Abstract
We show that the error of approximations of the solution y(x) of a linear system of ordinary differential equations, generated by using the Tau method, attain the same order as the best constant En(y):
where n is the degree of the Tau approximation yn(x), r is the semi-amplitude of the interval and v1 is related to the coefficients of the system.
We also show that the normalized error curves of successive Tau approximations of y(x) and of its derivative y′(x) are bounded by the simple curves
where L is a constant vector. These curves are reminiscent of curves already discussed by Ortiz and Rivlin in connection with intersections of orthogonal polynomials.
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References
Namasivayam, S. and Ortiz, E.L. An error analysis of the rational Tau method for ordinary differential equations, (submitted for publication).
Namasivayam, S. and Ortiz E.L. Dependence of the local truncation error on the choice of pertubation in the step-by-step Tau method for systems of differential equations, (submitted for publication).
Ortiz, E.L. The Tau method, SIAM J. Numer. Analysis, 6, pp. 480–492 (1969).
Ortiz, E.L. and Rivlin, T.J. Another look at Chebyshev polynomials, American Math. Monthly, 90, pp. 3–10 (1983).
Shohat, J. The best approximation of functions possessing derivatives, Duke Math. J., 8, pp. 376–385 (1941).
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© 1985 Springer-Verlag
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Namasivayam, S., Ortiz, E.L. (1985). On figures generated by normalized Tau approximation error curves. In: Brezinski, C., Draux, A., Magnus, A.P., Maroni, P., Ronveaux, A. (eds) Polynômes Orthogonaux et Applications. Lecture Notes in Mathematics, vol 1171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076573
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DOI: https://doi.org/10.1007/BFb0076573
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