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Approximation of analytic functions: a method of enhanced convergence

Authors: Oscar P. Bruno and Fernando Reitich
Journal: Math. Comp. 63 (1994), 195-213
MSC: Primary 30B10; Secondary 41A21, 41A25
MathSciNet review: 1240654
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Abstract: We deal with a method of enhanced convergence for the approximation of analytic functions. This method introduces conformal transformations in the approximation problems, in order to help extract the values of a given analytic function from its Taylor expansion around a point. An instance of this method, based on the Euler transform, has long been known; recently we introduced more general versions of it in connection with certain problems in wave scattering. In §2 we present a general discussion of this approach.

As is known in the case of the Euler transform, conformal transformations can enlarge the region of convergence of power series and can enhance substantially the convergence rates inside the circles of convergence. We show that conformal maps can also produce a rather dramatic improvement in the conditioning of Padé approximation. This improvement, which we discuss theoretically for Stieltjes-type functions, is most notorious in cases of very poorly conditioned Padé problems. In many instances, an application of enhanced convergence in conjunction with Padé approximation leads to results which are many orders of magnitude more accurate than those obtained by either classical Padé approximants or the summation of a truncated enhanced series.

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  • [1] G. A. Baker, The theory and application of the Padé approximant method, Advances in Theoretical Physics, Vol. I (K. A. Brueckner, ed.), Academic Press, New York, 1965. MR 0187807 (32:5253)
  • [2] G. A. Baker, J. L. Gammel, and J. G. Wills, An investigation of applicability of the Padé approximant method, J. Math. Anal. Appl. 2 (1961), 405-418. MR 0130093 (23:B3125)
  • [3] G. A. Baker and P. Graves-Morris, Padé approximants. Part I: Basic theory, Addison-Wesley, Reading, MA, 1981. MR 635619 (83a:41009a)
  • [4] -, Padé approximants. Part II: Extensions and applications, Addison-Wesley, Reading, MA, 1981. MR 635620 (83a:41009b)
  • [5] C. Brezinski, Procedures for estimating the error in Padé approximation, Math. Comp. 53 (1965), 639-648. MR 979935 (90b:65034)
  • [6] O. P. Bruno and F. Reitich, Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), 317-340. MR 1200203 (94e:35131)
  • [7] -, Numerical solution of diffraction problems: a method of variation of boundaries, J. Opt. Soc. Amer. A 10 (1993), 1168-1175.
  • [8] S. Cabay and D. Choi, Algebraic computations of scaled Padé fractions, SIAM J. Comput. 15 (1986), 243-270. MR 822203 (87e:30004)
  • [9] A. Edrei, Sur les déterminants récurrents et les singularités d'une fonction donnée par son développement de Taylor, Compositio Math. 7 (1939), 20-88. MR 0001285 (1:210c)
  • [10] G. E. Forsythe and C. B. Moler, Computer solution of linear algebraic systems, Prentice-Hall, Englewood Cliffs, NJ, 1967. MR 0219223 (36:2306)
  • [11] W. Gautschi, Construction of Gauss-Christoffel quadrature formulas, Math. Comp. 22 (1968), 251-270. MR 0228171 (37:3755)
  • [12] -, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (1982), 289-317. MR 667829 (84e:65022)
  • [13] P. Graves-Morris, The numerical calculation of Padé approximants, Lecture Notes in Math., vol. 765 (L. Wuytack, ed.), Springer-Verlag, Berlin and New York, 1979, pp. 231-245. MR 561453 (81j:65038)
  • [14] S. Gustafson, Convergence accerleration on a general class of power series, Computing 21 (1978), 53-69. MR 619912 (83m:65005)
  • [15] -, On stable calculation of linear functionals, Math. Comp. 33 (1979), 694-704. MR 521283 (80b:65070)
  • [16] C. Isenberg, Moment calculations in lattice dynamics. I. fcc lattice with nearest-neighbor interactions, Phys. Rev. 132 (1963), 2427-2433.
  • [17] -, Expansion of the vibrational spectrum at low frequencies, Phys. Rev. 150 (1966), 712-719.
  • [18] Y. L. Luke, Mathematical functions and their approximations, Academic Press, New York, 1977. MR 0501762 (58:19039)
  • [19] -, Algorithms for the computation of mathematical functions, Academic Press, New York, 1977. MR 0494840 (58:13624)
  • [20] -, Computations of coefficients in the polynomials of Padé approximations by solving systems of linear equations, J. Comput. Appl. Math. 6 (1980), 213-218. MR 594164 (81m:65026)
  • [21] L. R. Mead and N. Papanicolaou, Maximum entropy in the problem of moments, J. Math. Phys. 25 (1984), 2404-2417. MR 751523 (86h:82010)
  • [22] P. M. Morse and H. Feshbach, Methods of theoretical physics, Vol. 2, McGraw-Hill, New York, 1953. MR 0059774 (15:583h)
  • [23] R. E. Scraton, A note on the summation of divergent power series, Proc. Cambridge Philos. Soc. 66 (1969), 109-114. MR 0244667 (39:5981)
  • [24] -, The practical use of the Euler transformation, BIT 29 (1989), 356-360. MR 997541 (90e:65068)
  • [25] J. M. Taylor, The condition of gram matrices and related problems, Proc. Roy. Soc. Edinburgh 25 (1978), 45-56. MR 529568 (83e:65035)

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Keywords: Power series, enhanced convergence, Padé approximation, conditioning
Article copyright: © Copyright 1994 American Mathematical Society

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