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Convergence acceleration of alternating series. (English) Zbl 0972.11115
This paper treats several algorithms to calculate the sum of an alternating series \(\sum_{k=0}^{\infty} (-1)^ka_k\) from the first \(n\) entries. The \(a_k\) have to be either the moments of a positive measure on \([0,1]\) or those of a weight \(w(x)\) on \([0,1]\) that extends analytically into the complex domain in a certain manner.
In the first case the relative accuracy is of the order \(5.828^{-n}\); the algorithm uses \(O(1)\) storage and has a running time of \(O(1)\) for each \(a_k\) used. Chebyshev polynomials play a role in the construction of the algorithm.
The second algorithm requires storage \(O(n)\) and running time \(O(n^2)\); the relative accuracy is of the order \(7.89^{-n}\) for a large class of series and \(17.93^{-n}\) for a smaller class.
The conclusion is, that the choice of algorithm depends on the ease with which the terms of the series can be computed.
A very interesting and nicely written paper.

MSC:
11Y35 Analytic computations
65B10 Numerical summation of series
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References:
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