Accelerated polynomial approximation of finite order entire functions by growth reduction
HTML articles powered by AMS MathViewer
- by Jürgen Müller PDF
- Math. Comp. 66 (1997), 743-761 Request permission
Abstract:
Let $f$ be an entire function of positive order and finite type. The subject of this note is the convergence acceleration of polynomial approximants of $f$ by incorporating information about the growth of $f(z)$ for $z\to \infty$. We consider “near polynomial approximation” on a compact plane set $K$, which should be thought of as a circle or a real interval. Our aim is to find sequences $(f_n)_n$ of functions which are the product of a polynomial of degree $\le n$ and an “easy computable” second factor and such that $(f_n)_n$ converges essentially faster to $f$ on $K$ than the sequence $(P_n^*)_n$ of best approximating polynomials of degree $\le n$. The resulting method, which we call Reduced Growth method ($RG$-method) is introduced in Section 2. In Section 5, numerical examples of the $RG$-method applied to the complex error function and to Bessel functions are given.References
- Amos, D.E., A portable package for Bessel functions of a complex argument and nonnegative order, ACM Trans. Math. Softw., 12 (1986), 265-273
- Carlos A. Berenstein and Roger Gay, Complex analysis and special topics in harmonic analysis, Springer-Verlag, New York, 1995. MR 1344448, DOI 10.1007/978-1-4613-8445-8
- E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
- John P. Coleman and Nick J. Myers, The Faber polynomials for annular sectors, Math. Comp. 64 (1995), no. 209, 181–203, S1–S6. MR 1260127, DOI 10.1090/S0025-5718-1995-1260127-8
- S. W. Ellacott, Computation of Faber series with application to numerical polynomial approximation in the complex plane, Math. Comp. 40 (1983), no. 162, 575–587. MR 689474, DOI 10.1090/S0025-5718-1983-0689474-7
- David Elliott, Truncation errors in Padé approximations to certain functions: An alternative approach, Math. Comp. 21 (1967), 398–406. MR 227668, DOI 10.1090/S0025-5718-1967-0227668-6
- Bruno Gabutti, On two methods for accelerating convergence of series, Numer. Math. 43 (1984), no. 3, 439–461. MR 738387, DOI 10.1007/BF01390184
- B. Gabutti and J. N. Lyness, An acceleration method for the power series of entire functions of order $1$, Math. Comp. 39 (1982), no. 160, 587–597. MR 669651, DOI 10.1090/S0025-5718-1982-0669651-0
- D. Gaĭer, Lektsii po teorii approksimatsii v kompleksnoĭ oblasti, “Mir”, Moscow, 1986 (Russian). Translated from the German by L. M. Kartashov; Translation edited and with a preface by V. I. Belyĭ and P. M. Tamrazov. MR 894919
- Karin Gatermann, Christoph Hoffmann, and Gerhard Opfer, Explicit Faber polynomials on circular sectors, Math. Comp. 58 (1992), no. 197, 241–253, 51–56. MR 1106967, DOI 10.1090/S0025-5718-1992-1106967-4
- Walter Gautschi, Efficient computation of the complex error function, SIAM J. Numer. Anal. 7 (1970), 187–198. MR 293813, DOI 10.1137/0707012
- K. O. Geddes, Near-minimax polynomial approximation in an elliptical region, SIAM J. Numer. Anal. 15 (1978), no. 6, 1225–1233. MR 512695, DOI 10.1137/0715083
- K. O. Geddes and J. C. Mason, Polynomial approximation by projections on the unit circle, SIAM J. Numer. Anal. 12 (1975), 111–120. MR 364977, DOI 10.1137/0712011
- Peter Henrici, Applied and computational complex analysis. Vol. 2, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1991. Special functions—integral transforms—asymptotics—continued fractions; Reprint of the 1977 original; A Wiley-Interscience Publication. MR 1164865
- Rainer Hettich and Peter Zencke, Numerische Methoden der Approximation und semi-infiniten Optimierung, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks], B. G. Teubner, Stuttgart, 1982 (German). MR 653476, DOI 10.1007/978-3-322-93108-5
- William B. Jones and W. J. Thron, On the computation of incomplete gamma functions in the complex domain, Proceedings of the international conference on computational and applied mathematics (Leuven, 1984), 1985, pp. 401–417. MR 793971, DOI 10.1016/0377-0427(85)90034-2
- T. Kövari and Ch. Pommerenke, On Faber polynomials and Faber expansions, Math. Z. 99 (1967), 193–206. MR 227429, DOI 10.1007/BF01112450
- Lewin, B.J., Distribution of zeros of entire functions, Amer. Math. Soc., 1964.
- Lyness, J.N., Sande, G., Algorithm 413, Evaluation of normalized Taylor coefficients of an analytic function, Comm. ACM, 14 (1971), 669-675.
- A. I. Markushevich, Theory of functions of a complex variable. Vol. I, II, III, Second English edition, Chelsea Publishing Co., New York, 1977. Translated and edited by Richard A. Silverman. MR 0444912
- Müller, J., Convergence acceleration of polynomial expansions for finite order entire functions, Habilitationsschrift, Trier 1995.
- Müller, J., Convergence acceleration of Taylor sections for finite order entire functions, submitted.
- G. P. M. Poppe and C. M. J. Wijers, More efficient computation of the complex error function, ACM Trans. Math. Software 16 (1990), no. 1, 38–46. MR 1073408, DOI 10.1145/77626.77629
- John R. Rice, The degree of convergence for entire functions, Duke Math. J. 38 (1971), 429–440. MR 287010
- Theodore J. Rivlin, The Chebyshev polynomials, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0450850
- Software for numerical mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. Edited by D. J. Evans. MR 0353612
- Norbert Steinmetz, Exceptional values of solutions of linear differential equations, Math. Z. 201 (1989), no. 3, 317–326. MR 999730, DOI 10.1007/BF01214898
- Tsutomu Watanabe, Makoto Natori, and Tsutomu Oguni (eds.), Mathematical software for the P.C. and work stations, North-Holland Publishing Co., Amsterdam, 1994. A collection of Fortran 77 programs; Translated from the 1989 Japanese original; With 1 IBM-PC floppy disk (5.25 inch). MR 1300876
- J. A. C. Weideman, Computation of the complex error function, SIAM J. Numer. Anal. 31 (1994), no. 5, 1497–1518. MR 1293526, DOI 10.1137/0731077
- Paul Wild, Accelerating the convergence of power series of certain entire functions, Numer. Math. 51 (1987), no. 5, 583–595. MR 910867, DOI 10.1007/BF01400358
- T. Winiarski, Approximation and interpolation of entire functions, Ann. Polon. Math. 23 (1970/71), 259–273. (errata insert). MR 273032, DOI 10.4064/ap-23-3-259-273
Additional Information
- Jürgen Müller
- Affiliation: Fachbereich IV-Mathematik, Universität Trier, D-54286 Trier, Germany
- ORCID: 0000-0002-5872-0129
- Email: jmueller@uni-trier.de
- Received by editor(s): October 16, 1995
- Received by editor(s) in revised form: April 1, 1996
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 743-761
- MSC (1991): Primary 65B99; Secondary 30D10
- DOI: https://doi.org/10.1090/S0025-5718-97-00832-6
- MathSciNet review: 1401944