TY - JOUR

T1 - Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part II

T2 - Clenshaw-Lord type approximants

AU - Mason, John C

AU - Crampton, A.

PY - 2005/3

Y1 - 2005/3

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.

KW - Chebyshev series

KW - Chebyshev-Padé approximant

KW - Clenshaw-Lord approximant

KW - End-point singularities

KW - Laurent series

KW - Laurent-Padé approximant

UR - http://www.scopus.com/inward/record.url?scp=23944524636&partnerID=8YFLogxK

U2 - 10.1007/s11075-004-2855-2

DO - 10.1007/s11075-004-2855-2

M3 - Article

AN - SCOPUS:23944524636

VL - 38

SP - 19

EP - 29

JO - Numerical Algorithms

JF - Numerical Algorithms

SN - 1017-1398

IS - 1-3

ER -