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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Rational approximations to the exponential function are considered. Let , , and 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.

**[1]**George A. Baker Jr.,*Essentials of Padé approximants*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0454459****[2]**Byron L. Ehle and Zdenek Picel,*Two-parameter, arbitrary order, exponential approximations for stiff equations*, Math. Comp.**29**(1975), 501–511. MR**0375737**, 10.1090/S0025-5718-1975-0375737-7**[3]**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**, 10.1093/imanum/1.3.241**[4]**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****[5]**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**0260181****[6]**Syvert P. Nørsett,*𝐶-polynomials for rational approximation to the exponential function*, Numer. Math.**25**(1975/76), no. 1, 39–56. MR**0410189****[7]**Earl D. Rainville,*Special functions*, The Macmillan Co., New York, 1960. MR**0107725****[8]**G. Wanner, E. Hairer, and S. P. Nørsett,*Order stars and stability theorems*, BIT**18**(1978), no. 4, 475–489. MR**520756**, 10.1007/BF01932026

Retrieve articles in *Mathematics of Computation*
with MSC:
41A20,
65L05

Retrieve articles in all journals with MSC: 41A20, 65L05

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1983-0689470-X

Article copyright:
© Copyright 1983
American Mathematical Society