D04BBF generates abscissae about a target abscissa
${x}_{0}$ for use in a subsequent call to
D04BAF.
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
$\left\{{x}_{0},{x}_{0}\pm \left(2\mathit{j}-1\right)h\right\}$, 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
${x}_{0}$.
Lyness J N and Moler C B (1969) Generalised Romberg methods for integrals of derivatives Numer. Math. 14 1–14
None.
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.