NAG FL Interface
e01sgf (dim2_​scat_​shep)

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1 Purpose

e01sgf generates a two-dimensional interpolant to a set of scattered data points, using a modified Shepard method.

2 Specification

Fortran Interface
Subroutine e01sgf ( m, x, y, f, nw, nq, iq, liq, rq, lrq, ifail)
Integer, Intent (In) :: m, nw, nq, liq, lrq
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: iq(liq)
Real (Kind=nag_wp), Intent (In) :: x(m), y(m), f(m)
Real (Kind=nag_wp), Intent (Out) :: rq(lrq)
C Header Interface
#include <nag.h>
void  e01sgf_ (const Integer *m, const double x[], const double y[], const double f[], const Integer *nw, const Integer *nq, Integer iq[], const Integer *liq, double rq[], const Integer *lrq, Integer *ifail)
The routine may be called by the names e01sgf or nagf_interp_dim2_scat_shep.

3 Description

e01sgf constructs a smooth function Q(x,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
Q(x,y)=r=1mwr(x,y)qr r=1mwr(x,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 routine uses a modification (see Franke and Nielson (1980) and Renka (1988a)), whereby the method becomes local by adjusting each wr(x,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 qr(x,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 routine.
The efficiency of the routine 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 routine, and advice on alternatives is given in Section 9.2.
This routine is derived from the routine QSHEP2 described by Renka (1988b).
Values of the interpolant Q(x,y) generated by this routine, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to e01shf.

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: m Integer Input
On entry: m, the number of data points.
Constraint: m6.
2: x(m) Real (Kind=nag_wp) array Input
3: y(m) Real (Kind=nag_wp) array Input
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) Real (Kind=nag_wp) array Input
On entry: f(r) 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(19,m-1) is used instead.
Constraint: nwmin(40,m-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 nodal functions qr(x,y) (see Section 3). If nq0 the default value nq=min(13,m-1) is used instead.
Constraint: nq0 or 5nqmin(40,m-1).
7: iq(liq) Integer array Output
On exit: integer data defining the interpolant Q(x,y).
8: liq Integer Input
On entry: the dimension of the array iq as declared in the (sub)program from which e01sgf is called.
Constraint: liq2×m+1.
9: rq(lrq) Real (Kind=nag_wp) array Output
On exit: real data defining the interpolant Q(x,y).
10: lrq Integer Input
On entry: the dimension of the array rq as declared in the (sub)program from which e01sgf is called.
Constraint: lrq6×m+5.
11: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, liq is too small: liq=value.
On entry, lrq is too small: lrq=value.
On entry, m=value.
Constraint: m6.
On entry, nq=value.
Constraint: nq0 or nq5.
On entry, nq=value and m=value.
Constraint: nqmin(40,m-1).
On entry, nw=value and m=value.
Constraint: nwmin(40,m-1).
There are duplicate nodes in the dataset. (x(I),y(I))=(x(J),y(J)), for I=value and J=value. The interpolant cannot be derived.
All nodes are collinear. There is no unique solution.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL 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

Background information to multithreading can be found in the Multithreading documentation.
e01sgf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e01sgf 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 routine. 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 e01sgf 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 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 e01sgf 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 e01sgf 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(19,m-1) and nq = min(13,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).

9.3 Internal Changes

Internal changes have been made to this routine as follows:
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

10 Example

This program reads in a set of 30 data points and calls e01sgf to construct an interpolating function Q(x,y). It then calls e01shf 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 (e01sgfe.f90)

10.2 Program Data

Program Data (e01sgfe.d)

10.3 Program Results

Program Results (e01sgfe.r)