Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Accelerated polynomial approximation
of finite order entire functions
by growth reduction

Author: Jürgen Müller
Journal: Math. Comp. 66 (1997), 743-761
MSC (1991): Primary 65B99; Secondary 30D10
MathSciNet review: 1401944
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. Amos, D.E., A portable package for Bessel functions of a complex argument and nonnegative order, ACM Trans. Math. Softw., 12 (1986), 265-273 CMP 19:12
  • 2. Berenstein, C.A., Gay, R., Complex analysis and special topics in harmonic analysis, Springer, New York, 1995. MR 96j:30001
  • 3. Cheney, E.W., Approximation theory, McGraw-Hill, New York, 1966. MR 36:5568
  • 4. Coleman, J.P., Myers, N.J., The Faber polynomials for annular sectors, Math. Comp., 64 (1995), 181-203. MR 95c:30006
  • 5. Ellacott, S.W., Computation of Faber series with application to numerical polynomial approximation in the complex plane, Math. Comp., 40 (1983), 575-587. MR 84e:65021
  • 6. Elliott, D., Truncation error in Padé approximations to certain functions, an alternative approach, Math. Comp., 21 (1967), 398-406. MR 37:3252
  • 7. Gabutti, B., On two methods for accelerating convergence of series, Numer. Math., 43 (1984), 439-461. MR 85i:65005
  • 8. Gabutti, B., Lyness, J.N., An acceleration method for the power series of entire functions of order 1, Math. Comp., 39 (1982), 587-597. MR 83j:65011
  • 9. Gaier, D., Lectures on complex approximation, Birkhäuser, Boston, 1987. MR 88i:30059b
  • 10. Gatermann, K., Hoffmann, Ch., Opfer, G., Faber polynomials on circular sectors, Math. Comp., 58 (1992), 241-253. MR 92h:30011
  • 11. Gautschi, W., Efficient computation of the complex error function, SIAM J. Numer. Anal., 7 (1970), 187-198. MR 45:2889
  • 12. Geddes, K.O., Near minimax polynomial approximation in an elliptical region, SIAM J. Numer. Anal., 15 (1978), 1225-1233. MR 80a:65062
  • 13. Geddes, K.O., Mason, J.C., Polynomial approximation by projections on the unit circle, SIAM J. Numer. Anal., 12 (1975), 111-120. MR 51:1230
  • 14. Henrici, P., Applied and computational complex analysis, Vol. II, Wiley, New York, 1991. MR 93b:30001
  • 15. Hettich, R., Zencke, P., Numerische Methoden der Approximation und semi-infiniten Optimierung, Teubner, Stuttgart, 1982. MR 84a:90069
  • 16. Jones, W.B., Thron, W.F., On the computation of incomplete gamma functions in the complex domain, J. Comp. Appl. Math., 12 $\&$ 13 (1985), 401-417. MR 87c:65023
  • 17. Kövari, T., Pommerenke, Ch., On Faber polynomials and Faber expansions, Math. Z., 99 (1967), 193-206. MR 37:3013
  • 18. Lewin, B.J., Distribution of zeros of entire functions, Amer. Math. Soc., 1964.
  • 19. Lyness, J.N., Sande, G., Algorithm 413, Evaluation of normalized Taylor coefficients of an analytic function, Comm. ACM, 14 (1971), 669-675.
  • 20. Markushevich, A.I., The theory of functions, 2nd ed., Chelsea, New York, 1977. MR 56:3258
  • 21. Müller, J., Convergence acceleration of polynomial expansions for finite order entire functions, Habilitationsschrift, Trier 1995.
  • 22. Müller, J., Convergence acceleration of Taylor sections for finite order entire functions, submitted.
  • 23. Poppe; G.P.M., Wijers, C.M., More efficient computation of the complex error function, ACM Trans. Math. Softw., 16 (1990), 38-46. MR 91h:65068a
  • 24. Rice, J.R., The degree of convergence for entire functions, Duke Math. J., 38 (1971), 429-440. MR 44:4217
  • 25. Rivlin, T.J., The Chebyshev polynomials, Wiley, New York, 1974. MR 56:9142
  • 26. Schonfelder, J.L., Special functions in the NAG library. In: Software for numerical mathematics, D.J. Evans (Editor), Academic Press, New York, 1974. MR 50:6095
  • 27. Steinmetz, N., Exceptional values of solutions of linear differential equations, Math. Z., 201 (1989), 317-326. MR 90i:30044
  • 28. Watanabe, T., Natori, M., Oguni, T., Mathematical software for the P.C. and work stations, North-Holland, Amsterdam, 1994. MR 95g:65008
  • 29. Weideman, J.A.C., Computation of the complex error function, SIAM J. Numer. Anal., 31 (1994), 1497-1518. MR 95h:65012a
  • 30. Wild, P., Accelerating the convergence of power series of certain entire functions, Numer. Math., 51 (1987), 583-595. MR 89a:65007
  • 31. Winiarski, T., Approximation and interpolation of entire functions, Ann. Polon. Math. 23 (1970), 259-273. MR 42:7913

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65B99, 30D10

Retrieve articles in all journals with MSC (1991): 65B99, 30D10

Additional Information

Jürgen Müller
Affiliation: Fachbereich IV-Mathematik, Universität Trier, D-54286 Trier, Germany

Received by editor(s): October 16, 1995
Received by editor(s) in revised form: April 1, 1996
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society