NAG CL Interface
e01sjc (dim2_triang_interp)
1
Purpose
e01sjc generates a two-dimensional surface interpolating a set of scattered data points, using the method of Renka and Cline.
2
Specification
| void |
e01sjc (Integer m,
const double x[],
const double y[],
const double f[],
Integer triang[],
double grads[],
NagError *fail) |
|
The function may be called by the names: e01sjc, nag_interp_dim2_triang_interp or nag_2d_triang_interp.
3
Description
e01sjc constructs an interpolating surface
through a set of scattered data points , for , using a method due to Renka and Cline. In the plane, the data points must be distinct. The constructed surface is continuous and has continuous first derivatives.
The method involves firstly creating a triangulation with all the
data points as nodes, the triangulation being as nearly equiangular as possible (see
Cline and Renka (1984)). Then gradients in the
- and
-directions are estimated at node
, for
,
as the partial derivatives of a quadratic function of
and
which interpolates the data value
,
and which fits the data values at nearby nodes (those within a certain distance chosen by the algorithm) in a weighted least squares sense. The weights are chosen such that closer nodes have more influence than more distant nodes on derivative estimates at node
. The computed partial derivatives, with the
values, at the three nodes of each triangle define a piecewise polynomial surface of a certain form which is the interpolant on that triangle. See
Renka and Cline (1984) for more detailed information on the algorithm,
a development of that by
Lawson (1977). The code is derived from
Renka (1984).
The interpolant
can subsequently be evaluated at any point
inside or outside the domain of the data by a call to
e01skc.
Points outside the domain are evaluated by extrapolation.
4
References
Cline A K and Renka R L (1984) A storage-efficient method for construction of a Thiessen triangulation Rocky Mountain J. Math. 14 119–139
Lawson C L (1977) Software for surface interpolation Mathematical Software III (ed J R Rice) 161–194 Academic Press
Renka R L (1984) Algorithm 624: triangulation and interpolation of arbitrarily distributed points in the plane ACM Trans. Math. Software 10 440–442
Renka R L and Cline A K (1984) A triangle-based interpolation method Rocky Mountain J. Math. 14 223–237
5
Arguments
-
1:
– Integer
Input
-
On entry: , the number of data points.
Constraint:
.
-
2:
– const double
Input
-
3:
– const double
Input
-
4:
– const double
Input
-
On entry: the coordinates of the
th data point, for
. The data points are accepted in any order, but see
Section 9.
Constraint:
the nodes must not all be collinear, and each node must be unique.
-
5:
– Integer
Output
-
On exit: a data structure defining the computed triangulation, in a form suitable for passing to
e01skc.
-
6:
– double
Output
-
Note: the th element of the matrix is stored in .
On exit: the estimated partial derivatives at the nodes, in a form suitable for passing to
e01skc. The derivatives at node
with respect to
and
are contained in
and
respectively, for
.
-
7:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
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 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_DATA_NOT_UNIQUE
-
On entry, , for , , .
- NE_INT
-
On entry, .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
On successful exit, the computational errors should be negligible in most situations but you should always check the computed surface for acceptability, by drawing contours for instance. The surface always interpolates the input data exactly.
8
Parallelism and Performance
e01sjc is not threaded in any implementation.
The time taken for a call of
e01sjc is approximately proportional to the number of data points,
. The function is more efficient if, before entry, the values in
x,
y and
f are arranged so that the
x array is in ascending order.
10
Example
This example reads in a set of
data points and calls
e01sjc
to construct an interpolating surface. It then calls
e01skc
to evaluate the interpolant at a sample of points on a rectangular grid.
Note that this example is not typical of a realistic problem: the number of data points would normally be larger, and the interpolant would need to be evaluated on a finer grid to obtain an accurate plot, say.
10.1
Program Text
10.2
Program Data
10.3
Program Results