Numerical linear algebra and optimization. Vol. 1.

*(English)*Zbl 0714.65063
Redwood City, CA: Addison-Wesley Publishing Company. xvii, 426 p. DM 95.00 (1991).

The authors of this book are well-known researchers in numerical optimization methods. The book is intended for teaching purpose and can be considered as a thorough up-to-date course in numerical linear algebra and optimization. Volume 1 consists of eight chapters. After the Introduction, Chapter 2 is devoted to a brief background in linear algebra with concepts and definitions of direct relevance to subsequent chapters.

Chapter 3 deals with some of the fundamental issues that arise in analysing both the nature of mathematical problems and the effect of computation in finite precision. The authors emphasize that mathematically equivalent methods may differ dramatically in their sensitivity to rounding errors and that much of numerical analysis is directed to developing methods that reduce the undesirable consequences of finite precision calculation.

Chapter 4 contains the details about the methods for solving the linear equations system \(Ax=b\), with a nonsingular square matrix A. The basic idea of most of these methods is to find a triangular system with the same solution, that is easy to solve. It is shown that A can be reduced to upper-triangular form through either elimination or direct factorization. Then, the pivoting strategies, the different factorization methods and techniques for updating matrix factorization which have become an essential part of modern linear algebra and optimization are discussed.

Chapter 5 considers general linear compatible systems. These cases are solved by methods that ultimately rely on reduction to a related nonsingular system. It is shown how can be extended the techniques from Chapter 4 to transform a general matrix A to triangular rank-revealing form.

The case of incompatible system \(Ax=b\) is treated in Chapter 6. The linear least-squares problem makes the introduction in the optimization by asking for a vector x that provides the best fit between Ax and b.

Chapters 7 and 8 are devoted to linear programming. Chapter 7 presents background material on linear functions and linear constraints. The primary subject of Chapter 8 is the simplex method invented in 1947 by George B. Dantzig. A class of methods traditionally called nonsimplex but referred here as active-set methods for linear programming are also discussed. In their treatment of the subject, the authors try to emphasize the connection of the linear programming to the general optimization problems as well as to linear algebra.

The presentation of the material, in these two chapters, is introductory, theorems and proofs are worked in detail and the pace is leisurely.

All the presented algorithms in the book are illustrated with detailed examples to enable one to write one’s own computer code and to understand numerical methods. Volume 1 contains 221 exercises of varying difficulty, distributed approximately uniformly among the seven chapters of the volume. Many of these are intended to be solved using an interactive package for matrix computations.

Each chapter has a Further Reading section intended to provide more detailed coverage of relevant topics.

An index round out this useful publication.

In summary, this volume is well organized and almost self-contained. It certainly will be of great value both for teaching purposes as well as for anyone who works in the broad area of numerical methods and software.

Chapter 3 deals with some of the fundamental issues that arise in analysing both the nature of mathematical problems and the effect of computation in finite precision. The authors emphasize that mathematically equivalent methods may differ dramatically in their sensitivity to rounding errors and that much of numerical analysis is directed to developing methods that reduce the undesirable consequences of finite precision calculation.

Chapter 4 contains the details about the methods for solving the linear equations system \(Ax=b\), with a nonsingular square matrix A. The basic idea of most of these methods is to find a triangular system with the same solution, that is easy to solve. It is shown that A can be reduced to upper-triangular form through either elimination or direct factorization. Then, the pivoting strategies, the different factorization methods and techniques for updating matrix factorization which have become an essential part of modern linear algebra and optimization are discussed.

Chapter 5 considers general linear compatible systems. These cases are solved by methods that ultimately rely on reduction to a related nonsingular system. It is shown how can be extended the techniques from Chapter 4 to transform a general matrix A to triangular rank-revealing form.

The case of incompatible system \(Ax=b\) is treated in Chapter 6. The linear least-squares problem makes the introduction in the optimization by asking for a vector x that provides the best fit between Ax and b.

Chapters 7 and 8 are devoted to linear programming. Chapter 7 presents background material on linear functions and linear constraints. The primary subject of Chapter 8 is the simplex method invented in 1947 by George B. Dantzig. A class of methods traditionally called nonsimplex but referred here as active-set methods for linear programming are also discussed. In their treatment of the subject, the authors try to emphasize the connection of the linear programming to the general optimization problems as well as to linear algebra.

The presentation of the material, in these two chapters, is introductory, theorems and proofs are worked in detail and the pace is leisurely.

All the presented algorithms in the book are illustrated with detailed examples to enable one to write one’s own computer code and to understand numerical methods. Volume 1 contains 221 exercises of varying difficulty, distributed approximately uniformly among the seven chapters of the volume. Many of these are intended to be solved using an interactive package for matrix computations.

Each chapter has a Further Reading section intended to provide more detailed coverage of relevant topics.

An index round out this useful publication.

In summary, this volume is well organized and almost self-contained. It certainly will be of great value both for teaching purposes as well as for anyone who works in the broad area of numerical methods and software.

Reviewer: F.Luban

##### MSC:

65K05 | Numerical mathematical programming methods |

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

90C05 | Linear programming |

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

90-04 | Software, source code, etc. for problems pertaining to operations research and mathematical programming |

15-04 | Software, source code, etc. for problems pertaining to linear algebra |

65G50 | Roundoff error |