# 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 ${x}_{0}$.

## 4References

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

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.