NAG Library Routine Document

Integer, Intent (In)	::	m, iq(2*m+1), n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(4,m), f(m), rq(15*m+9), xe(4,n)
Real (Kind=nag_wp), Intent (Out)	::	q(n), qx(4,n)

C Header Interface

#include <nagmk26.h>

void	e01tlf_ (const Integer m, const double x[], const double f[], const Integer iq[], const double rq[], const Integer n, const double xe[], double q[], double qx[], Integer *ifail)

3

Description

e01tlf takes as input the interpolant

Q (x)

x \in ℝ^{4}

of a set of scattered data points

(x_{r}, f_{r})

, for

r = 1, 2, \dots, m

, as computed by e01tkf, and evaluates the interpolant and its first partial derivatives at the set of points

x_{i}

, for

i = 1, 2, \dots, n

e01tlf must only be called after a call to e01tkf.

e01tlf is derived from the new implementation of QS3GRD described by Renka (1988). It uses the modification for high-dimensional interpolation described by Berry and Minser (1999).

4

References

Berry M W, Minser K S (1999) Algorithm 798: high-dimensional interpolation using the modified Shepard method ACM Trans. Math. Software 25 353–366

Renka R J (1988) Algorithm 661: QSHEP3D: Quadratic Shepard method for trivariate interpolation of scattered data ACM Trans. Math. Software 14 151–152

5

Arguments

1: $m$ – IntegerInput: On entry: must be the same value supplied for argument m in the preceding call to e01tkf.

Constraint: $m \geq 16$ .
2: $x (4, m)$ – Real (Kind=nag_wp) arrayInput: Note: the coordinates of $x_{r}$ are stored in $x (1, r) \dots x (4, r)$ .

On entry: must be the same array supplied as argument x in the preceding call to e01tkf. It must remain unchanged between calls.
3: $f (m)$ – Real (Kind=nag_wp) arrayInput: On entry: must be the same array supplied as argument f in the preceding call to e01tkf. It must remain unchanged between calls.
4: $iq (2 \times m + 1)$ – Integer arrayInput: On entry: must be the same array returned as argument iq in the preceding call to e01tkf. It must remain unchanged between calls.
5: $rq (15 \times m + 9)$ – Real (Kind=nag_wp) arrayInput: On entry: must be the same array returned as argument rq in the preceding call to e01tkf. It must remain unchanged between calls.
6: $n$ – IntegerInput: On entry: $n$ , the number of evaluation points.

Constraint: $n \geq 1$ .
7: $xe (4, n)$ – Real (Kind=nag_wp) arrayInput: On entry: $xe (1 : 4, i)$ must be set to the evaluation point $x_{i}$ , for $i = 1, 2, \dots, n$ .
8: $q (n)$ – Real (Kind=nag_wp) arrayOutput: On exit: $q (i)$ contains the value of the interpolant, at $x_{i}$ , 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 the largest machine representable number (see x02alf), and e01tlf returns with $ifail = 3$ .
9: $qx (4, n)$ – Real (Kind=nag_wp) arrayOutput: On exit: $qx (j, i)$ contains the value of the partial derivatives with respect to $x_{j}$ of the interpolant $Q (x)$ at $x_{i}$ , for $i = 1, 2, \dots, n$ , and for each of the four partial derivatives $j = 1, 2, 3, 4$ . If any of these evaluation points lie outside the region of definition of the interpolant, the corresponding entries in qx are set to the largest machine representable number (see x02alf), and e01tlf returns with $ifail = 3$ .
10: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . 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, $m = 〈value〉$ .
Constraint: $m \geq 16$ .

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 e01tkf and e01tlf.

On entry, values in rq appear to be invalid. Check that rq has not been corrupted between calls to e01tkf and e01tlf.

$ifail = 3$: 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 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

Computational errors should be negligible in most practical situations.

8

Parallelism and Performance

e01tlf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

e01tlf 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 e01tlf will depend in general on the distribution of the data points. If the data points 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, has been 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 list.

10

Example

This program evaluates the function

f (x) = \frac{(1.25 + \cos (5.4 x_{4})) \cos (6 x_{1}) \cos (6 x_{2})}{6 + 6 {(3 x_{3} - 1)}^{2}}

at a set of

30

randomly generated data points and calls e01tkf to construct an interpolating function

Q (x)

. It then calls e01tlf to evaluate the interpolant at a set of random points.

To reduce the time taken by this example, the number of data points is limited to

30

. Increasing this value improves the interpolation accuracy at the expense of more time.

NAG Library Routine Document

e01tlf (dim4_scat_shep_eval)

▸▿ 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

9.1 Internal Changes

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification