NDLI: On rational function approximations to the exponential function with application to the practical solution of linear differential difference equation with constant coefficients

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On rational function approximations to the exponential function with application to the practical solution of linear differential difference equation with constant coefficients

On rational function approximations to the exponential function with application to the practical solution of linear differential difference equation with constant coefficients

Content Provider	ACM Digital Library
Author	Luke, Yudell L.
Abstract	In connection with (15), it is important to know the nature of the zeros of Gn(z). All are simple and lie in the left half plane outside the circle of radius 2 (n&equil;1 is an exception where the zero is on the latter circle). If n is even, there are no real zeros, while if n is odd, there is only one real zero. For complete details see [4]* and [5]. The work of Nassif [6] shows that if n > 1, the zeros of Gn (z ) lie on or outside a circle of radius 2 [(2n-1)/(n-1)] 1/2, which is an improvement over the above result. Thomson [7]* has tabulated the zeros of Gn (2z) for n &equil; 1(1)9 to 4 decimals. Examination of these data shows that the zeros of Gn (z) lie outside the circle of radius 4n/3 and we conjecture that this is true for all n. In fact, the magnitude of the smallest zero(s) is very nearly (4n+2)/3, for n &equil; 2(1)9, and the latter formula is exact for n&equil;1. Thus the applicable range of the rational approximation (α&equil;β&equil;0) appears to be limited by the location of the zeros of Gn (z). However, an analysis of (15) reveals that to obtain good accuracy n must be considerably larger than z, so that (15) is dominant in deciding the choice of n for a pre-selected z.
Starting Page	177
Ending Page	178
Page Count	2
File Format	PDF
DOI	10.1145/800258.808979
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	1956-01-01
Publisher Place	New York
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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