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Mathematics of Computation

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Frequency fitting of rational approximations to the exponential functions

Authors: A. Iserles and S. P. Nørsett
Journal: Math. Comp. 40 (1983), 547-559
MSC: Primary 41A20; Secondary 65L05
MathSciNet review: 689470
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Abstract: Rational approximations to the exponential function are considered. Let $R = P/Q$, $\deg P = \deg Q = n$, $R(z) = \exp (z) + \mathcal {O}({z^{2n - 1}})$ and $R( \pm iT) = \exp ( \pm iT)$ for a given positive number T. We show that this approximation is A-acceptable if and only if T belongs to one of intervals, whose endpoints are related to zeros of certain Bessel functions. The existence of this type of approximation and its connection to diagonal Padé approximations is studied. Approximations which interpolate the exponential on the imaginary axis are important in the numerical analysis of highly-oscillatory ordinary differential systems.

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Article copyright: © Copyright 1983 American Mathematical Society