NAG Library Routine Document
E01SGF
1 Purpose
E01SGF generates a two-dimensional interpolant to a set of scattered data points, using a modified Shepard method.
2 Specification
| SUBROUTINE E01SGF ( | M, X, Y, F, NW, NQ, IQ, LIQ, RQ, LRQ, IFAIL) |
| INTEGER | M, NW, NQ, IQ(LIQ), LIQ, LRQ, IFAIL |
| double precision | X(M), Y(M), F(M), RQ(LRQ) |
3 Description
E01SGF 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
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
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 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 parameters are provided by the routine, and advice on alternatives is given in
Section 8.2.
This routine is derived from the routine QSHEP2 described by
Renka (1988b).
Values of the interpolant
Qx,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 Parameters
- 1: M – INTEGERInput
On entry:
m, the number of data points.
Constraint:
M≥6.
- 2: X(M) – double precision arrayInput
- 3: Y(M) – double precision arrayInput
On entry: the Cartesian co-ordinates of the data points xr,yr, for r=1,2,…,m.
Constraint:
these co-ordinates must be distinct, and must not all be collinear.
- 4: F(M) – double precision 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
NW≤0 the default value
NW=min19,M-1 is used instead.
Constraint:
NW≤min40,M-1.
- 6: NQ – INTEGERInput
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
NQ≤0 the default value
NQ=min13,M-1 is used instead.
Constraint:
NQ≤0 or 5≤NQ≤min40,M-1.
- 7: IQ(LIQ) – INTEGER arrayOutput
On exit: integer data defining the interpolant Qx,y.
- 8: LIQ – INTEGERInput
On entry: the dimension of the array
IQ as declared in the (sub)program from which E01SGF is called.
Constraint:
LIQ≥2×M+1.
- 9: RQ(LRQ) – double precision arrayOutput
On exit: real data defining the interpolant Qx,y.
- 10: LRQ – INTEGERInput
On entry: the dimension of the array
RQ as declared in the (sub)program from which E01SGF is called.
Constraint:
LRQ≥6×M+5.
- 11: 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.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
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.
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:
- IFAIL=1
-
| On entry, | M<6, |
| or | 0<NQ<5, |
| or |
NQ
>
min40,M-1
, |
| or |
NW
>
min40,M-1
, |
| or | LIQ<2×M+1, |
| or | LRQ<6×M+5. |
- IFAIL=2
-
| On entry, | Xi,Yi=Xj,Yj for some i≠j. |
- IFAIL=3
-
| On entry, | all the data points are collinear. No unique solution exists. |
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 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
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 E01SGF with NW≤0 and NQ≤0. These default values may well be satisfactory for many applications.
If nondefault values are required they must be supplied to E01SGF through positive values of
NW and
NQ. Increasing these parameters 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 parameter values. For further advice on the choice of these parameters see
Renka (1988a).
9 Example
This program reads in a set of
30 data points and calls E01SGF to construct an interpolating function
Qx,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.
9.1 Program Text
Program Text (e01sgfe.f)
9.2 Program Data
Program Data (e01sgfe.d)
9.3 Program Results
Program Results (e01sgfe.r)