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 > High-Performance Intelligent Computing School of Computing and Engineering > High-Performance Intelligent Computing > Planning, Autonomy and Representation of Knowledge School of Computing and Engineering > High-Performance Intelligent Computing > Planning, Autonomy and Representation of Knowledge ?? tserg ?? School of Computing and Engineering > High-Performance Intelligent Computing > Information and Systems Engineering 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:	31 Mar 2018 12:16
URI:	http://eprints.hud.ac.uk/id/eprint/238

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