A Formal Extension of the Padé Table to Include Two Point Padé Quotients

McCabe, John

doi:10.1093/imamat/15.3.363

Cited by 62 publications

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“…The proof of Case 1 follows from [8] while Cases 2 and 3 are simple consequences of a result in [7]. Case 4 then follows as a consequence of the other three cases, as is now shown.…”

Section: Two Further Continuedmentioning

confidence: 54%

“…These rational functions are discussed briefly in the next section. An example of such a continued fraction expansion is / \ -ax u 2m 3m a + x -a + I -u -a + 2 + u -a + 3 + wDetails of the properties of this expansion when p -2 are given in [7].…”

Section: Two Further Continuedmentioning

confidence: 99%

“…The Two-Point Padé Table. The alternative form of the quotient-difference algorithm (7) first arose out of a study of the continued fractions associated with the two-point Padé table, the points being the origin and the point at 00. See [7] for details. Briefly, suppose, in addition to the series (1), we also have the series *>0, + 1 + rf** + is a rational function which agrees with ir + k) terms of (1) and (r -k) terms of (30) when expanded accordingly.…”

Section: Two Further Continuedmentioning

confidence: 99%

“…All the previous continued fraction expansions relate to the two-point Padé table and have the described paths through it. The correspondence properties of these expansions with the two series follow from the structure of the two-point Padé table, described in [7].…”

Section: Two Further Continuedmentioning

confidence: 99%

“…Specifically The alternate version of the q-d algorithm, as given by Eqs. (7), was first obtained in a study of two-point Padé approximations by McCabe [7]. The rhombus rules (11) were also given by Bussonnais [2].…”

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

McCabe¹

1983

Math. Comp.

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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. Introduction. In [4] Gragg gives a substantial review of some of the properties of the Padé table for a given power series ?df=0ckxk and the relations between Padé approximants and other areas of numerical analysis. The earliest and, as Gragg suggests, perhaps the most significant of these related areas is the theory of continued fractions. In particular it is the so-called corresponding fractions for the power series that then provide the link between the Padé table and the quotient-difference algorithm of Rutishauser. One of the many applications of the quotientdifference algorithm is to obtain the coefficients of the corresponding fractions from those of the power series, and the convergents of these continued fractions form staircases in the Padé table for the series.In this paper the quotient-difference algorithm is taken in a form that is slightly different from the usual and is then used to provide, very simply, the coefficients of six types of continued fractions whose convergents form ordered sequences of Padé approximants, including those mentioned above. It is then shown that the table of coefficients generated by the revised quotient-difference algorithm also provide the partial numerators and denominators of a 'general' continued fraction whose convergents are a sequence of Padé approximants that is made up of any chosen horizontal or vertical connected subsequences. This freedom in choosing the path in the Padé table means that some of the existing algorithms based on particular paths can be regarded as special cases of the revised quotient-difference algorithm. Two such instances are described.The revised quotient-difference algorithm used in this paper is essentially the algorithm that has already been used to construct the continued fractions whose convergents form the so called two-point Padé table for two given power series, the only difference being the necessity for an additional rule for defining certain coefficients in the absence of a second series. However, by looking at the case where

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“…The proof of Case 1 follows from [8] while Cases 2 and 3 are simple consequences of a result in [7]. Case 4 then follows as a consequence of the other three cases, as is now shown.…”

Section: Two Further Continuedmentioning

confidence: 54%

Section: Two Further Continuedmentioning

confidence: 99%

Section: Two Further Continuedmentioning

confidence: 99%

Section: Two Further Continuedmentioning

confidence: 99%

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confidence: 99%

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