. ACM 517–523 Brandon/Systems Press Inc., Princeton

5 Arguments

1: 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 m4000.
Constraint: m23.
2: x[5×m] const double Input
Note: the (i,j)th element of the matrix X is stored in x[(j-1)×5+i-1].
On entry: x[(r-1)×5] ,, x[(r-1)×5+4] must be set to the Cartesian coordinates of the data point xr, for r=1,2,,m.
Constraint: these coordinates must be distinct, and must not all lie on the same four-dimensional hypersurface.
3: f[m] const double Input
On entry: f[r-1] must be set to the data value fr, for r=1,2,,m.
4: nw Integer Input
On entry: the number Nw of data points that determines each radius of influence Rw, appearing in the definition of each of the weights wr, for r=1,2,,m (see Section 3). Note that Rw is different for each weight. If nw0 the default value nw=min(32,m-1) is used instead.
Constraint: nwmin(50,m-1).
5: nq Integer Input
On entry: the number Nq of data points to be used in the least squares fit for coefficients defining the quadratic functions qr (x) (see Section 3). If nq0 the default value nq=min(50,m-1) is used instead.
Constraint: nq0 or 20nqmin(70,m-1).
6: iq[2×m+1] Integer Output
On exit: integer data defining the interpolant Q(x).
7: rq[21×m+11] double Output
On exit: real data defining the interpolant Q(x).
8: 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

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, all the data points lie on the same four-dimensional hypersurface. No unique solution exists.
There are duplicate nodes in the dataset. x[(k-1)×5+i-1]=x[(k-1)×5+j-1], for i=value, j=value and k=1,2,,5. The interpolant cannot be derived.
On entry, m=value.
Constraint: m23.
On entry, nq=value.
Constraint: nq0 or nq20.
On entry, nq=value and m=value.
Constraint: nqmin(70,m-1).
On entry, nw=value and m=value.
Constraint: nwmin(50,m-1).
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.
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

Background information to multithreading can be found in the Multithreading documentation.
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.

9 Further Comments

9.1 Timing

The time taken for a call to e01tmc will depend in general on the distribution of the data points and on the choice of Nw and Nq parameters. If the data points are uniformly randomly distributed, then the time taken should be O(m). At worst O(m2) time will be required.

9.2 Choice of Nw and Nq

Default values of the arguments Nw and Nq may be selected by calling e01tmc with nw0 and nq0. 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 nw = min(32,m-1) and nq = min(50,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 Nq, but to be little affected by the choice of Nw. 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 (x) . 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 e01tnc.

10.1 Program Text

Program Text (e01tmce.c)

10.2 Program Data

Program Data (e01tmce.d)

10.3 Program Results

Program Results (e01tmce.r)