NAG CL Interface
e01tgc (dim3_​scat_​shep)

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1 Purpose

e01tgc generates a three-dimensional interpolant to a set of scattered data points, using a modified Shepard method.

2 Specification

#include <nag.h>
void  e01tgc (Integer m, const double x[], const double y[], const double z[], const double f[], Integer nw, Integer nq, Integer iq[], double rq[], NagError *fail)
The function may be called by the names: e01tgc, nag_interp_dim3_scat_shep or nag_3d_shep_interp.

3 Description

e01tgc constructs a smooth function Q(x,y,z) which interpolates a set of m scattered data points (xr,yr,zr,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,y,z)=r=1mwr(x,y,z)qr r=1mwr(x,y,z) ,  
where
qr=fr ​ and ​ wr(x,y,z)= 1dr2 ​ and ​ dr2= (x-xr) 2+ (y-yr) 2+ (z-zr) 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 wr(x,y,z) to be zero outside a sphere with centre (xr,yr,zr) and some radius Rw. Also, to improve the performance of the basic method, each qr above is replaced by a function qr(x,y,z), which is a quadratic fitted by weighted least squares to data local to (xr,yr,zr) and forced to interpolate (xr,yr,zr,fr). In this context, a point (x,y,z) is defined to be local to another point if it lies within some distance Rq 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 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 by the function, and advice on alternatives is given in Section 9.2.
This function is derived from the function QSHEP3 described by Renka (1988b).
Values of the interpolant Q(x,y,z) 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 e01thc.

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 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.
Constraint: m10.
2: x[m] const double Input
3: y[m] const double Input
4: z[m] const double Input
On entry: x[r-1], y[r-1], z[r-1] must be set to the Cartesian coordinates of the data point (xr,yr,zr), for r=1,2,,m.
Constraint: these coordinates must be distinct, and must not all be coplanar.
5: f[m] const double Input
On entry: f[r-1] must be set to the data value fr, for r=1,2,,m.
6: 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(40,m-1).
7: nq Integer Input
On entry: the number Nq of data points to be used in the least squares fit for coefficients defining the nodal functions qr(x,y,z) (see Section 3). If nq0 the default value nq=min(17,m-1) is used instead.
Constraint: nq0 or 9nqmin(40,m-1).
8: iq[dim] Integer Output
On exit: integer data defining the interpolant Q(x,y,z).
9: rq[dim] double Output
On exit: real data defining the interpolant Q(x,y,z).
10: 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

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 value had an illegal value.
NE_DATA_COPLANAR
All nodes are coplanar. There is no unique solution.
NE_DUPLICATE_NODE
There are duplicate nodes in the dataset. (x[I-1],y[I-1],z[I-1])=(x[J-1],y[J-1],z[J-1]) for: I=value and J=value. The interpolant cannot be derived.
NE_INT
On entry, m=value.
Constraint: m10.
On entry, nq=value.
Constraint: nq0 or nq9.
NE_INT_2
On entry, nq=value and m=value.
Constraint: nqmin(40,m-1).
On entry, nw=value and m=value.
Constraint: nwmin(40,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.

7 Accuracy

On successful exit, the function generated interpolates the input data exactly and has quadratic accuracy.

8 Parallelism and Performance

e01tgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e01tgc 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 e01tgc will depend in general on the distribution of the data points. If x, y and z 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 e01tgc 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 e01tgc 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 nw = min(32,m-1) and nq = min(17,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 Nw and Nq, 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 e01tgc to construct an interpolating function Q(x,y,z). It then calls e01thc 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 (e01tgce.c)

10.2 Program Data

Program Data (e01tgce.d)

10.3 Program Results

Program Results (e01tgce.r)