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**Numerical methods for unconstrained optimization and nonlinear equations.**
*(English)*
Zbl 0579.65058

Prentice-Hall Series in Computational Mathematics. Englewood Cliffs, New Jersey: Prentice-Hall, Inc. XIII, 378 p. $ 48.55 (1983).

This textbook is concerned with the methods, algorithms and analysis of three important nonlinear problems: solving systems of nonlinear equations, unconstrained minimization of a nonlinear functional, and parameter selection by nonlinear least squares. It has the character of a self-contained introduction and is written primarily for students, engineers and scientists, who do not have the necessary mathematical background to go directly to the research literature. Though, for the lack of space, all the material of this research area is not covered in full generality, a sufficient background is provided to make it accessible to a wide scientific readership and it introduces several topics which are at the forefront of current research.

This book is organized as follows. The first four chapters serve as the background material for the further study. Chapter 1 provides an introduction to the subject in terms of simple examples and describes the typical characteristics of ”real-world” problems and introduces the features of finite-precision computer arithmetic. Chapter 2 develops some algorithms for nonlinear problems in one variable. Chapters 3 and 4 contain some relevant notions and results in numerical linear algebra and multivariable calculus. Chapter 5 discusses Newton’s method for nonlinear equations and unconstrained minimization and its computer implementation. Chapter 6 is devoted to the globally convergent modifications of Newton’s method. For proceeding when the Newton step is unsatisfactory the two major ideas are considered: the method of line searches and the model- trust region approach.

Chapter 7 describes the procedures of scaling, stopping and testing which are essential to the computer solution of actual problems. Chapters 8 and 9 deal with quasi-Newton methods like Broyden’s or Powell’s methods termed here as secant methods. Relying on certain results in previous chapters chapter 10 provides a very comprehensive study of algorithms for solving least squares problem. In this chapter two different approaches for nonlinear least squares are considered: (i) Gauss-Newton-type methods which ignore second derivative terms; (ii) Full Newton-type methods which take second derivative terms into account. Chapter 11 is devoted to more complicated computational aspects and in a way serves as a guide for the further research. It treats methods for solving problems with special structure including ones with the sparse Jacobian or Hessian.

The second part of the book (written by the second author) maintains the same high standards in the treatment of algorithms taken to the stage of implementation in the form of computer programs. Appendix A gives an account of a modular system of algorithms for unconstrained minimization and nonlinear equations. It describes programming considerations and options. Appendix B contains corresponding test problems. Appendices A and B can be used to create class projects. The book contains many figures and exercises. It has author and subject indices and a bibliography containing 134 titles.

The material of the book is a good blend of classical results and modern developments. Its clear and precise style makes the book easy to read. It is ideally suited for students, but it is an essential aid for engineers, computer scientists and mathematicians as well who are interested in numerical analysis and design computer programs.

This book is organized as follows. The first four chapters serve as the background material for the further study. Chapter 1 provides an introduction to the subject in terms of simple examples and describes the typical characteristics of ”real-world” problems and introduces the features of finite-precision computer arithmetic. Chapter 2 develops some algorithms for nonlinear problems in one variable. Chapters 3 and 4 contain some relevant notions and results in numerical linear algebra and multivariable calculus. Chapter 5 discusses Newton’s method for nonlinear equations and unconstrained minimization and its computer implementation. Chapter 6 is devoted to the globally convergent modifications of Newton’s method. For proceeding when the Newton step is unsatisfactory the two major ideas are considered: the method of line searches and the model- trust region approach.

Chapter 7 describes the procedures of scaling, stopping and testing which are essential to the computer solution of actual problems. Chapters 8 and 9 deal with quasi-Newton methods like Broyden’s or Powell’s methods termed here as secant methods. Relying on certain results in previous chapters chapter 10 provides a very comprehensive study of algorithms for solving least squares problem. In this chapter two different approaches for nonlinear least squares are considered: (i) Gauss-Newton-type methods which ignore second derivative terms; (ii) Full Newton-type methods which take second derivative terms into account. Chapter 11 is devoted to more complicated computational aspects and in a way serves as a guide for the further research. It treats methods for solving problems with special structure including ones with the sparse Jacobian or Hessian.

The second part of the book (written by the second author) maintains the same high standards in the treatment of algorithms taken to the stage of implementation in the form of computer programs. Appendix A gives an account of a modular system of algorithms for unconstrained minimization and nonlinear equations. It describes programming considerations and options. Appendix B contains corresponding test problems. Appendices A and B can be used to create class projects. The book contains many figures and exercises. It has author and subject indices and a bibliography containing 134 titles.

The material of the book is a good blend of classical results and modern developments. Its clear and precise style makes the book easy to read. It is ideally suited for students, but it is an essential aid for engineers, computer scientists and mathematicians as well who are interested in numerical analysis and design computer programs.

Reviewer: O.Vaarmann

### MSC:

65K05 | Numerical mathematical programming methods |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

90C30 | Nonlinear programming |

65H10 | Numerical computation of solutions to systems of equations |

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