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Laurent-Pade approximants to four kinds of Chebyshev polynomial expansion. Part 1. Maehly type approximants

Mason, John C. and Crampton, Andrew (2005) Laurent-Pade approximants to four kinds of Chebyshev polynomial expansion. Part 1. Maehly type approximants. Journal of Numerical Algorithms, 38 (1). pp. 3-18. ISSN 1572-9265

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    Abstract

    Laurent Padé–Chebyshev rational approximants, A m (z,z –1)/B n (z,z –1), whose Laurent series expansions match that of a given function f(z,z –1) up to as high a degree in z,z –1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z –1)B n (z,z –1) and A m (z,z –1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.

    Item Type: Article
    Additional Information: UoA 23 (Computer Science and Informatics) © Springer. Part of Springer Science+Business Media
    Subjects: Q Science > Q Science (General)
    Q Science > QA Mathematics > QA75 Electronic computers. Computer science
    Q Science > QA Mathematics
    Schools: School of Computing and Engineering
    School of Computing and Engineering > Automotive Engineering Research Group
    School of Computing and Engineering > Informatics Research Group
    School of Computing and Engineering > Informatics Research Group > Knowledge Engineering and Intelligent Interfaces
    School of Computing and Engineering > Informatics Research Group > Software Engineering Research Group
    School of Computing and Engineering > Informatics Research Group > XML, Database and Information Retrieval Research Group
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    References:

    [1] J.S.R. Chisholm and A.K. Common, Generalisations of Padé approximation for Chebyshev and Fourier
    ser[5] H.J. Maehly, Rational approximation for transcendental functions, in: Proc. of IFIP Conference, Butterworths
    (1960).
    [6] J.C. Mason, Some applications and drawbacks of Padé approximants, in: Approximation Theory and
    Applications, ed. Z. Ziegler (1982).
    [7] J.C. Mason and A Crampton, Laurent–Padé approximants to four kinds of Chebyshev polynomial
    expansions. Part II: Clenshaw–Lord type approximants, Numer. Algorithms 38 (2005) 19–29.
    [8] J.C. Mason and D.C. Handscomb, Chebyshev Polynomials – Theory and Applications (Chapman and
    Hall/CRC Press, London, 2003).ies, in: Proc. of Internat. Christoffel Symposium (1979) pp. 212–231.
    [2] C.W. Clenshaw and K. Lord, Rational approximations from Chebyshev series, in: Studies in Numerical
    Analysis, ed. B.K.P. Skaife (Academic Press, New York, 1974) pp. 95–113.
    [3] J.A. Fromme and M.A. Golberg, Convergence and stability of a collocation method for the generalised
    airfoil equation, Appl. Math. Comput. 8 (1981) 281–292.
    [4] W.B. Gragg and G.D. Johnson, The Laurent–Padé table, in: Information Processing (North-Holland,
    Amsterdam, 1974) pp. 632–637.

    Depositing User: Sara Taylor
    Date Deposited: 22 Jun 2007
    Last Modified: 10 Dec 2010 13:29
    URI: http://eprints.hud.ac.uk/id/eprint/238

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