D04 Chapter Contents
D04 Chapter Introduction
NAG Library Manual

NAG Library Routine DocumentD04BBF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

1  Purpose

D04BBF generates abscissae about a target abscissa ${x}_{0}$ for use in a subsequent call to D04BAF.

2  Specification

 SUBROUTINE D04BBF ( X_0, HBASE, XVAL)
 REAL (KIND=nag_wp) X_0, HBASE, XVAL(21)

3  Description

D04BBF may be used to generate the necessary abscissae about a target abscissa ${x}_{0}$ for the calculation of derivatives using D04BAF.
For a given ${x}_{0}$ and $h$, the abscissae correspond to the set $\left\{{x}_{0},{x}_{0}±\left(2\mathit{j}-1\right)h\right\}$, for $\mathit{j}=1,2,\dots ,10$. These $21$ points will be returned in ascending order in XVAL. In particular, ${\mathbf{XVAL}}\left(11\right)$ will be equal to ${x}_{0}$.

4  References

Lyness J N and Moler C B (1969) Generalised Romberg methods for integrals of derivatives Numer. Math. 14 1–14

5  Parameters

1:     X_0 – REAL (KIND=nag_wp)Input
On entry: the abscissa ${x}_{0}$ at which derivatives are required.
2:     HBASE – REAL (KIND=nag_wp)Input
On entry: the chosen step size $h$. If $h<10\epsilon$, where $\epsilon ={\mathbf{X02AJF}}\left(\right)$, then the default $h={\epsilon }^{\left(1/4\right)}$ will be used.
3:     XVAL($21$) – REAL (KIND=nag_wp) arrayOutput
On exit: the abscissae for passing to D04BAF.

None.

7  Accuracy

Not applicable.

The results computed by D04BAF depend very critically on the choice of the user-supplied step length $h$. The overall accuracy is diminished as $h$ becomes small (because of the effect of round-off error) and as $h$ becomes large (because the discretization error also becomes large). If the process of calculating derivatives is repeated four or five times with different values of $h$ one can find a reasonably good value. A process in which the value of $h$ is successively halved (or doubled) is usually quite effective. Experience has shown that in cases in which the Taylor series for for the objective function about ${x}_{0}$ has a finite radius of convergence $R$, the choices of $h>R/19$ are not likely to lead to good results. In this case some function values lie outside the circle of convergence.