NAG CL Interface
e01sjc (dim2_​triang_​interp)

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1 Purpose

e01sjc generates a two-dimensional surface interpolating a set of scattered data points, using the method of Renka and Cline.

2 Specification

#include <nag.h>
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 F(x,y) through a set of m scattered data points (xr,yr,fr), for r=1,2,,m, using a method due to Renka and Cline. In the (x,y) 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 (x,y) data points as nodes, the triangulation being as nearly equiangular as possible (see Cline and Renka (1984)). Then gradients in the x- and y-directions are estimated at node r, for r=1,2,,m, as the partial derivatives of a quadratic function of x and y which interpolates the data value fr, 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 r. The computed partial derivatives, with the fr 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 F(x,y) can subsequently be evaluated at any point (x,y) 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 C1 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 C1 interpolation method Rocky Mountain J. Math. 14 223–237

5 Arguments

1: m Integer Input
On entry: m, the number of data points.
Constraint: m3.
2: x[m] const double Input
3: y[m] const double Input
4: f[m] const double Input
On entry: the coordinates of the rth data point, for r=1,2,,m. The data points are accepted in any order, but see Section 9.
Constraint: the (x,y) nodes must not all be collinear, and each node must be unique.
5: triang[7×m] Integer Output
On exit: a data structure defining the computed triangulation, in a form suitable for passing to e01skc.
6: grads[2×m] double Output
Note: the (i,j)th element of the matrix is stored in grads[(j-1)×2+i-1].
On exit: the estimated partial derivatives at the nodes, in a form suitable for passing to e01skc. The derivatives at node r with respect to x and y are contained in grads[(r-1)×2] and grads[(r-1)×2+1] respectively, for r=1,2,,m.
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

All nodes are collinear. There is no unique solution.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, (x[I-1],y[I-1])=(x[J-1],y[J-1]), for I,J=valuevalue, x[I-1], y[I-1]=valuevalue.
On entry, m=value.
Constraint: m3.
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.
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.

9 Further Comments

The time taken for a call of e01sjc is approximately proportional to the number of data points, m. 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 30 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

Program Text (e01sjce.c)

10.2 Program Data

Program Data (e01sjce.d)

10.3 Program Results

Program Results (e01sjce.r)