A lag-averaged generalization of Euler's method for accelerating series

摘要

Euler's method is an efficient accelerator for an alternating series, that is, one whose termsaj are such thatsign(aj) = −sign(aj+1). However, its effectiveness deteriorates rapidly as the periodP of the oscillations in degreej increases. To reduce this nonuniformity, we introduce an algorithm which begins with the recursive computation of a triangular array of numbers. The diagonal elements define a generalization of Euler's method. Each horizontal row defines a “delayed Euler” method which is even more effective. The key element is that the array is generated by repeatedly averaging partial sums of the original series with a “lag,” that is, a distance (in degree) between the partial sums that are combined, which optimally is half the periodP of the oscillations in the series.We illustrate the algorithm with the Fourier series for a function with a shock wave (jump discontinuity). The termsaj = (−1)jsin(jx) vary from strictly alternating (P = 2) atx = 0 to monotonic (P ⇒ ∞) asx ⇒ π. The lag-averaged Euler scheme, with and without delay, is much more efficient than its classical counterpart, especially near the discontinuities. We present an analytical convergence theory for both the standard and generalized Euler methods.