NAG Library Function Document

nag_numdiff_1d_real_absci (d04bbc)

nag_numdiff_1d_real_absci (d04bbc) generates abscissae about a target abscissa $x_0$ for use in a subsequent call to nag_numdiff_1d_real_eval (d04bac).

nag_numdiff_1d_real_absci (d04bbc) may be used to generate the necessary abscissae about a target abscissa $x_0$ for the calculation of derivatives using nag_numdiff_1d_real_eval (d04bac).

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, $\mathbf{xval}\left[10\right]$ 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

None.

Not applicable.

Not applicable.

The results computed by nag_numdiff_1d_real_eval (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.

See Section 10 in nag_numdiff_1d_real_eval (d04bac).