, 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

f_{r}

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

C^{1}

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

C^{1}

interpolation method Rocky Mountain J. Math. 14 223–237

5 Arguments

1: $m$ – Integer Input: On entry: $m$ , the number of data points.

Constraint: $m \geq 3$ .
2: $x [m]$ – const double Input
3: $y [m]$ – const double Input
4: $f [m]$ – const double Input: On entry: the coordinates of the $r$ th data point, for $r = 1, 2, \dots, m$ . 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: $triang [7 \times m]$ – Integer Output: On exit: a data structure defining the computed triangulation, in a form suitable for passing to e01skc.
6: $grads [2 \times m]$ – double Output: Note: the $(i, j)$ th element of the matrix is stored in $grads [(j - 1) \times 2 + i - 1]$ .

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 $grads [(r - 1) \times 2]$ and $grads [(r - 1) \times 2 + 1]$ respectively, for $r = 1, 2, \dots, m$ .
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

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 $⟨ value ⟩$ had an illegal value.
NE_DATA_NOT_UNIQUE: On entry, $(x [I - 1], y [I - 1]) = (x [J - 1], y [J - 1])$ , for $I, J = ⟨ value ⟩ ⟨ value ⟩$ , $x [I - 1]$ , $y [I - 1] = ⟨ value ⟩ ⟨ value ⟩$ .
NE_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 3$ .
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

Background information to multithreading can be found in the Multithreading documentation.

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.

e01sj: FL CL CPP AD PY MB

NAG CL Interfacee01sjc (dim2_​triang_​interp)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
e01sjc (dim2_triang_interp)