Integer, Intent (In)	::	m, iq(liq), liq, lrq, n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(m), y(m), f(m), rq(lrq), u(n), v(n)
Real (Kind=nag_wp), Intent (Out)	::	q(n), qx(n), qy(n)

C Header Interface

#include <nag.h>

void	e01shf_ (const Integer m, const double x[], const double y[], const double f[], const Integer iq[], const Integer liq, const double rq[], const Integer lrq, const Integer n, const double u[], const double v[], double q[], double qx[], double qy[], Integer *ifail)

The routine may be called by the names e01shf or nagf_interp_dim2_scat_shep_eval.

3 Description

e01shf takes as input the interpolant

Q (x, y)

of a set of scattered data points

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

, for

r = 1, 2, \dots, m

, as computed by e01sgf, and evaluates the interpolant and its first partial derivatives at the set of points

(u_{i}, v_{i})

, for

i = 1, 2, \dots, n

e01shf must only be called after a call to e01sgf.

This routine is derived from the routine QS2GRD described by Renka (1988).

4 References

Renka R J (1988) Algorithm 660: QSHEP2D: Quadratic Shepard method for bivariate interpolation of scattered data ACM Trans. Math. Software 14 149–150

5 Arguments

1: $m$ – Integer Input
2: $x (m)$ – Real (Kind=nag_wp) array Input
3: $y (m)$ – Real (Kind=nag_wp) array Input
4: $f (m)$ – Real (Kind=nag_wp) array Input: On entry: m, x, y and f must be the same values as were supplied in the preceding call to e01sgf.
5: $iq (liq)$ – Integer array Input: On entry: must be unchanged from the value returned from a previous call to e01sgf.
6: $liq$ – Integer Input: On entry: the dimension of the array iq as declared in the (sub)program from which e01shf is called.

Constraint: $liq \geq 2 \times m + 1$ .
7: $rq (lrq)$ – Real (Kind=nag_wp) array Input: On entry: must be unchanged from the value returned from a previous call to e01sgf.
8: $lrq$ – Integer Input: On entry: the dimension of the array rq as declared in the (sub)program from which e01shf is called.

Constraint: $lrq \geq 6 \times m + 5$ .
9: $n$ – Integer Input: On entry: $n$ , the number of evaluation points.

Constraint: $n \geq 1$ .
10: $u (n)$ – Real (Kind=nag_wp) array Input
11: $v (n)$ – Real (Kind=nag_wp) array Input: On entry: the evaluation points $(u_{i}, v_{i})$ , for $i = 1, 2, \dots, n$ .
12: $q (n)$ – Real (Kind=nag_wp) array Output: On exit: the values of the interpolant at $(u_{i}, v_{i})$ , for $i = 1, 2, \dots, n$ . If any of these evaluation points lie outside the region of definition of the interpolant the corresponding entries in q are set to an extrapolated approximation, and e01shf returns with $ifail = 3$ .
13: $qx (n)$ – Real (Kind=nag_wp) array Output
14: $qy (n)$ – Real (Kind=nag_wp) array Output: On exit: the values of the partial derivatives of the interpolant $Q (x, y)$ at $(u_{i}, v_{i})$ , for $i = 1, 2, \dots, n$ . If any of these evaluation points lie outside the region of definition of the interpolant, the corresponding entries in qx and qy are set to extrapolated approximations to the partial derivatives, and e01shf returns with $ifail = 3$ .
15: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, liq is too small: $liq = ⟨ value ⟩$ .

On entry, lrq is too small: $lrq = ⟨ value ⟩$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 6$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = 2$: On entry, values in iq appear to be invalid. Check that iq has not been corrupted between calls to e01sgf and e01shf.

On entry, values in rq appear to be invalid. Check that rq has not been corrupted between calls to e01sgf and e01shf.

$ifail = 3$: On entry, at least one evaluation point lies outside the region of definition of the interpolant. At such points the corresponding values in q and qx contain extrapolated approximations. Points should be evaluated one by one to identify extrapolated values.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Computational errors should be negligible in most practical situations.

8 Parallelism and Performance

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

e01shf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

e01shf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken for a call to e01shf will depend in general on the distribution of the data points. If x and y are approximately uniformly distributed, then the time taken should be only

O (n)

. At worst

O (m n)

time will be required.

9.1 Internal Changes

Internal changes have been made to this routine as follows:

At Mark 26.0: The algorithm used by this routine, based on a Modified Shepard method, was changed to produce more reliable results for some data sets which were previously not well handled. In addition, handling of evaluation points which are far away from the original data points has been improved by use of an extrapolation method which returns useful results rather than just an error message as was done at earlier Marks.
At Mark 26.1: The algorithm has undergone further changes which enable it to work better on certain data sets, for example data presented on a regular grid. The results returned when evaluating the function at points which are not in the original data set used to construct the interpolating function are now likely to be slightly different from those returned at previous Marks of the Library, but the function still interpolates the original data.

For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.