NAG Library Function Document
nag_5d_shep_interp (e01tmc)
1 Purpose
nag_5d_shep_interp (e01tmc) generates a five-dimensional interpolant to a set of scattered data points, using a modified Shepard method.
2 Specification
#include <nag.h> |
#include <nage01.h> |
void |
nag_5d_shep_interp (Integer m,
const double x[],
const double f[],
Integer nw,
Integer nq,
Integer iq[],
double rq[],
NagError *fail) |
|
3 Description
nag_5d_shep_interp (e01tmc) constructs a smooth function , which interpolates a set of scattered data points , for , 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
,
and
.
The basic method is global in that the interpolated value at any point depends on all the data, but nag_5d_shep_interp (e01tmc) uses a modification (see
Franke and Nielson (1980) and
Renka (1988a)), whereby the method becomes local by adjusting each
to be zero outside a hypersphere with centre
and some radius
. Also, to improve the performance of the basic method, each
above is replaced by a function
, which is a quadratic fitted by weighted least squares to data local to
and forced to interpolate
. In this context, a point
is defined to be local to another point if it lies within some distance
of it.
The efficiency of nag_5d_shep_interp (e01tmc) is enhanced by using a cell method for nearest neighbour searching due to
Bentley and Friedman (1979) with a cell density of
.
The radii
and
are chosen to be just large enough to include
and
data points, respectively, for user-supplied constants
and
. Default values of these arguments are provided, and advice on alternatives is given in
Section 9.2.
nag_5d_shep_interp (e01tmc) is derived from the new implementation of QSHEP3 described by
Renka (1988b). It uses the modification for five-dimensional interpolation described by
Berry and Minser (1999).
Values of the interpolant
generated by nag_5d_shep_interp (e01tmc), and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to
nag_5d_shep_eval (e01tnc).
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 Arguments
- 1:
m – IntegerInput
On entry:
, the number of data points.
Note: on the basis of experimental results reported in
Berry and Minser (1999), it is recommended to use
.
Constraint:
.
- 2:
x[] – const doubleInput
-
Note: the th element of the matrix is stored in .
On entry: must be set to the Cartesian coordinates of the data point , for .
Constraint:
these coordinates must be distinct, and must not all lie on the same four-dimensional hypersurface.
- 3:
f[m] – const doubleInput
On entry: must be set to the data value , for .
- 4:
nw – IntegerInput
On entry: the number
of data points that determines each radius of influence
, appearing in the definition of each of the weights
, for
(see
Section 3). Note that
is different for each weight. If
the default value
is used instead.
Constraint:
.
- 5:
nq – IntegerInput
On entry: the number
of data points to be used in the least squares fit for coefficients defining the quadratic functions
(see
Section 3). If
the default value
is used instead.
Constraint:
or .
- 6:
iq[] – IntegerOutput
On exit: integer data defining the interpolant .
- 7:
rq[] – doubleOutput
On exit: real data defining the interpolant .
- 8:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_DATA_HYPERSURFACE
-
On entry, all the data points lie on the same four-dimensional hypersurface.
No unique solution exists.
- NE_DUPLICATE_NODE
-
There are duplicate nodes in the dataset. , for , and . The interpolant cannot be derived.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: or
.
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- 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 precision. Overall accuracy of the interpolant is affected by the choice of arguments
nw and
nq as well as the smoothness of the function represented by the input data.
Berry and Minser (1999) report on the results obtained for a set of test functions.
8 Parallelism and Performance
nag_5d_shep_interp (e01tmc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_5d_shep_interp (e01tmc) 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
Users' Note for your implementation for any additional implementation-specific information.
The time taken for a call to nag_5d_shep_interp (e01tmc) will depend in general on the distribution of the data points and on the choice of and parameters. If the data points are uniformly randomly distributed, then the time taken should be . At worst time will be required.
Default values of the arguments and may be selected by calling nag_5d_shep_interp (e01tmc) with and . These default values may well be satisfactory for many applications.
If non-default values are required they must be supplied to nag_5d_shep_interp (e01tmc) through positive values of
nw and
nq. Increasing these argument 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
and
have been chosen on the basis of experimental results reported in
Berry and Minser (1999). In these experiments the error norm was found to increase with the decrease of
, but to be little affected by the choice of
. The choice of both, directly affected the time taken by the function. For further advice on the choice of these arguments see
Berry and Minser (1999).
10 Example
This program reads in a set of
data points and calls nag_5d_shep_interp (e01tmc) to construct an interpolating function
. It then calls
nag_5d_shep_eval (e01tnc) 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 10 in nag_5d_shep_eval (e01tnc).
10.1 Program Text
Program Text (e01tmce.c)
10.2 Program Data
Program Data (e01tmce.d)
10.3 Program Results
Program Results (e01tmce.r)