NAG CL Interface
e01zmc (dimn_​scat_​shep)

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1 Purpose

e01zmc generates a multidimensional interpolant to a set of scattered data points, using a modified Shepard method. When the number of dimensions is no more than five, there are corresponding functions in Chapter E01 which are specific to the given dimensionality. e01sgc generates the two-dimensional interpolant, while e01tgc, e01tkc and e01tmc generate the three-, four- and five-dimensional interpolants respectively.

2 Specification

#include <nag.h>
void  e01zmc (Integer d, 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: e01zmc, nag_interp_dimn_scat_shep or nag_nd_shep_interp.

3 Description

e01zmc constructs a smooth function Q (x) , xd 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) = 1 x-xr2 2 .
The basic method is global in that the interpolated value at any point depends on all the data, but e01zmc 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 e01zmc 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 parameters are provided, and advice on alternatives is given in Section 9.2.
e01zmc is derived from the new implementation of QSHEP3 described by Renka (1988b). It uses the modification for high-dimensional interpolation described by Berry and Minser (1999).
Values of the interpolant Q (x) generated by e01zmc, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to e01znc.

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: d Integer Input
On entry: d, the number of dimensions.
Constraint: d2.
2: m Integer Input
On entry: m, the number of data points.
Note: on the basis of experimental results reported in Berry and Minser (1999), when d5 it is recommended to use m4000.
Constraint: m (d+1)× (d+2)/ 2+2.
3: x[d×m] const double Input
Note: the (i,j)th element of the matrix X is stored in x[(j-1)×d+i-1].
On entry: the d components of the first data point must be stored in elements 0,1,,d-1 of x. The second data point must be stored in elements d,d+1,,2×d-1 of x, and so on. In general, the m data points must be stored in x[i×d+j], for i=0,1,,m-1 and j=0,1,,d-1.
Constraint: these coordinates must be distinct, and must not all lie on the same (d-1)-dimensional hypersurface.
4: f[m] const double Input
On entry: f[r-1] must be set to the data value fr, for r=1,2,,m.
5: 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( 2× (d+1)× (d+2) ,m-1) is used instead.
Suggested value: nw=−1.
Constraint: nwm-1.
6: 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((d+1)×(d+2)×6/5,m-1) is used instead.
Suggested value: nq=−1.
Constraint: nq0 or (d+1)×(d+2)/2-1nqm-1.
7: iq[2×m+1] Integer Output
On exit: integer data defining the interpolant Q(x).
8: rq[dim] double Output
Note: the dimension, dim, of the array rq must be at least ((d+1)×(d+2)/2)×m+2×d+1.
On exit: real data defining the interpolant Q(x).
9: 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 hypersurface. No unique solution exists.
There are duplicate nodes in the dataset. x[(i-1)×d+k-1] = x[(j-1)×d+k-1] , for i=value, j=value and k=1,2,,d. The interpolant cannot be derived.
On entry, d=value.
Constraint: d2.
On entry, ((d+1)×(d+2)/2)×m+2×d+1 exceeds the largest machine integer.
d=value and m=value.
On entry, m=value and d=value.
Constraint: m (d+1)× (d+2)/ 2+2.
On entry, nq=value and d=value.
Constraint: nq0 or nq(d+1)×(d+2)/2-1.
On entry, nq=value and m=value.
Constraint: nqm-1.
On entry, nw=value and m=value.
Constraint: nwm-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

In experiments undertaken by Berry and Minser (1999), the accuracies obtained for a conditional function resulting in sharp functional transitions were of the order of 10−1 at best. In other cases in these experiments, the function generated interpolates the input data with maximum absolute error of the order of 10−2.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
e01zmc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e01zmc 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 e01zmc 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 parameters Nw and Nq may be selected by calling e01zmc 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 e01zmc 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( 2× (d+1)× (d+2) ,m-1) and nq = min( (d+1)× (d+2)× 6/5,m-1) have been chosen on the basis of experimental results reported in Renka (1988a) and Berry and Minser (1999). For further advice on the choice of these arguments see Renka (1988a) and Berry and Minser (1999).

10 Example

This program reads in a set of 30 data points and calls e01zmc to construct an interpolating function Q (x) . It then calls e01znc 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 very much larger.
See also e01znc.

10.1 Program Text

Program Text (e01zmce.c)

10.2 Program Data

Program Data (e01zmce.d)

10.3 Program Results

Program Results (e01zmce.r)