TY - JOUR
T1 - Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part II
T2 - Clenshaw-Lord type approximants
AU - Mason, J. C.
AU - Crampton, A.
N1 - Publisher Copyright:
© 2005, Springer.
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/BF02810613
DO - 10.1007/BF02810613
M3 - Article
AN - SCOPUS:23944524636
VL - 38
SP - 19
EP - 29
JO - Numerical Algorithms
JF - Numerical Algorithms
SN - 1017-1398
IS - 1-3
ER -