The quotient-difference algorithm and the Padé table: an alternative form and a general continued fraction

Author:
J. H. McCabe

Journal:
Math. Comp. **41** (1983), 183-197

MSC:
Primary 30B70; Secondary 10A30, 10F20, 41A21

DOI:
https://doi.org/10.1090/S0025-5718-1983-0701633-3

MathSciNet review:
701633

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Abstract | References | Similar Articles | Additional Information

Abstract: The quotient-difference algorithm is applied to a given power series in a modified way, and various continued fractions provided by the algorithm are described in terms of their relationships with the Padé table for the power series. In particular a general continued fraction whose convergents form any chosen combination of horizontal or vertical connected sequences of Padé approximants is introduced.

**[1]**G. A. Baker, Jr., "The Padé approximant method and some related generalizations,"*The Padé Approximant in Theoretical Physics*(G. A. Baker,Jr. and J. L. Gammel, eds.), Academic Press, New York, 1970, pp. 1-39.**[2]**D. Bussonnais,*Tous les Algorithms de Calcul par Recurrence des Approximations de Padé d'une Serie. Construction des Fractions Continues Correspondantes*, Conference on Padé Approximation, Lille, 1978.**[3]**I. Gargantini and P. Henrici,*A continued fraction algorithm for the computation of higher transcendental functions in the complex plane*, Math. Comp.**21**(1967), 18–29. MR**0240950**, https://doi.org/10.1090/S0025-5718-1967-0240950-1**[4]**W. B. Gragg,*The Padé table and its relation to certain algorithms of numerical analysis*, SIAM Rev.**14**(1972), 1–16. MR**0305563**, https://doi.org/10.1137/1014001**[5]***Mathematical methods for digital computers. Vol. II*, Edited by Anthony Ralston and Herbert S. Wilf, John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0211638****[6]**William B. Jones, W. J. Thron, and Haakon Waadeland,*A strong Stieltjes moment problem*, Trans. Amer. Math. Soc.**261**(1980), no. 2, 503–528. MR**580900**, https://doi.org/10.1090/S0002-9947-1980-0580900-4**[7]**J. H. McCabe,*A formal extension of the Padé table to include two point Padé quotionts*, J. Inst. Math. Appl.**15**(1975), 363–372. MR**0381246****[8]**J. H. McCabe,*Two-point Padé approximants and the quotient difference algorithm*, J. Comput. Appl. Math.**7**(1981), no. 2, 151–153. MR**636008**, https://doi.org/10.1016/0771-050X(81)90049-8**[9]**Heinz Rutishauser,*Eine Formel von Wronski und ihre Bedeutung für den Quotienten-Differenzen-Algorithmus*, Z. Angew. Math. Phys.**7**(1956), 164–169 (German). MR**0081546**, https://doi.org/10.1007/BF01600787**[10]**A. Sri Ranga,*The Strong Hamburger Moment Problem*, Internal Report, University of St. Andrews, 1982.**[11]**A. N. Stokes,*A stable quotient-difference algorithm*, Math. Comp.**34**(1980), no. 150, 515–519. MR**559199**, https://doi.org/10.1090/S0025-5718-1980-0559199-4

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0701633-3

Keywords:
Continued fractions,
Padé approximants

Article copyright:
© Copyright 1983
American Mathematical Society