Rational approximations for the cosine function; P-acceptability and order. (English) Zbl 0785.65093

Coleman, John P.

Numer. Algorithms 3, No. 1-4, 143-158 (1992).

The paper deals with rational approximations \(R_{nm}(z^ 2)\) (with the numerator of the degree \(n\) and the denominator of the degree \(m\)) of the function \(\cos z\). It is proved that the order of accuracy of a \(P\)- acceptable \(R_{nm}\) (i.e. \(| R_{nm}(\nu^ 2)| < 1\) for all \(\nu > 0\)) cannot exceed \(2m\). Further, if the poles of \(R_{nm}(z^ 2)\) are pure-imaginary, then the maximum attainable order is \(2n+2\) for \(m\geq 1\).

Reviewer: M.Bartušek (Brno)

MSC:

65L20	Stability and convergence of numerical methods for ordinary differential equations
65L05	Numerical methods for initial value problems involving ordinary differential equations
41A20	Approximation by rational functions
65D20	Computation of special functions and constants, construction of tables
34A34	Nonlinear ordinary differential equations and systems

Keywords:

cosine function; \(P\)-acceptability; stability analysis; rational approximations; order of accuracy

References:

[1]	G. Baker, V.A. Dougalis and S.M. Serbin, An approximation theorem for second-order evolution equations, Numer. Math. 35 (1980) 127–142. · Zbl 0445.65075
[2]	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. · Zbl 0658.65017
[3]	J.R. Cash, High order P-stable formulae for the numerical integration of periodic initial value problems, Numer. Math. 37 (1981) 355–370. · Zbl 0488.65029
[4]	J.P. Coleman, Numerical methods fory”=f(x, y) via rational approximations for the cosine, IMA J. Numer. Anal. 9 (1989) 145–165. · Zbl 0675.65072
[5]	V.A. Dougalis and S.M. Serbin, Some remarks on a class of rational approximations to the cosine, BIT 20 (1980) 204–211. · Zbl 0425.65006
[6]	E. Hairer, Unconditionally stable methods for second order differential equations, Numer. Math. 32 (1979) 373–379. · Zbl 0393.65035
[7]	E. Hairer and G. Wanner,Solving Ordinary Differential Equations II (Springer, 1991). · Zbl 0729.65051
[8]	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. · Zbl 0731.65071
[9]	A. Iserles, Order stars, approximations and finite differences. I: The general theory of order stars, SIAM J. Math. Anal. 16 (1985) 559–576. · Zbl 0589.30038
[10]	A. Iserles and S.P. Nørsett,Order Stars (Chapman and Hall, 1991).
[11]	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. · Zbl 0472.41013
[12]	L. Kramarz, Stability of collocation methods for the numerical solution ofy”=f(x, y), BIT 20 (1980) 215–222. · Zbl 0425.65043
[13]	C. Lanczos, Trigonometric interpolation of empirical and analytical functions, J. Math. and Phys. 17 (1938) 123–199. · Zbl 0020.01301
[14]	S.P. Nørsett and A. Wolfbrandt, Attainable order of rational approximations to the exponential function with only real poles, BIT 17 (1977) 200–208. · Zbl 0361.41011
[15]	G. Wanner, E. Hairer and S.P. Nørsett, Order stars and stability theorems, BIT 18 (1978) 475–489. · Zbl 0444.65039