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

C Header Interface

#include <nag.h>

void	e01znf_ (const Integer d, 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)

The routine may be called by the names e01znf or nagf_interp_dimn_scat_shep_eval.

3 Description

e01znf takes as input the interpolant

Q (x)

x \in ℝ^{d}

of a set of scattered data points

(x_{r}, f_{r})

, for

r = 1, 2, \dots, m

, as computed by e01zmf, and evaluates the interpolant and its first partial derivatives at the set of points

x_{i}

, for

i = 1, 2, \dots, n

e01znf must only be called after a call to e01zmf.

e01znf 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: $d$ – Integer Input: On entry: must be the same value supplied for argument d in the preceding call to e01zmf.

Constraint: $d \geq 2$ .
2: $m$ – Integer Input: On entry: must be the same value supplied for argument m in the preceding call to e01zmf.

Constraint: $m \geq (d + 1) \times (d + 2) / 2 + 2$ .
3: $x (d, m)$ – Real (Kind=nag_wp) array Input: Note: the $i$ th ordinate of the point $x_{j}$ is stored in $x (i, j)$ .

On entry: must be the same array supplied as argument x in the preceding call to e01zmf. It must remain unchanged between calls.
4: $f (m)$ – Real (Kind=nag_wp) array Input: On entry: must be the same array supplied as argument f in the preceding call to e01zmf. It must remain unchanged between calls.
5: $iq (2 \times m + 1)$ – Integer array Input: On entry: must be the same array returned as argument iq in the preceding call to e01zmf. It must remain unchanged between calls.
6: $rq (*)$ – Real (Kind=nag_wp) array Input: Note: the dimension of the array rq must be at least $((d + 1) \times (d + 2) / 2) \times m + 2 \times d + 1$ .

On entry: must be the same array returned as argument rq in the preceding call to e01zmf. It must remain unchanged between calls.
7: $n$ – Integer Input: On entry: $n$ , the number of evaluation points.

Constraint: $n \geq 1$ .
8: $xe (d, n)$ – Real (Kind=nag_wp) array Input: Note: the $i$ th ordinate of the point $x_{j}$ is stored in $xe (i, j)$ .

On entry: $xe (1 : d, j)$ must be set to the evaluation point $x_{j}$ , for $j = 1, 2, \dots, n$ .
9: $q (n)$ – Real (Kind=nag_wp) array Output: 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 an extrapolated approximation, and e01znf returns with $ifail = 3$ .
10: $qx (d, n)$ – Real (Kind=nag_wp) array Output: On exit: $qx (i, j)$ contains the value of the partial derivatives with respect to the $i$ th independent variable (dimension) of the interpolant $Q (x)$ at $x_{j}$ , for $j = 1, 2, \dots, n$ , and for each of the partial derivatives $i = 1, 2, \dots, d$ . If any of these evaluation points lie outside the region of definition of the interpolant, the corresponding entries in qx are set to extrapolated approximations to the partial derivatives, and e01znf returns with $ifail = 3$ .
11: $ifail$ – 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, $d = ⟨ value ⟩$ .
Constraint: $d \geq 2$ .

On entry, $((d + 1) \times (d + 2) / 2) \times m + 2 \times d + 1$ exceeds the largest machine integer.
$d = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .

On entry, $m = ⟨ value ⟩$ and $d = ⟨ value ⟩$ .
Constraint: $m \geq (d + 1) \times (d + 2) / 2 + 2$ .

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 e01zmf and e01znf.

On entry, values in rq appear to be invalid. Check that rq has not been corrupted between calls to e01zmf and e01znf.

$ifail = 3$: 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.

e01znf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

e01znf 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 e01znf 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, 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.

10 Example

This program evaluates the function (in six variables)

f (x) = \frac{x_{1} x_{2} x_{3}}{1 + 2 x_{4} x_{5} x_{6}}

at a set of randomly generated data points and calls e01zmf to construct an interpolating function

Q_{x}

. It then calls e01znf to evaluate the interpolant at a set of points on the line

x_{i} = x

, for

i = 1, 2, \dots, 6

. To reduce the time taken by this example, the number of data points is limited. Increasing this value to the suggested minimum of

4000

improves the interpolation accuracy at the expense of more time.

e01zn: FL CL CPP AD PY MB

NAG FL Interfacee01znf (dimn_​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

NAG FL Interface
e01znf (dimn_scat_shep_eval)