E01ZMF (PDF version)
E01 Chapter Contents
E01 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

E01ZMF generates a multi-dimensional 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 routines in Chapter E01 which are specific to the given dimensionality. E01SGF generates the two-dimensional interpolant, while E01TGF, E01TKF and E01TMF generate the three-, four- and five-dimensional interpolants respectively.

2  Specification

REAL (KIND=nag_wp)  X(D,M), F(M), RQ(*)

3  Description

E01ZMF 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 - xr 2 2 .
The basic method is global in that the interpolated value at any point depends on all the data, but E01ZMF 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 E01ZMF 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 8.2.
E01ZMF 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 E01ZMF, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to E01ZNF.

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  Parameters

1:     D – INTEGERInput
On entry: d, the number of dimensions.
Constraint: D2.
2:     M – INTEGERInput
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) – REAL (KIND=nag_wp) arrayInput
On entry: X1:Dr 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 d-1-dimensional hypersurface.
4:     F(M) – REAL (KIND=nag_wp) arrayInput
On entry: Fr must be set to the data value fr, for r=1,2,,m.
5:     NW – INTEGERInput
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 – INTEGERInput
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=minD+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 arrayOutput
On exit: integer data defining the interpolant Q x.
8:     RQ(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension 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:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

On entry, D+1×D+2/2×M+2×D+1 exceeds the largest machine integer.
D=value and M=value.
On entry, D=value.
Constraint: D2.
On entry, M=value and D=value.
Constraint: MD+1D+2/2+2.
On entry, NQ=value and D=value.
Constraint: NQ0 or NQD+1D+2/2-1.
On entry, NQ=value and M=value.
Constraint: NQM-1.
On entry, NW=value and M=value.
Constraint: NWM-1.
There are duplicate nodes in the dataset. Xik=Xjk, for i=value, j=value and k=1,2,,D. The interpolant cannot be derived.
On entry, all the data points lie on the same hypersurface. No unique solution exists.
Dynamic memory allocation failed.

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  Further Comments

8.1  Timing

The time taken for a call to E01ZMF 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 Om. At worst Om2 time will be required.

8.2  Choice of Nw and Nq

Default values of the parameters Nw and Nq may be selected by calling E01ZMF with NW0 and NQ0. These default values may well be satisfactory for many applications.
If nondefault values are required they must be supplied to E01ZMF through positive values of NW and NQ. Increasing these parameter 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 parameters see Renka (1988a) and Berry and Minser (1999).

9  Example

This program reads in a set of 30 data points and calls E01ZMF to construct an interpolating function Q x . It then calls E01ZNF 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 Section 9 in E01ZNF.

9.1  Program Text

Program Text (e01zmfe.f90)

9.2  Program Data

Program Data (e01zmfe.d)

9.3  Program Results

Program Results (e01zmfe.r)

E01ZMF (PDF version)
E01 Chapter Contents
E01 Chapter Introduction
NAG Library Manual

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