# NAG CL Interfaced04bbc (sample)

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## 1Purpose

d04bbc generates abscissae about a target abscissa ${x}_{0}$ for use in a subsequent call to d04bac.

## 2Specification

 #include
 void d04bbc (double x_0, double hbase, double xval[])
The function may be called by the names: d04bbc, nag_numdiff_sample or nag_numdiff_1d_real_absci.

## 3Description

d04bbc may be used to generate the necessary abscissae about a target abscissa ${x}_{0}$ for the calculation of derivatives using d04bac.
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, ${\mathbf{xval}}\left[10\right]$ 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

## 5Arguments

1: $\mathbf{x_0}$double Input
On entry: the abscissa ${x}_{0}$ at which derivatives are required.
2: $\mathbf{hbase}$double Input
On entry: the chosen step size $h$. If $h<10\epsilon$, where $\epsilon ={\mathbf{nag_machine_precision}}$, the default $h={\epsilon }^{\left(1/4\right)}$ will be used.
3: $\mathbf{xval}\left[21\right]$double Output
On exit: the abscissae for passing to d04bac.

None.

Not applicable.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
d04bbc is not threaded in any implementation.

The results computed by d04bac 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.