Abstract
The aim of this paper is to verify efficiency of two acceleration methods for orthogonal series (more strictly, for series defined at the beginning of Section 1). These methods are quite different although they use the same transform of such a series given there. The first method (Section 3) has some features common with Levin’s and Weniger’s methods. It may be profitably used in numerical calculations for a vast class of series. The second one (Sections 4 and 5) is somewhat similar to the Euler–Knopp transform of power series. Also this method is numerically realizable but more important is that for a narrower class of series, including some ones having applications in physics, it gives explicit analytic formulae of their transform.
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References
Björck, Å., Dahlquist, G.: Numerical Methods. Prentice-Hall (1974)
Brezinski, C., Redivo-Zaglia, M.: Extrapolation Methods. Theory and Practice. North-Holland (1991)
Gabutti, B., Lyness, J.N.: Some generalizations of the Euler–Knopp transformation. Numer. Math. 48, 199–220 (1986)
Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products, 6th edn. In: Jeffrey, A. and Zwillinger, D. (eds.). Academic Press (2000)
Holdeman, Jr., J.T.: A method for the approximation of functions defined by formal series expansions in orthogonal polynomials. Math. Comput. 23, 275–287 (1969)
Homeier, H.H.H.: A Levin-type algorithm for accelerating the convergence of Fourier series. Numer. Algebra 3, 245–254 (1992)
Homeier, H.H.H.: Some applications of nonlinear convergence accelerators. Int. J. Quantum Chem. 45, 545–562 (1993)
Homeier, H.H.H.: Nonlinear convergence acceleration for orthogonal series. In: Proc. 6th Joint EPS-APS Intern. Conf. on Physics Computing PC ‘94
Lewanowicz, S., Paszkowski, S.: An analytic method for convergence acceleration of certain hypergeometric series. Math. Comput. 64, 691–713 (1995)
Lorentzen, L., Waadeland, H.: Continued Fractions with Applications. North-Holland (1992)
Oleksy, Cz.: A convergence acceleration method of Fourier series. Comput. Phys. Comm. 96, 17–26 (1996)
Paszkowski, S.: Fast convergent quasipower series for some elementary and special functions. Comput. Math. Appl. 33, 181–191 (1997)
Szegö, G.: Orthogonal polynomials. In: AMS Colloq. Publ., vol. 23, Providence, R.I. (1959)
Weniger, E.J.: Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10, 189–371 (1989)
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Paszkowski, S. Convergence acceleration of orthogonal series. Numer Algor 47, 35–62 (2008). https://doi.org/10.1007/s11075-007-9146-7
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DOI: https://doi.org/10.1007/s11075-007-9146-7