Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part II: Clenshaw-Lord type approximants

John C Mason, A. Crampton

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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 m ≥ 0, n ≥ 0. 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.

LanguageEnglish
Pages19-29
Number of pages11
JournalNumerical Algorithms
Volume38
Issue number1-3
DOIs
Publication statusPublished - Mar 2005

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Laurent Series
Padé Approximants
Chebyshev Polynomials
Chebyshev
Polynomials
Term
Linear equations
Coefficient
System of Linear Equations
Numerical Results
Series
Approximation
Knowledge

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title = "Laurent-Pad{\'e} approximants to four kinds of Chebyshev polynomial expansions. Part II: Clenshaw-Lord type approximants",
abstract = "Laurent-Pad{\'e} (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 Pad{\'e}-Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for all m ≥ 0, n ≥ 0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Pad{\'e}-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.",
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Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part II : Clenshaw-Lord type approximants. / Mason, John C; Crampton, A.

In: Numerical Algorithms, Vol. 38, No. 1-3, 03.2005, p. 19-29.

Research output: Contribution to journalArticle

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AU - Crampton, A.

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N2 - 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 m ≥ 0, n ≥ 0. 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.

AB - 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 m ≥ 0, n ≥ 0. 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.

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