Frequency fitting of rational approximations to the exponential functions

Iserles, A.; Nørsett, S. P.

doi:10.1090/S0025-5718-1983-0689470-X

Frequency fitting of rational approximations to the exponential functions
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by A. Iserles and S. P. Nørsett PDF

Math. Comp. 40 (1983), 547-559 Request permission

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.

References

George A. Baker Jr., Essentials of Padé approximants, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0454459
Byron L. Ehle and Zdenek Picel, Two-parameter, arbitrary order, exponential approximations for stiff equations, Math. Comp. 29 (1975), 501–511. MR 375737, DOI 10.1090/S0025-5718-1975-0375737-7
A. Iserles and M. J. D. Powell, On the $A$-acceptability of rational approximations that interpolate the exponential function, IMA J. Numer. Anal. 1 (1981), no. 3, 241–251. MR 641308, DOI 10.1093/imanum/1.3.241
J. D. Lambert, Frequency fitting in the numerical solution of ordinary differential equations, Numerical treatment of differential equations in applications (Proc. Meeting, Math. Res. Center, Oberwolfach, 1977) Lecture Notes in Math., vol. 679, Springer, Berlin, 1978, pp. 65–72. MR 515570
Werner Liniger and Ralph A. Willoughby, Efficient integration methods for stiff systems of ordinary differential equations, SIAM J. Numer. Anal. 7 (1970), 47–66. MR 260181, DOI 10.1137/0707002
Syvert P. Nørsett, $C$-polynomials for rational approximation to the exponential function, Numer. Math. 25 (1975/76), no. 1, 39–56. MR 410189, DOI 10.1007/BF01419527
Earl D. Rainville, Special functions, The Macmillan Company, New York, 1960. MR 0107725
G. Wanner, E. Hairer, and S. P. Nørsett, Order stars and stability theorems, BIT 18 (1978), no. 4, 475–489. MR 520756, DOI 10.1007/BF01932026