Fast Summation of Power Series with Coefficients Analytic at Infinity |
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Authors: | Alvise Sommariva Marco Vianello Renato Zanovello |
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Institution: | (1) Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy |
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Abstract: | We propose a fast summation algorithm for slowly convergent power series of the form
j=j
0
z
j
j
j
![mgr](/content/m026p21107216065/xxlarge956.gif)
i=1
s
(j+
i
)–
i
, where ![mgr](/content/m026p21107216065/xxlarge956.gif) R,
i
0 and
i
C, 1 i s, are known parameters, and
j
= (j), being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of . The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests. |
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Keywords: | slowly convergent power series coefficients analytic at infinity numerical differentiation of analytic functions asymptotic summation method fast summation |
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