NAG Library Function Document
nag_nd_shep_interp (e01zmc)
1 Purpose
nag_nd_shep_interp (e01zmc) generates a multi-dimensional interpolant to a set of scattered data points, using a modified Shepard method. When the number of dimensions is no more than five, there are corresponding functions in
Chapter e01 which are specific to the given dimensionality.
nag_2d_shep_interp (e01sgc) generates the two-dimensional interpolant, while
nag_3d_shep_interp (e01tgc),
nag_4d_shep_interp (e01tkc) and
nag_5d_shep_interp (e01tmc) generate the three-, four- and five-dimensional interpolants respectively.
2 Specification
#include <nag.h> |
#include <nage01.h> |
void |
nag_nd_shep_interp (Integer d,
Integer m,
const double x[],
const double f[],
Integer nw,
Integer nq,
Integer iq[],
double rq[],
NagError *fail) |
|
3 Description
nag_nd_shep_interp (e01zmc) 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
,
.
The basic method is global in that the interpolated value at any point depends on all the data, but nag_nd_shep_interp (e01zmc) 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_nd_shep_interp (e01zmc) 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 parameters are provided, and advice on alternatives is given in
Section 9.2.
nag_nd_shep_interp (e01zmc) 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
generated by nag_nd_shep_interp (e01zmc), and its first partial derivatives, can subsequently be evaluated for points in the domain of the data by a call to
nag_nd_shep_eval (e01znc).
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:
d – IntegerInput
On entry: , the number of dimensions.
Constraint:
.
- 2:
m – IntegerInput
On entry:
, the number of data points.
Note: on the basis of experimental results reported in
Berry and Minser (1999), when
it is recommended to use
.
Constraint:
.
- 3:
x[] – const doubleInput
-
Note: the th element of the matrix is stored in .
On entry:
the
d components of the first data point must be stored in elements
of
x. The second data point must be stored in elements
of
x, and so on. In general, the
m data points must be stored in
, for
and
.
Constraint:
these coordinates must be distinct, and must not all lie on the same -dimensional hypersurface.
- 4:
f[m] – const doubleInput
On entry: must be set to the data value , for .
- 5:
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.
Suggested value:
Constraint:
.
- 6:
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.
Suggested value:
Constraint:
or .
- 7:
iq[] – IntegerOutput
On exit: integer data defining the interpolant .
- 8:
rq[] – doubleOutput
-
Note: the dimension,
dim, of the array
rq
must be at least
.
On exit: real data defining the interpolant .
- 9:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_DATA_HYPERSURFACE
-
On entry, all the data points lie on the same 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: .
- NE_INT_2
-
On entry, exceeds the largest machine integer.
and .
On entry, and .
Constraint: .
On entry, and .
Constraint: or .
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
In experiments undertaken by
Berry and Minser (1999), the accuracies obtained for a conditional function resulting in sharp functional transitions were of the order of
at best. In other cases in these experiments, the function generated interpolates the input data with maximum absolute error of the order of
.
8 Parallelism and Performance
nag_nd_shep_interp (e01zmc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_nd_shep_interp (e01zmc) 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_nd_shep_interp (e01zmc) 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 parameters and may be selected by calling nag_nd_shep_interp (e01zmc) with and . These default values may well be satisfactory for many applications.
If non-default values are required they must be supplied to nag_nd_shep_interp (e01zmc) 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
Renka (1988a) and
Berry and Minser (1999). For further advice on the choice of these arguments see
Renka (1988a) and
Berry and Minser (1999).
10 Example
This program reads in a set of
data points and calls nag_nd_shep_interp (e01zmc) to construct an interpolating function
. It then calls
nag_nd_shep_eval (e01znc) 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 very much larger.
See also
Section 10 in nag_nd_shep_eval (e01znc).
10.1 Program Text
Program Text (e01zmce.c)
10.2 Program Data
Program Data (e01zmce.d)
10.3 Program Results
Program Results (e01zmce.r)