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Error analysis of the Tau method: Dependence of the error on the degree and on the length of the interval of approximation. (English) Zbl 0772.65054
For the Tau method it is known that the norm of the error function and the sum of the absolute values of the Tau method parameters have the same rate of convergence. In this paper the authors investigate the speed of convergence of the approximation error by concentrating on the behaviour of these parameters.
Basic results are: the parameters decay exponentially in terms of $$n$$, for $$n$$ fixed parameters decay as $$(h/2)^ n$$ where $$h$$ is the length of the interval on which the approximation is sought. Two examples from initial and boundary value problems for ordinary differential equations are given.

##### MSC:
 65L70 Error bounds for numerical methods for ordinary differential equations 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L05 Numerical methods for initial value problems 34A34 Nonlinear ordinary differential equations and systems, general theory 34B15 Nonlinear boundary value problems for ordinary differential equations
##### Keywords:
error analysis; Tau method; rate of convergence
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##### References:
 [1] Ortiz, E.L., The tau method, SIAM J. numer. anal., 6, 480-492, (1969) · Zbl 0195.45701 [2] Birkhoff, G.; Rota, G., Ordinary differential equations, (1962), Ginn & Company Boston · Zbl 0183.35601 [3] Namasivayam, S.; Ortiz, E.L., A hierarchy of truncation error estimates for the numerical solution of a system of ordinary differential equations with techniques based on the tau method, (), 113-121, Teubner, Leipzig · Zbl 0688.65055 [4] Namasivayam, S.; Ortiz, E.L., Error analysis of the tau method: dependence of the approximation error on the choice of the perturbation term, Comput. & maths. appls., (1992), (to appear) · Zbl 0769.65045 [5] El-Daou, M.K.; Ortiz, E.L.; Samara, H., A unified approach to the tau method and Chebyshev series expansions techniques, Comput. & maths. appls., (1992), (to appear) · Zbl 0777.65051
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