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Practical extrapolation methods. Theory and applications. (English) Zbl 1041.65001
Cambridge Monographs on Applied and Computational Mathematics 10. Cambridge: Cambridge University Press (ISBN 0-521-66159-5/hbk). xxii, 519 p. £ 70.00; \$ 95.00 (2003).

The book contains the following four parts:

I. The Richardson extrapolation method: the first generalization of the Richardson extrapolation process; the extrapolation method $GREP$; the D-transformation for infinite-range integrals; the d-transformation for infinite series and sequences; two recursive algorithms for $GREP$: W-algorithm for $GRE{P}^{\left(1\right)}$ and ${W}^{\left(m\right)}$-algorithm for $GRE{P}^{\left(m\right)}$ as well as an efficient algorithm, $EW$-algorithm, for a special case of an extension of $GREP$; the analytic study of $GRE{P}^{\left(1\right)}$; acceleration of convergence of power series by d-transformation; acceleration of convergence of Fourier and generalized Fourier series by the d-transformation.

II. Sequence transformations: Euler transformation; Aitken process; Lubkin W-transformation; Shanks transformation; Padé table; Levin and Sidi transformations; Brezinsky $\theta$-algorithm; the transformations of Overholt and Wimp; confluent forms of sequence transformations: confluent $\epsilon$- algorithm, confluent $\rho$- algorithm, confluent Overholt method, confluent ${D}^{\left(m\right)}$- transformations.

III. Applications of the extrapolation methods and sequence transformations: multidimensional numerical quadrature, ordinary differential equations, the computation of the inverse Laplace transforms, convergence acceleration of infinite products, ill-posed problems.

IV. This part contains a review of several concepts and results required by the previous chapters and a Fortran 77 implementation for ${d}^{\left(m\right)}$- transformation of real infinite series (via ${W}^{\left(m\right)}$-algorithm) and this implementation can be adapted to complex series.

The book is an excellent support for the theoretical and practical studies of the speed-up methods based on extrapolation. It is an useful book for mathematicians interested in this field of research, but it can be used successfully by computer scientists and engineers.

##### MSC:
 65B05 Extrapolation to the limit, deferred corrections 65-02 Research monographs (numerical analysis) 65B10 Summation of series (numerical analysis) 65D25 Numerical differentiation 65D32 Quadrature and cubature formulas (numerical methods) 65R10 Integral transforms (numerical methods) 44A10 Laplace transform 42A20 Convergence and absolute convergence of Fourier and trigonometric series 65T40 Trigonometric approximation and interpolation (numerical methods)