The selection of interpolation points in numerical differentiation

Abstract

Abstract.The relationships between various numerical methods for obtaining polynomial approximations to the first derivative of a known function are investigated, and their computational advantages discussed. Optimum sequences of interpolation points are then selected with the objective of minimising the relative contribution of rounding errors to the total error, and geometric sequences, though non-optimal in this sense, are considered for computational reasons.

On numerical differentiation

“…In fact the sequence (3.7) is explicitly recommended. The choice h i--fl-jho, also considered in [4], produces a distribution of the opposite character. Thus in making ~his choice one may be increasing the magnitude of %he roundoff error in an effort to make the effect easier to recognize quantitatively.…”

“…Heuristic reasoning indicates %hat the distribution of abscissas should become slightly denser as one moves away from the origin. This problem has been extensively treated in [4] where optimal allocations are derived for the approximation of the first derivative. In fact the sequence (3.7) is explicitly recommended.…”

A characterisation of certain optimal collocation points for numerical differentiation

“…Differentiation of the usual expression for f(x)-p(x) gives a truncation error A. RUFFHEAD AND J. OLIVER As in [2], we define a scaled error magnification factor S. by…”

“…n), to x l -X o as shown in [2], rather than on xa -Xo itself which may therefore be used to vary the truncation error (1) for any given set of ratios without altering the value of S.. Consequently we may regard the product 1"]7= 1 ( x i -Xo) as constant and minimise S., and hence IR.I, with respect to the ratios, and in the light Of (1) and (4) this is equivalent to minimising the potential effect of rounding error relative to a fixed truncation error, provided ft.+l)(~) is regarded as locally constant. .…”

Adaptive numerical differentiation

“…If h is chosen too small, the finite precision of the computer can cause cancellation and other "rounding" errors to be made that result in poor answers; while if h is chosen too large, d(h) need not be a good approximation to /' even in "exact" arithmetic. Several authors have given algorithms that attempt to deal with this situation, see, e.g., Curtis and Reid [1], Dahlquist and Bjorck [2], Dumontet and Vignes [3], and Oliver and Ruffhead [4]. Often they are based on estimating the amount of rounding and truncation error made.…”