Neville's and Romberg's processes: a fresh appraisal with extensions

Abstract
In this paper Neville’s process for the repetitive linear combination of numerical estimates is re-examined and exhibited as a process for term-by-term elimination of error, expressed as a power series; this point of view immediately suggests a wide range of applications—other than interpolation, for which the process was originally developed, and which is barely mentioned in this paper—for example, to the evaluation of finite or infinite integrals in one or more variables, to the evaluation of sums, etc. A matrix formulation is also developed, suggesting further extensions, for example, to the evaluation of limits, derivatives, sums of series with alternating signs, and so on. It is seen also that Neville’s process may be readily applied in Romberg Integration; each suggests extensions of the other. Several numerical examples exhibit various applications, and are accompanied by comments on the behaviour of truncation and rounding errors as exhibited in each Neville tableau, to show how these provide evidence of progress in the improvement of the approximation, and internal numerical evidence of the nature of the truncation error. A fuller and more connected account of the behaviour of truncation errors and rounding errors is given in a later section, and suggestions are also made for choosing suitable specific original estimates, i.e. for choosing suitable tabular arguments in the elimination variable, in order to produce results as precise and accurate as possible.