nag_2d_shep_interp (e01sgc) (PDF version)
e01 Chapter Contents
e01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_2d_shep_interp (e01sgc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_2d_shep_interp (e01sgc) generates a two-dimensional interpolant to a set of scattered data points, using a modified Shepard method.

2  Specification

#include <nag.h>
#include <nage01.h>
void  nag_2d_shep_interp (Integer m, const double x[], const double y[], const double f[], Integer nw, Integer nq, Integer iq[], double rq[], NagError *fail)

3  Description

nag_2d_shep_interp (e01sgc) constructs a smooth function Qx,y which interpolates a set of m scattered data points xr,yr,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 (1968) method interpolates the input data with the weighted mean
Qx,y=r=1mwrx,yqr r=1mwrx,y ,
where qr = fr , wr x,y = 1dr2  and dr2 = x-xr 2 + y-yr 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 wrx,y to be zero outside a circle with centre xr,yr and some radius Rw. Also, to improve the performance of the basic method, each qr above is replaced by a function qrx,y, which is a quadratic fitted by weighted least squares to data local to xr,yr and forced to interpolate xr,yr,fr. In this context, a point x,y 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 8.2.
This function is derived from the function QSHEP2 described by Renka (1988b).
Values of the interpolant Qx,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 nag_2d_shep_eval (e01shc).

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 660: QSHEP2D: Quadratic Shepard method for bivariate interpolation of scattered data ACM Trans. Math. Software 14 149–150
Shepard D (1968) A two-dimensional interpolation function for irregularly spaced data Proc. 23rd Nat. Conf. ACM 517–523 Brandon/Systems Press Inc., Princeton

5  Arguments

1:     mIntegerInput
On entry: m, the number of data points.
Constraint: m6.
2:     x[m]const doubleInput
3:     y[m]const doubleInput
On entry: the Cartesian coordinates of the data points xr,yr, for r=1,2,,m.
Constraint: these coordinates must be distinct, and must not all be collinear.
4:     f[m]const doubleInput
On entry: f[r-1] must be set to the data value fr, for r=1,2,,m.
5:     nwIntegerInput
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=min19,m-1 is used instead.
Constraint: nwmin40,m-1.
6:     nqIntegerInput
On entry: the number Nq of data points to be used in the least squares fit for coefficients defining the nodal functions qrx,y (see Section 3). If nq0 the default value nq=min13,m-1 is used instead.
Constraint: nq0 or 5nqmin40,m-1.
7:     iq[2×m+1]IntegerOutput
On exit: integer data defining the interpolant Qx,y.
8:     rq[6×m+5]doubleOutput
On exit: real data defining the interpolant Qx,y.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALL_DATA_COLLINEAR
All nodes are collinear. There is no unique solution.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_DATA_NOT_UNIQUE
There are duplicate nodes in the dataset. x[i-1],y[i-1]=x[j-1],y[j-1], for i=value and j=value. The interpolant cannot be derived.
NE_INT
On entry, m=value.
Constraint: m6.
On entry, nq=value.
Constraint: nq0 or nq5.
NE_INT_2
On entry, nq=value and m=value.
Constraint: nqmin40,m-1.
On entry, nw=value and m=value.
Constraint: nwmin40,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.

7  Accuracy

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

8  Further Comments

8.1  Timing

The time taken for a call to nag_2d_shep_interp (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 Om. At worst Om2 time will be required.

8.2  Choice of Nw and Nq

Default values of the arguments Nw and Nq may be selected by calling nag_2d_shep_interp (e01sgc) with nw0 and nq0. These default values may well be satisfactory for many applications.
If nondefault values are required they must be supplied to nag_2d_shep_interp (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 nw = min19,m-1  and nq = min13,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).

9  Example

This program reads in a set of 30 data points and calls nag_2d_shep_interp (e01sgc) to construct an interpolating function Qx,y. It then calls nag_2d_shep_eval (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.

9.1  Program Text

Program Text (e01sgce.c)

9.2  Program Data

Program Data (e01sgce.d)

9.3  Program Results

Program Results (e01sgce.r)


nag_2d_shep_interp (e01sgc) (PDF version)
e01 Chapter Contents
e01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012