NAG Library Function Document
nag_2d_shep_interp (e01sgc)
1
Purpose
nag_2d_shep_interp (e01sgc) generates a two-dimensional interpolant to a set of scattered data points, using a modified Shepard method.
2
Specification
#include <nag.h> |
#include <nage01.h> |
void |
nag_2d_shep_interp (Integer m,
const double x[],
const double y[],
const double f[],
Integer nw,
Integer nq,
Integer iq[],
double rq[],
NagError *fail) |
|
3
Description
nag_2d_shep_interp (e01sgc) 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 (1968) method 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 this function uses a modification (see
Franke and Nielson (1980) and
Renka (1988a)), whereby the method becomes local by adjusting each
to be zero outside a circle 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. Computation of these quadratics constitutes the main work done by this function.
The efficiency of the function is further enhanced by using a cell method for nearest neighbour searching due to
Bentley and Friedman (1979).
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 by the function, and advice on alternatives is given in
Section 9.2.
This function is derived from the function QSHEP2 described by
Renka (1988b).
Values of the interpolant
generated by this function, and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to
nag_2d_shep_eval (e01shc).
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:
– IntegerInput
-
On entry: , the number of data points.
Constraint:
.
- 2:
– const doubleInput
- 3:
– const doubleInput
-
On entry: the Cartesian coordinates of the data points
, for .
Constraint:
these coordinates must be distinct, and must not all be collinear.
- 4:
– const doubleInput
-
On entry: must be set to the data value , for .
- 5:
– 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:
.
- 6:
– IntegerInput
-
On entry: the number
of data points to be used in the least squares fit for coefficients defining the nodal functions
(see
Section 3). If
the default value
is used instead.
Constraint:
or .
- 7:
– IntegerOutput
-
On exit: integer data defining the interpolant .
- 8:
– doubleOutput
-
On exit: real data defining the interpolant .
- 9:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALL_DATA_COLLINEAR
-
All nodes are collinear. There is no unique solution.
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_DATA_NOT_UNIQUE
-
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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
On successful exit, the function generated interpolates the input data exactly and has quadratic accuracy.
8
Parallelism and Performance
nag_2d_shep_interp (e01sgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_2d_shep_interp (e01sgc) 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 function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The time taken for a call to
nag_2d_shep_interp (e01sgc) 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
. At worst
time will be required.
Default values of the arguments and may be selected by calling nag_2d_shep_interp (e01sgc) with and . These default values may well be satisfactory for many applications.
If non-default values are required they must be supplied to
nag_2d_shep_interp (e01sgc) 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
and
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
and
, 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).
10
Example
This program reads in a set of
data points and calls
nag_2d_shep_interp (e01sgc) to construct an interpolating function
. It then calls
nag_2d_shep_eval (e01shc) 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 (e01sgce.c)
10.2
Program Data
Program Data (e01sgce.d)
10.3
Program Results
Program Results (e01sgce.r)