Numerical continuation methods. An introduction.

*(English)*Zbl 0717.65030
Springer Series in Computational Mathematics, 13. Berlin etc.: Springer- Verlag. xiv, 388 p. DM 128.00 (1990).

The book deals with two important techniques for determining numerical solutions to systems of nonlinear equations. These two techniques are the predictor-corrector continuation methods (PC methods) and the piecewise linear continuation methods (PL methods), as they are referred to by the authors. Basically, both techniques are methods for numerically following curves c(s), \(s\in J\) (J is some interval), implicitly defined by the equation \(H(c(s))=0,\) \(s\in J\), where H: \({\mathbb{R}}^{N+1}\to {\mathbb{R}}^ N\) is a given map. These classes include, as an important case, a series of homotopy algorithms for solving systems of nonlinear equations \(F(x)=0\) with F: \({\mathbb{R}}^ N\to {\mathbb{R}}^ N.\)

After a general introduction of the basic principles in chapters 1 and 2, various aspects of PC methods are presented in chapters 3-10, also including methods for calculating special points on the curve c(s), such as zero points \((f(c(s))=0\), necessary in connection with homotopy methods), extremal points (e.g. turning points) and bifurcation points. Chapters 12-16 treat the PL methods, also including a discussion of implicitly defined manifolds of dimension \(K\geq 1\). Chapter 11 deals with numerically implementable existence proofs and bridges the two techniques.

The algorithms presented in the book are given in the form of pseudo codes using a PASCAL syntax, which helps the reader to proceed to write programs. In the appendix, 5 FORTRAN programs are listed and the package SCOUT is described, together with numerical examples for illustration. The authors also offer these programs to be available for a limited time via electronic mail.

Throughout the book it will become evident to the reader that the two seemingly distinct techniques are closely related in a number of ways. On the other hand, the particular advantages for each of the two techniques are worked out.

The book is written in a carefully chosen notation, consequently used throughout the manuscript. It is an excellent book for both the beginner (with an adequate background in elementary analysis and elementary linear algebra) and the expert. Both will particularly appreciate the concise presentation and the rather extensive up-to-date bibliography, which helps to complete the information.

After a general introduction of the basic principles in chapters 1 and 2, various aspects of PC methods are presented in chapters 3-10, also including methods for calculating special points on the curve c(s), such as zero points \((f(c(s))=0\), necessary in connection with homotopy methods), extremal points (e.g. turning points) and bifurcation points. Chapters 12-16 treat the PL methods, also including a discussion of implicitly defined manifolds of dimension \(K\geq 1\). Chapter 11 deals with numerically implementable existence proofs and bridges the two techniques.

The algorithms presented in the book are given in the form of pseudo codes using a PASCAL syntax, which helps the reader to proceed to write programs. In the appendix, 5 FORTRAN programs are listed and the package SCOUT is described, together with numerical examples for illustration. The authors also offer these programs to be available for a limited time via electronic mail.

Throughout the book it will become evident to the reader that the two seemingly distinct techniques are closely related in a number of ways. On the other hand, the particular advantages for each of the two techniques are worked out.

The book is written in a carefully chosen notation, consequently used throughout the manuscript. It is an excellent book for both the beginner (with an adequate background in elementary analysis and elementary linear algebra) and the expert. Both will particularly appreciate the concise presentation and the rather extensive up-to-date bibliography, which helps to complete the information.

Reviewer: W.Zulehner

##### MSC:

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

65K05 | Numerical mathematical programming methods |

65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |

65H17 | Numerical solution of nonlinear eigenvalue and eigenvector problems |

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

90C30 | Nonlinear programming |