Numerical methods in scientific computing. Vol. 1.

*(English)*Zbl 1153.65001
Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898716-44-3/hbk; 978-0-89871-778-5/ebook). xxvii, 717 p. (2008).

In 1974, the book “Numerical Methods” was published by the present authors in the Prentice-Hall series in Automatic Computation, edited by George Forsythe. It was one of the most successful books in numerical mathematics and as late as 2003 it was reprinted by Dover Publications [Zbl 1029.65002].

The present 717 page book is the first part (Volume I) of an heroic effort to write an updated version of the 1974 book to take into account the rapid development of the field since that time. Though Germund Dahlquist tragically died in 2005, the book was then already at a stage where the other author, Åke Björck, could finish it. We can expect that it will be an important reference work for many years.

Chapter 1, ”Principles of Numerical Calculations”, is written in a spirit somewhat different from the other 5 Chapters. It gives a broad introduction to the most common problems in numerical mathematics and basic ideas about how to solve them. This includes topics such as power series expansions, sparse matrices, the singular value decomposition, pseudorandom numbers, ordinary differential equations and Monte Carlo methods. The Chapter is written in a very lucid, pleasant style but the reader should be aware that this is only an introduction. For example, least squares are mentioned but no QR-decomposition. The reader is warned, of course, that a substantial treatment of numerical linear algebra and differential equations is deferred to later volumes.

Chapter 2, ”How to obtain and estimate accuracy” is an in-depth treatment of error theory, floating-point arithmetic, condition numbers, stability and backward stability of algorithms. It contains also discussions on less common topics such as multiple precision arithmetic, statistical methods for rounding errors, interval arithmetic and interval matrix computations.

The remaining Chapters 3,4,5 and 6 thoroughly treat well-defined subfields of numerical mathematics in the best tradition, i.e. from a rigorous mathematical viewpoint but always aimed to algorithmic, i.e. computer implementation.

Chapter 3, ”Series, Operators, and Continued Fractions” discusses power series, the Cauchy fast Fourier transform (FFT)-method, difference operators, acceleration of convergence but also divergent or semiconvergent series, continued fractions and Padé approximants.

Chapter 4, ”Interpolation and Approximation” is an encyclopaedic-style treatment of this subject with a stong emphasis on piecewise polynomial approximation, Fourier methods and FFT. Obvious omissions are wavelets and radial basis functions; the authors exlicitly note in their Preface that the emphasis is on traditional and well-developed methods.

Chapter 5, ”Numerical Integration” contains the usual material on this topic but also multidimensional integration, Monte Carlo methods and lattice methods.

Chapter 6, ”Solving scalar nonlinear equations” is an unusually thorough treatment of a subject that is usually supposed to be well-understood. Apart from the classical bisection method, method of false position, the secant method and their hybrids (which are discussed in a very lucid style), it also discusses Newton’s method and an interval Newton method. A separate section is devoted to the problem of finding a minimum of a function and another to algebraic equations. The book is well-illustrated and prepared carefully with respect to Lists of Figures, Tables, Conventions, the Bibliography and the Index. It further contains a wealth of Examples, Review Questions, Problems and Computer Exercises, examples of Matlab code, Notes and references, and interesting biographical notes.

In addition, a supplementary Web page contains three appendices. The first is an introduction to matrix computations and the second contains documentation and m-files of Mulprec, the multiple precision Matlab package developed by Germund Dahlquist. The third is a guide to literature, algorithms and software in numerical analysis.

The book will obviously be a reference work for researchers in numerical mathematics and for those who use scientific computing in applied sciences. It should be available in every scientific library. Parts of it can certainly be selected for various courses in numerical mathematics but the teacher will have to make choices, since the material presented in the book is overwhelming.

The present 717 page book is the first part (Volume I) of an heroic effort to write an updated version of the 1974 book to take into account the rapid development of the field since that time. Though Germund Dahlquist tragically died in 2005, the book was then already at a stage where the other author, Åke Björck, could finish it. We can expect that it will be an important reference work for many years.

Chapter 1, ”Principles of Numerical Calculations”, is written in a spirit somewhat different from the other 5 Chapters. It gives a broad introduction to the most common problems in numerical mathematics and basic ideas about how to solve them. This includes topics such as power series expansions, sparse matrices, the singular value decomposition, pseudorandom numbers, ordinary differential equations and Monte Carlo methods. The Chapter is written in a very lucid, pleasant style but the reader should be aware that this is only an introduction. For example, least squares are mentioned but no QR-decomposition. The reader is warned, of course, that a substantial treatment of numerical linear algebra and differential equations is deferred to later volumes.

Chapter 2, ”How to obtain and estimate accuracy” is an in-depth treatment of error theory, floating-point arithmetic, condition numbers, stability and backward stability of algorithms. It contains also discussions on less common topics such as multiple precision arithmetic, statistical methods for rounding errors, interval arithmetic and interval matrix computations.

The remaining Chapters 3,4,5 and 6 thoroughly treat well-defined subfields of numerical mathematics in the best tradition, i.e. from a rigorous mathematical viewpoint but always aimed to algorithmic, i.e. computer implementation.

Chapter 3, ”Series, Operators, and Continued Fractions” discusses power series, the Cauchy fast Fourier transform (FFT)-method, difference operators, acceleration of convergence but also divergent or semiconvergent series, continued fractions and Padé approximants.

Chapter 4, ”Interpolation and Approximation” is an encyclopaedic-style treatment of this subject with a stong emphasis on piecewise polynomial approximation, Fourier methods and FFT. Obvious omissions are wavelets and radial basis functions; the authors exlicitly note in their Preface that the emphasis is on traditional and well-developed methods.

Chapter 5, ”Numerical Integration” contains the usual material on this topic but also multidimensional integration, Monte Carlo methods and lattice methods.

Chapter 6, ”Solving scalar nonlinear equations” is an unusually thorough treatment of a subject that is usually supposed to be well-understood. Apart from the classical bisection method, method of false position, the secant method and their hybrids (which are discussed in a very lucid style), it also discusses Newton’s method and an interval Newton method. A separate section is devoted to the problem of finding a minimum of a function and another to algebraic equations. The book is well-illustrated and prepared carefully with respect to Lists of Figures, Tables, Conventions, the Bibliography and the Index. It further contains a wealth of Examples, Review Questions, Problems and Computer Exercises, examples of Matlab code, Notes and references, and interesting biographical notes.

In addition, a supplementary Web page contains three appendices. The first is an introduction to matrix computations and the second contains documentation and m-files of Mulprec, the multiple precision Matlab package developed by Germund Dahlquist. The third is a guide to literature, algorithms and software in numerical analysis.

The book will obviously be a reference work for researchers in numerical mathematics and for those who use scientific computing in applied sciences. It should be available in every scientific library. Parts of it can certainly be selected for various courses in numerical mathematics but the teacher will have to make choices, since the material presented in the book is overwhelming.

Reviewer: Willy Govaerts (Gent)

##### MSC:

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

65H05 | Numerical computation of solutions to single equations |

65Q05 | Numerical methods for functional equations (MSC2000) |

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

65Gxx | Error analysis and interval analysis |

65Fxx | Numerical linear algebra |

65C05 | Monte Carlo methods |

65C10 | Random number generation in numerical analysis |

65B05 | Extrapolation to the limit, deferred corrections |

65T50 | Numerical methods for discrete and fast Fourier transforms |

65Hxx | Nonlinear algebraic or transcendental equations |

65Y15 | Packaged methods for numerical algorithms |