Continued fractions. Vol. 1: Convergence theory
The volume under review deals mainly with continued fractions having complex number partial numerators and denominators and offers interpretations of some special results of interest to the authors. In this general account there are heavy casualties. It is difficult, in a short review, to deal adequately with perceived shortcomings, but some specific matters may be mentioned. It is suggested (on p.~11 and on p.~260 et seq.) that to avoid instability in the use of forward recursion in the computation of convergents, systematic use should be made of backward recursion. It is true that instability can occur, but its effect can be monitored. A preliminary error analysis is given in Section~2.6 on p.~277 of a paper by the reviewer [\textit{P. Wynn}, Numer. Math. 1, 272--307, 308--320 (1959; Zbl 0092.05101)] to which the authors do refer. This instability is partnered by that present in the computation of continued fraction coefficients, e.g.\ in the use of the \(q\)-\(d\)-algorithm [\textit{H. Rutishauser}, Der Quotienten-Differenzen-Algorithmus, Mitt\ Inst. Angew. Math. Techn. Hochschule Zürich. No.~7. Birkhäuser Verlag (1957; Zbl 0077.11103)] which is simply not mentioned. Analyses of stability may be referred to a number of representative cases just as has recently been done with regard to convergence ([\textit{K. A. Driver} and \textit{D. S. Lubinsky}, Aequationes Math. 42, No.~1, 85--106 (1991; Zbl 0746.41021), Approximation theory, Proc. Conf. Kecskemet/Hung. 1990, Colloq. Math. Soc. Janos Bolyai 58, 221--239 (1991; Zbl 0788.41010), Aequationes Math. 45, No.~1, 1--23 (1993; Zbl 0767.41019); \textit{M. Prevost}, Appl. Numer. Math. 17, No.~4, 461--469 (1995; Zbl 0840.65004)]; these papers are again simply not mentioned). Convergence acceleration of continued fractions is dealt with, but may be carried out by the use of stable methods which are once again not mentioned (for a gracious Zentralblatt review, see [\textit{P. Wynn}, Rev. Franc. Traitement Inform., Chiffres 8, 23--62; Errata 156 (1965; Zbl 0132.36903)]. The selection of references may amuse the reader. One would expect, in a definitive encyclopaedia, references to original work to be made. But to Jacobi, Tschebyscheff, Kronecker, Frobenius, Markoff, Christoffel, \dots\ all of whom contributed to the theories of the continued fractions and approximating rational functions dealt with, no reference is made at all. Nevertheless, the name in its past or present form of one of the authors occurs thirty-four times in the reference list, that of the other occurring twenty-one times. The volume is one of a number of encyclopaedias, dealing in particular with mathematics, which have recently been made available to the interested public. They are on offer from Wikipedia (\url{http://www.wikipedia.com}), Planet Math (\url{http://planetmath.org/encyclopedia}), Citizendium (\url{http://www.citizendium.org}), the Cambridge University Press, World Scientific, soon from Google's Knoll (\url{http://knol.google.com}), from Springer (\url{http://eom.springer.de} for a work taken over from Kluwer) and so on. The standards are not uniformly high. Fortunately ultra-inexpensive electronically stored copies of many standard works are currently being made available and in this way standards are being maintained. [For a review of the first edition, published by North-Holland in 1992, see Zbl 0782.40001.]
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