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Laurent-Pade approximants to four kinds of Chebyshev polynomial expansion. Part 2. Clenshaw-Lord type approximants

Mason, John C. and Crampton, Andrew (2005) Laurent-Pade approximants to four kinds of Chebyshev polynomial expansion. Part 2. Clenshaw-Lord type approximants. Journal of Numerical Algorithms, 38 (1). pp. 19-29. ISSN 1572-9265

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    Abstract

    Laurent–Padé (Chebyshev) rational approximants P m (w,w –1)/Q n (w,w –1) of Clenshaw–Lord type [2,1] are defined, such that the Laurent series of P m /Q n matches that of a given function f(w,w –1) up to terms of order w ±(m+n), based only on knowledge of the Laurent series coefficients of f up to terms in w ±(m+n). This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series of P m matches that of Q n f up to terms of order w ±(m+n), but based on knowledge of the series coefficients of f up to terms in w ±(m+2n). The Clenshaw–Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé–Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for all m0, n0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé–Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw–Lord type methods, thus validating the use of either.

    Item Type: Article
    Additional Information: © Springer 2005
    Uncontrolled Keywords: Chebyshev series - Laurent series - Laurent–Padé approximant - Chebyshev–Padé approximant - Clenshaw–Lord approximant - end-point singularities
    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 approximations for Chebyshev and
    Fourier series, 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] K.O. Geddes, Block structure of the Chebyshev–Padé table, SIAMJ. Numer. Anal. 18 (1981) 844–861.
    [4] W.B. Gragg and G.D. Johnson, The Laurent–Padé table, in: Information Processing (1974) pp. 632–
    637.
    [5] H.J. Maehly, Rational approximations for transcendental functions, in: Proc. of IFIP Congress (1960).
    [6] J.C. Mason and A. Crampton, Laurent–Padé approximants to four kinds of Chebyshev polynomial
    expansions. Part I: Maehly type approximants, Numer. Algorithms 38 (2005) 3–18.
    [7] J.C.Mason and D.C. Handscomb, Chebyshev Polynomials (Chapman and Hall/CRC Press, Boca Raton,
    FL, 2003).

    Depositing User: Sara Taylor
    Date Deposited: 22 Jun 2007
    Last Modified: 10 Dec 2010 13:29
    URI: http://eprints.hud.ac.uk/id/eprint/240

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