NAG Library Routine Document
E01TKF
1 Purpose
E01TKF generates a four-dimensional interpolant to a set of scattered data points, using a modified Shepard method.
2 Specification
INTEGER |
M, NW, NQ, IQ(2*M+1), IFAIL |
REAL (KIND=nag_wp) |
X(4,M), F(M), RQ(15*M+9) |
|
3 Description
E01TKF constructs a smooth function
Q
x
, x∈ℝ4 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
where
qr
=
fr
,
wr
x
=
1dr2
and
dr2
=
x-xr2
2
.
The basic method is global in that the interpolated value at any point depends on all the data, but E01TKF 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 E01TKF 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 by the routine, and advice on alternatives is given in
Section 8.2.
E01TKF 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 E01TKF, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to
E01TLF.
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: M – INTEGERInput
On entry: m, the number of data points.
Constraint:
M≥16.
- 2: X(4,M) – REAL (KIND=nag_wp) arrayInput
On entry: X1:4r 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 three-dimensional hypersurface.
- 3: F(M) – REAL (KIND=nag_wp) arrayInput
On entry: Fr must be set to the data value fr, for r=1,2,…,m.
- 4: 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=min32,M-1 is used instead.
Constraint:
NW≤min50,M-1.
- 5: 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
NQ≤0 the default value
NQ=min38,M-1 is used instead.
Constraint:
NQ≤0 or 14≤NQ≤min50,M-1.
- 6: IQ(2×M+1) – INTEGER arrayOutput
On exit: integer data defining the interpolant Q x .
- 7: RQ(15×M+9) – REAL (KIND=nag_wp) arrayOutput
On exit: real data defining the interpolant Q x .
- 8: 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
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<16, |
or | 0<NQ<14, |
or | NQ > min50,M-1 , |
or | NW > min50,M-1 . |
- IFAIL=2
On entry, X1:4i = X1:4j for some i≠j. The interpolant cannot be derived.
- IFAIL=3
On entry, all the data points lie on the same three-dimensional hypersurface. No unique solution exists.
7 Accuracy
On successful exit, the routine generated interpolates the input data exactly and has quadratic precision. Overall accuracy of the interpolant is affected by the choice of parameters
NW and
NQ as well as the smoothness of the function represented by the input data.
8 Further Comments
8.1 Timing
The time taken for a call to E01TKF 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 E01TKF 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 E01TKF 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.
9 Example
This program reads in a set of
30 data points and calls E01TKF to construct an interpolating function
Q
x
. It then calls
E01TLF 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.
See also
Section 9 in E01TLF.
9.1 Program Text
Program Text (e01tkfe.f90)
9.2 Program Data
Program Data (e01tkfe.d)
9.3 Program Results
Program Results (e01tkfe.r)