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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



An acceleration method for the power series of entire functions of order $1$

Authors: B. Gabutti and J. N. Lyness
Journal: Math. Comp. 39 (1982), 587-597
MSC: Primary 65B10; Secondary 30B10
MathSciNet review: 669651
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Abstract: When $f(z)$ is given by a known power series expansion, it is possible to construct the power series expansion for $f(z;p) = {e^{ - pz}}f(z)$. We define ${p_{{\text {opt}}}}$ to be the value of p for which the expansion for $f(z;p)$ converges most rapidly. When $f(z)$ is an entire function of order 1, we show that ${p_{{\text {opt}}}}$ is uniquely defined and may be characterized in terms of the set of singularities ${z_i} = 1/{\sigma _i}$ of an associated function $h(z)$. Specifically, it is the center of the smallest circle in the complex plane which contains all points ${\sigma _i}$.

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Article copyright: © Copyright 1982 American Mathematical Society