Numerical mathematics.

*(English)*Zbl 0957.65001
Texts in Applied Mathematics. 37. New York, NY: Springer. xx, 654 p. (2000).

This is an excellent and modern textbook in numerical mathematics! It is primarily addressed to undergraduate students in mathematics, physics, computer science and engineering. But you will need a weekly 4 hour lecture for 3 terms lecture to teach all topics treated in this book! Well known methods as well as very new algorithms are given. The methods and their performances are demonstrated by illustrative examples and computer examples. Exercises shall help the reader to understand the theory and to apply it. MATLAB-software satisfies the need of user-friendliness. “The spread of numerical software presents an enrichment for the scientific community. However, the user has to make the correct choice of the method which best suits at hand. As a matter of fact, no black-box methods or algorithms exist that can effectively and accurately solve all kinds of problems.” All MATLAB-programs are available by internet.

The content of the textbook is organized into four parts and 13 chapters. In Part I the authors review basic linear algebra and introduce to consistency, stability and convergence of numerical methods and they explain basic elements of computer arithmetic. Part II is about numerical linear algebra, and it is devoted to the numerical solution of linear systems, eigenvalue and eigenvectors computation, results on sparse matrices are given.

Part III deals with rootfinding for nonlinear equations, nonlinear systems and numerical optimization, polynomial interpolation including approximation by splines, numerical integration including singular integrals and multidimensional numerical integration.

Part IV is concerned with approximation, integration and transforms based on orthogonal polynomials. There are solutions of initial value problems and boundary value problems for ordinary differential equations. The reviewer only misses the effective multiple shooting method. The final chapter of the book is devoted to the approximation of time-dependent partial differential equations. Parabolic and hyperbolic initial-boundary value problems will be addressed and either finite differences and finite elements are considered for their discretization.

There are a lot of numerical examples and impressing figures and very useful applications, as for instance: Regularization of a triangular grid, analysis of an electric network and of a nonlinear electrical circuit, finite difference analysis of beam bending, analysis of the buckling of a beam, free dynamic vibration of a bridge, analysis of the state equation for a real gas, solution of a nonlinear system arising from semiconductor device simulation, finite element analysis of a clamped beam, geometric reconstruction based on computer tomographies, computation of the wind action on a sailboat mast, numerical solution of blackbody radiation, compliance of arterial walls, lubrication of a slider, heat conduction in a bar, a hyperbolic model for blood flow interaction with arterial walls.

It is a joy to read the book, it is carefully written and well printed. Only in the references (more than 200) are some printing mistakes, and there are no references to modern German and Russian papers and textbooks.

In the reviewers opinion, the presented book is the best textbook in numerical mathematics edited in the last ten years.

The content of the textbook is organized into four parts and 13 chapters. In Part I the authors review basic linear algebra and introduce to consistency, stability and convergence of numerical methods and they explain basic elements of computer arithmetic. Part II is about numerical linear algebra, and it is devoted to the numerical solution of linear systems, eigenvalue and eigenvectors computation, results on sparse matrices are given.

Part III deals with rootfinding for nonlinear equations, nonlinear systems and numerical optimization, polynomial interpolation including approximation by splines, numerical integration including singular integrals and multidimensional numerical integration.

Part IV is concerned with approximation, integration and transforms based on orthogonal polynomials. There are solutions of initial value problems and boundary value problems for ordinary differential equations. The reviewer only misses the effective multiple shooting method. The final chapter of the book is devoted to the approximation of time-dependent partial differential equations. Parabolic and hyperbolic initial-boundary value problems will be addressed and either finite differences and finite elements are considered for their discretization.

There are a lot of numerical examples and impressing figures and very useful applications, as for instance: Regularization of a triangular grid, analysis of an electric network and of a nonlinear electrical circuit, finite difference analysis of beam bending, analysis of the buckling of a beam, free dynamic vibration of a bridge, analysis of the state equation for a real gas, solution of a nonlinear system arising from semiconductor device simulation, finite element analysis of a clamped beam, geometric reconstruction based on computer tomographies, computation of the wind action on a sailboat mast, numerical solution of blackbody radiation, compliance of arterial walls, lubrication of a slider, heat conduction in a bar, a hyperbolic model for blood flow interaction with arterial walls.

It is a joy to read the book, it is carefully written and well printed. Only in the references (more than 200) are some printing mistakes, and there are no references to modern German and Russian papers and textbooks.

In the reviewers opinion, the presented book is the best textbook in numerical mathematics edited in the last ten years.

Reviewer: Werner H.Schmidt (Greifswald)

##### MSC:

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

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |

65D32 | Numerical quadrature and cubature formulas |

65D05 | Numerical interpolation |

65D07 | Numerical computation using splines |

65F05 | Direct numerical methods for linear systems and matrix inversion |

65F10 | Iterative numerical methods for linear systems |

65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |

65F50 | Computational methods for sparse matrices |

65Gxx | Error analysis and interval analysis |

65Hxx | Nonlinear algebraic or transcendental equations |

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 |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

78A55 | Technical applications of optics and electromagnetic theory |

82D37 | Statistical mechanics of semiconductors |

92C35 | Physiological flow |

92C55 | Biomedical imaging and signal processing |