Brezinski, Claude; Redivo Zaglia, Michela Extrapolation methods theory and practice. (English) Zbl 0744.65004 Studies in Computational Mathematics. 2. Amsterdam etc.: North-Holland. ix, 464 p. with floppy disk (1991). 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 0111.13001)] of Romberg’s principle is not mentioned. Van Wijngaarden’s transformation [for an accessible account, see J. W. Daniel, Math. Comput. 23, 91–96 (1969; Zbl 0183.44101)] 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 0061.27109)] 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 0096.09502)] 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”. Reviewer: Peter Wynn (Mexico) Cited in 3 ReviewsCited in 228 Documents MSC: 65B05 Extrapolation to the limit, deferred corrections 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis 65F05 Direct numerical methods for linear systems and matrix inversion 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65L05 Numerical methods for initial value problems involving ordinary differential equations 65R10 Numerical methods for integral transforms 65D05 Numerical interpolation 65C99 Probabilistic methods, stochastic differential equations 65C05 Monte Carlo methods 65D32 Numerical quadrature and cubature formulas 65D25 Numerical differentiation 65B10 Numerical summation of series Keywords:extrapolation methods; sequence transformation; monograph; numerical examples; floppy disk; programs; convergence acceleration; continued fractions Citations:Zbl 0111.13001; Zbl 0183.44101; Zbl 0061.27109; Zbl 0096.09502; Zbl 0531.40002; Zbl 0228.65044; Zbl 0228.65042 × Cite Format Result Cite Review PDF Digital Library of Mathematical Functions: §3.9(iii) Aitken’s Δ²-Process ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods §3.9(iv) Shanks’ Transformation ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods §3.9(vi) Applications and Further Transformations ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods §3.9(v) Levin’s and Weniger’s Transformations ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods