Accelerated Convergence, Divergence, Iteration, Extrapolation, and Curve Fitting

Abstract

This paper discusses some applications of the epsilon algorithm (EA), a sequential procedure for calculating Padé approximants. The EA may be used to: (1) accelerate the convergence of slowly converging series and iterations; (2) obtain useful results from divergent series and iterations; (3) obtain the limits of iterated vector and matrix sequences; (4) aid in the solution of differential and integral equations; (5) carry out numerical integration in a new way; (6) extrapolate; (7) fit a curve to a polynomial or to a constant plus sum of exponentials. As an illustration of curve fitting and extrapolation, we present results obtained with exact polynomial data plus random noise combined additively or proportionately. For such nonstationary data, the results are comparable, and in some cases superior, to least squares in yielding good estimates of the exact polynomial coefficients. One important advantage of the EA is that it builds up polynomials whose lower‐order coefficients are independent of higher‐order ones. This property is valuable when the degree of the polynomial is unknown. Finally, a simple empirical equation is given relating the precision of least‐squares‐calculated polynomial coefficients to the degree of the fitted polynomial and the number of effective decimal digits carried in the calculation.