Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part I. Maehly type approximants

J. C. Mason, A. Crampton

Research output: Contribution to journalArticle

7 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)3-18
Number of pages16
JournalNumerical Algorithms
Volume38
Issue number1-3
DOIs
Publication statusPublished - Mar 2005

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Chebyshev Polynomials
Chebyshev
Polynomials
Laurent Series
Coefficient
Chebyshev Series
Denominator
Linear systems
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Laurent Expansion
Series
Series Representation
Linear system of equations
Partial Sums
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Power series
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@article{24d2a6661a4f41f6a190c355f3e6fbed,
title = "Laurent-Pad{\'e} approximants to four kinds of Chebyshev polynomial expansions. Part I. Maehly type approximants",
abstract = "Laurent Pad{\'e}-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 Pad{\'e} 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, Pad{\'e}-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 Pad{\'e}-Chebyshev coefficients are similar to that for a traditional Pad{\'e} 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) Pad{\'e}-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 Pad{\'e}-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] Pad{\'e}-Chebyshev approximants of Clenshaw-Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.",
keywords = "Chebyshev series, Chebyshev-Pad{\'e} approximant, End-point singularities, Laurent series, Laurent-Pad{\'e} approximant, Maehly approximant",
author = "Mason, {J. C.} and A. Crampton",
year = "2005",
month = "3",
doi = "10.1007/BF02810612",
language = "English",
volume = "38",
pages = "3--18",
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}

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part I. Maehly type approximants. / Mason, J. C.; Crampton, A.

In: Numerical Algorithms, Vol. 38, No. 1-3, 03.2005, p. 3-18.

Research output: Contribution to journalArticle

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AU - Mason, J. C.

AU - Crampton, A.

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N2 - 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.

AB - 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.

KW - Chebyshev series

KW - Chebyshev-Padé approximant

KW - End-point singularities

KW - Laurent series

KW - Laurent-Padé approximant

KW - Maehly approximant

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