Integer, Intent (In)	::	n, m
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(n), f(n), d(n), px(m)
Real (Kind=nag_wp), Intent (Out)	::	pf(m), pd(m)

C Header Interface

#include <nag.h>

void	e01bgf_ (const Integer n, const double x[], const double f[], const double d[], const Integer m, const double px[], double pf[], double pd[], Integer *ifail)

The routine may be called by the names e01bgf or nagf_interp_dim1_monotonic_deriv.

3 Description

e01bgf evaluates a piecewise cubic Hermite interpolant, as computed by e01bef, at the points

px (i)

, for

i = 1, 2, \dots, m

. The first derivatives at the points are also computed. If any point lies outside the interval from

x (1)

x (n)

, values of the interpolant and its derivative are extrapolated from the nearest extreme cubic, and a warning is returned.

If values of the interpolant only, and not of its derivative, are required, e01bff should be used.

The routine is derived from routine PCHFD in Fritsch (1982).

4 References

Fritsch F N (1982) PCHIP final specifications Report UCID-30194 Lawrence Livermore National Laboratory

5 Arguments

1: $n$ – Integer Input
2: $x (n)$ – Real (Kind=nag_wp) array Input
3: $f (n)$ – Real (Kind=nag_wp) array Input
4: $d (n)$ – Real (Kind=nag_wp) array Input: On entry: n, x, f and d must be unchanged from the previous call of e01bef.
5: $m$ – Integer Input: On entry: $m$ , the number of points at which the interpolant is to be evaluated.

Constraint: $m \geq 1$ .
6: $px (m)$ – Real (Kind=nag_wp) array Input: On entry: the $m$ values of $x$ at which the interpolant is to be evaluated.
7: $pf (m)$ – Real (Kind=nag_wp) array Output: On exit: $pf (i)$ contains the value of the interpolant evaluated at the point $px (i)$ , for $i = 1, 2, \dots, m$ .
8: $pd (m)$ – Real (Kind=nag_wp) array Output: On exit: $pd (i)$ contains the first derivative of the interpolant evaluated at the point $px (i)$ , for $i = 1, 2, \dots, m$ .
9: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface 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, $n = 〈value〉$ .
Constraint: $n \geq 2$ .

$ifail = 2$: On entry, $r = 〈value〉$ , $x (r - 1) = 〈value〉$ and $x (r) = 〈value〉$ .
Constraint: $x (r - 1) < x (r)$ for all $r$ .

$ifail = 3$: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

$ifail = 4$: Warning – some points in array px lie outside the range $x (1) \dots x (n)$ . Values at these points are unreliable because computed by extrapolation.

$ifail = - 99$: 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

The computational errors in the arrays pf and pd should be negligible in most practical situations.

8 Parallelism and Performance

e01bgf is not threaded in any implementation.

9 Further Comments

The time taken by e01bgf is approximately proportional to the number of evaluation points,

m

. The evaluation will be most efficient if the elements of px are in nondecreasing order (or, more generally, if they are grouped in increasing order of the intervals

[x (r - 1), x (r)]

). A single call of e01bgf with

m > 1

is more efficient than several calls with

m = 1

10 Example

This example reads in values of n, x, f and d, and calls e01bgf to compute the values of the interpolant and its derivative at equally spaced points.