Abstract
Properties of order stars corresponding to rational approximations for cos z are derived and are used to prove that the order of accuracy of a P-acceptable approximantR nm(z 2), with numerator of degreen and denominator of degreem, cannot exceed 2m. It is shown that if the poles ofR nm(z 2) are restricted to pure-imaginary values ofz the maximum attainable order is 2n+2, whatever the value ofm≥1. A study of rational approximations for the cosine function produced by symmetric one-step collocation methods, applied to the differential equationy n=−ω2 y, provides the answer to a question posed by Kramarz [BIT 20 (1980) 215–222]; there are no P-stable methods of that type.
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References
G. Baker, V.A. Dougalis and S.M. Serbin, An approximation theorem for second-order evolution equations, Numer. Math. 35 (1980) 127–142.
L.A. Bales, O.A. Karakashian and S.M. Serbin, On the stability of rational approximations to the cosine with only imaginary poles, BIT 28 (1988) 652–658.
J.R. Cash, High order P-stable formulae for the numerical integration of periodic initial value problems, Numer. Math. 37 (1981) 355–370.
J.P. Coleman, Numerical methods fory″=f(x, y) via rational approximations for the cosine, IMA J. Numer. Anal. 9 (1989) 145–165.
V.A. Dougalis and S.M. Serbin, Some remarks on a class of rational approximations to the cosine, BIT 20 (1980) 204–211.
E. Hairer, Unconditionally stable methods for second order differential equations, Numer. Math. 32 (1979) 373–379.
E. Hairer and G. Wanner,Solving Ordinary Differential Equations II (Springer, 1991).
P.J. van der Houwen, B.P. Sommeijer and Nguyen Huu Cong, Stability of collocation-based Runge-Kutta-Nyström methods, BIT 31 (1991) 469–481.
A. Iserles, Order stars, approximations and finite differences. I: The general theory of order stars, SIAM J. Math. Anal. 16 (1985) 559–576.
A. Iserles and S.P. Nørsett,Order Stars (Chapman and Hall, 1991).
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) 242–251.
L. Kramarz, Stability of collocation methods for the numerical solution ofy″=f(x, y), BIT 20 (1980) 215–222.
C. Lanczos, Trigonometric interpolation of empirical and analytical functions, J. Math. and Phys. 17 (1938) 123–199.
S.P. Nørsett and A. Wolfbrandt, Attainable order of rational approximations to the exponential function with only real poles, BIT 17 (1977) 200–208.
G. Wanner, E. Hairer and S.P. Nørsett, Order stars and stability theorems, BIT 18 (1978) 475–489.
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Coleman, J.P. Rational approximations for the cosine function; P-acceptability and order. Numer Algor 3, 143–158 (1992). https://doi.org/10.1007/BF02141924
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DOI: https://doi.org/10.1007/BF02141924