NAG Library Function Document

nag_2d_triang_eval (e01skc)

+− 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

1 Purpose

nag_2d_triang_eval (e01skc) evaluates at a given point the two-dimensional interpolant function computed by nag_2d_triang_interp (e01sjc).

2 Specification

#include <nag.h>

#include <nage01.h>

void	nag_2d_triang_eval (Integer m, const double x[], const double y[], const double f[], const Integer triang[], const double grads[], double px, double py, double pf, NagError fail)

3 Description

nag_2d_triang_eval (e01skc) takes as input the arguments defining the interpolant

F (x, y)

of a set of scattered data points

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

, for

r = 1, 2, \dots, m

, as computed by nag_2d_shep_interp (e01sgc), and evaluates the interpolant at the point

(p x, p y)

(p x, p y)

is equal to

(x_{r}, y_{r})

for some value of

r

, the returned value will be equal to

f_{r}

(p x, p y)

is not equal to

(x_{r}, y_{r})

for any

r

, the derivatives in grads will be used to compute the interpolant. A triangle is sought which contains the point

(p x, p y)

, and the vertices of the triangle along with the partial derivatives and

f_{r}

values at the vertices are used to compute the value

F (p x, p y)

. If the point

(p x, p y)

lies outside the triangulation defined by the input arguments, the returned value is obtained by extrapolation. In this case, the interpolating function

f

is extended linearly beyond the triangulation boundary. The method is described in more detail in Renka and Cline (1984) and the code is derived from Renka (1984).

nag_2d_triang_eval (e01skc) must only be called after a call to nag_2d_shep_interp (e01sgc).

4 References

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
2: x[m] – const doubleInput
3: y[m] – const doubleInput
4: f[m] – const doubleInput
5: triang[ $7 \times m$ ] – const IntegerInput
6: grads[ $2 \times m$ ] – const doubleInput: On entry: m, x, y, f, triang and grads must be unchanged from the previous call of nag_2d_triang_interp (e01sjc).
7: px – doubleInput
8: py – doubleInput: On entry: the point $(p x, p y)$ at which the interpolant is to be evaluated.
9: pf – double *Output: On exit: the value of the interpolant evaluated at the point $(p x, p y)$ .
10: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal 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.
NE_TRIANG_INVALID: On entry, triang does not contain a valid data point triangulation; triang may have been corrupted since the call to nag_2d_triang_interp (e01sjc).
NW_VALUE_EXTRAPOLATED: Warning – the evaluation point $(⟨value⟩, ⟨value⟩)$ lies outside the triangulation boundary. The returned value was computed by extrapolation.

7 Accuracy

Computational errors should be negligible in most practical situations.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken for a call of nag_2d_triang_eval (e01skc) is approximately proportional to the number of data points,

m

The results returned by this function are particularly suitable for applications such as graph plotting, producing a smooth surface from a number of scattered points.