*(English)*Zbl 0744.65004

There is a spectrum of convergence acceleration methods. At one end the methods are exercises in simple analysis; they are the subject of much academic activity; they are not of much use. At the other, the methods derive from results of considerable profundity in the theories of power and function series and of continued fractions; they suggest advances, which are relatively difficult to obtain, in these subjects; their numerical behaviour is subject to control and they are powerful. The book gives a pleasantly written and equitable treatment of the broad spectrum of methods.

There are, of course, omissions. *H. Rutishauser’s* extension [Numer. Math. 5, 48-54 (1963; Zbl 0111.130)] of Romberg’s principle is not mentioned. Van Wijngaarden’s transformation [for an accessible account, see *J. W. Daniel*, Math. Comp. 23, 91-96 (1969; Zbl 0183.441)] which is possibly the most successful known method for the treatment of extremely slowly convergent monotonic sequences is not dealt with. It might have been suggested that *J. R. Schmidt’s* transformation [Philos. Mag., VII. Ser. 32, 369-383 (1941; Zbl 0061.271)] may be derived in a few lines from a result contained in a classic paper [*C. G. J. Jacobi*, J. Reine Angew. Math. 30, 127-156 (1845)] (not mentioned); in this way the theory of the transformation may be based upon the convergence theory of continued fractions which, apart from some recently published work of dubious utility, is not dealt with at all. Reference to the reviewer’s paper [Arch. Math. 11, 223-236 (1960; Zbl 0096.095)] which is of no practical use, might have been discarded in favour of another paper of the reviewer [Calcolo 15, No. 4, Suppl., 1-103 (1978; Zbl 0531.40002)] which contains many numerical examples and some convergence results.

To have repaired these omissions would have destroyed the balance of the book. The literature index is selective. For example, the first author’s paper [C. R. Acad. Sci., Paris, Sér. A 272, 145-148 (1971; Zbl 0228.65044)] is included but the slightly earlier and equivalent paper by *E. Gekeler* [Z. Angew. Math. Mech. 51, T53–T54 (1971; Zbl 0228.65042)] is not, and there are many similar examples. The book is perhaps more remarkable for this sort of selectivity than for any other reason.

As a minor point: in a later edition of this book, the numerous allusions (p. 152 et seq.) to “monotonous sequences” might be directed to “monotonic sequences”.

##### MSC:

65B05 | Extrapolation to the limit, deferred corrections |

65-02 | Research monographs (numerical analysis) |

65F05 | Direct methods for linear systems and matrix inversion (numerical linear algebra) |

65F15 | Eigenvalues, eigenvectors (numerical linear algebra) |

65L05 | Initial value problems for ODE (numerical methods) |

65R10 | Integral transforms (numerical methods) |

65D05 | Interpolation (numerical methods) |

65C99 | Probabilistic methods, simulation and stochastic differential equations (numerical analysis) |

65C05 | Monte Carlo methods |

65D32 | Quadrature and cubature formulas (numerical methods) |

65D25 | Numerical differentiation |

65B10 | Summation of series (numerical analysis) |