Adaptive numerical differentiation

Abstract:

It is well known that the calculation of an accurate approximate derivative $f\prime (x)$ of a nontabular function $f(x)$ on a finite-precision computer by the formula $d(h) = (f(x + h) - f(x - h))/2h$ is a delicate task. If h is too large, truncation errors cause poor answers, while if h is too small, cancellation and other "rounding" errors cause poor answers. We will show that by using simple results on the nature of the asymptotic convergence of $d(h)$ to $f\prime$, a reliable numerical method can be obtained which can yield efficiently the theoretical maximum number of accurate digits for the given machine precision.