NAG Library Function Document

nag_2d_triang_interp (e01sjc)

+− 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

1 Purpose

nag_2d_triang_interp (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>

#include <nage01.h>

void	nag_2d_triang_interp (Integer m, const double x[], const double y[], const double f[], Integer triang[], double grads[], NagError *fail)

3 Description

nag_2d_triang_interp (e01sjc) constructs an interpolating surface

F (x, y)

through a set of

m

scattered data points

(x_{r}, y_{r}, f_{r})

, for

r = 1, 2, \dots, 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, \dots, m

, as the partial derivatives of a quadratic function of

x

and

y

which interpolates the data value

f_{r}

, 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 nag_2d_triang_eval (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 – IntegerInput: On entry: $m$ , the number of data points.
Constraint: $m \geq 3$ .
2: x[m] – const doubleInput
3: y[m] – const doubleInput
4: f[m] – const doubleInput: 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[ $7 \times m$ ] – IntegerOutput: On exit: a data structure defining the computed triangulation, in a form suitable for passing to nag_2d_triang_eval (e01skc).
6: grads[ $2 \times m$ ] – doubleOutput: 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 nag_2d_triang_eval (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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALL_DATA_COLLINEAR: All nodes are collinear. There is no unique solution.
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.

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

Not applicable.

9 Further Comments

The time taken for a call of nag_2d_triang_interp (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 nag_2d_triang_interp (e01sjc) to construct an interpolating surface. It then calls nag_2d_triang_eval (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.

NAG Library Function Documentnag_2d_triang_interp (e01sjc)