*(English)*Zbl 1041.65001

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) |