##
**Numerical methods. Textbook. (Chislennye metody. Uchebnoe posobie).**
*(Russian)*
Zbl 0638.65001

Moskva: “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. 600 p. R. 1.60 (1987).

The book is a very good and detailed textbook on numerical methods. It consists of eleven chapters, references and the index. The contents of chapters is briefly presented in the following.

Chapter one contains some general remarks on accuracy of results in numerical solutions of problems. In the second chapter the authors present a number of methods in interpolation and numerical differentiation. This chapter discusses, among others, the interpolation by Lagrange, Newton and Chebyshev polynomials, gives theorems on convergence of interpolation and differentiation formulas, and presents rational interpolation. Chapter three is devoted to numerical integration. Apart from the well-known integration formulas of Newton- Cotes and Gauss, the authors discuss the case of integration of nonregular function, present principles to achieve a high accuracy and algorithms with an automatic step-size correction. Moreover, there are a number of very interesting results to optimize the choice of integration nodes. The next two chapters are dealt with approximation of functions and multidimensional problems. The authors discuss in detail, among others, fast Fourier transformations, the Monte Carlo method, and give a lot useful, from practical point of view, theorems on convergence of the methods. Chapters six and seven present numerical methods of linear algebra and optimization problems. The next three chapters consider methods for solving the initial and boundary value problems for ordinary differential equations and methods for solving partial differential equations. The final chapter presents numerical methods for solving integral equations.

It is not enough place here to mention all the methods which the authors discuss in the book. But it should be noted that for all the methods the accuracy, efficiency, convergence and reliability are discussed in detail, clear and systematically. The book is meticulously organized and very well written. I am sure that this important book will be intended not only for graduate students and scientists in numerical analyis, but will be also useful for all other scientists who face the need to apply numerical methods in their practice.

Chapter one contains some general remarks on accuracy of results in numerical solutions of problems. In the second chapter the authors present a number of methods in interpolation and numerical differentiation. This chapter discusses, among others, the interpolation by Lagrange, Newton and Chebyshev polynomials, gives theorems on convergence of interpolation and differentiation formulas, and presents rational interpolation. Chapter three is devoted to numerical integration. Apart from the well-known integration formulas of Newton- Cotes and Gauss, the authors discuss the case of integration of nonregular function, present principles to achieve a high accuracy and algorithms with an automatic step-size correction. Moreover, there are a number of very interesting results to optimize the choice of integration nodes. The next two chapters are dealt with approximation of functions and multidimensional problems. The authors discuss in detail, among others, fast Fourier transformations, the Monte Carlo method, and give a lot useful, from practical point of view, theorems on convergence of the methods. Chapters six and seven present numerical methods of linear algebra and optimization problems. The next three chapters consider methods for solving the initial and boundary value problems for ordinary differential equations and methods for solving partial differential equations. The final chapter presents numerical methods for solving integral equations.

It is not enough place here to mention all the methods which the authors discuss in the book. But it should be noted that for all the methods the accuracy, efficiency, convergence and reliability are discussed in detail, clear and systematically. The book is meticulously organized and very well written. I am sure that this important book will be intended not only for graduate students and scientists in numerical analyis, but will be also useful for all other scientists who face the need to apply numerical methods in their practice.

Reviewer: A.Marciniak

### MSC:

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |

65C05 | Monte Carlo methods |

65Dxx | Numerical approximation and computational geometry (primarily algorithms) |

65T40 | Numerical methods for trigonometric approximation and interpolation |

65Fxx | Numerical linear algebra |

65Kxx | Numerical methods for mathematical programming, optimization and variational techniques |

65Lxx | Numerical methods for ordinary differential equations |

65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

65Nxx | Numerical methods for partial differential equations, boundary value problems |

65Rxx | Numerical methods for integral equations, integral transforms |