##
**Fixed points, zeros and Newton’s method.
(Points fixes, zéros et la méthode de Newton.)**
*(English)*
Zbl 1095.65047

Mathématiques & Applications (Berlin) 54. Berlin: Springer (ISBN 3-540-30995-0/pbk). xii, 196 p. (2006).

In the introductory Chapter 1 of the book under review, the author states that the book has its origin in a university course about methods of solving systems of nonlinear equations; but, while making up the redaction of that course more topics have been added, and so the final result is a book containing much more than the things that he had originally in mind. Hence, the author has produced a book of 196 pages, structured as follows: Chapter 1: Introduction (pp. 1–3); Chapter 2: Fixed points (pp. 5–74); Chapter 3: Newton’s method (pp. 75–110); Chapter 4: Newton’s method for underdetermined systems (pp. 111–144); Chapter 5: The method of Newton-Gauss for overdetermined systems (pp. 145–175); Chapter 6: Appendices (pp. 177–190). The book ends with a list of 57 references (pp. 191–193) and an index (pp. 195–196).

Returning to the introductory Chapter 1, the author states that the book is intended for students doing higher level studies in mathematics, and for researchers. What is needed for reading it, he says, is a good knowledge of linear algebra, general topology and differential calculus, and some knowledge of functional analysis and of complex variables. The potential reader should also know that the two general frames in which the results are stated are either Banach spaces or Hilbert spaces (a few results are stated in complete metric spaces). In order to make the book in some sense “self-contained”, the author has added the appendices in Chapter 6, containing a.o. some definitions and results about differential calculus on Banach spaces, on Hilbert spaces and on Euclidean spaces.

The author describes the contents of the book as consisting of three parts. The first part is devoted to fixed points (chapter 2). By way of example, for differentiable maps on Banach spaces three categories of fixed points are considered: attractive, repulsive and hyperbolic fixed-points (connected to the theorem of Grobman-Hartman when the map is nonlinear); conntected to a fixed-point, we can consider stable and instable sets (going back to a theorem of Perron). Of those topics, the author describes results and gives proofs in a setting that may be more general than before; for this, he follows or adapts expositions from some of the more recent papers and books mentioned in the references. The second part consists of the chapters 3 and 4, and it treats the solution of systems of nonlinear equations by Newton’s method. In case we can speak of a “well-determined” system, the results are given for Banach spaces (chapter 3); on the other hand, chapter 4 treats the case of underdetermined systems using the notion of generalised inverse of a continuous linear operator (with closed range), and hence the presentation is given in Hilbert spaces. This last fact is also true for part three (chapter 5) of the paper, treating the case of overdetermined systems. The definition of generalised inverse and its immediate properties are given in Section 4.2; a continuation of properties for injective operateurs appears in Section 5.2.1. As for the contents of chapters 3,4 and 5 in which the so-called alpha-theory of S. Smale is presented the author states that at present this theory has for the most part not yet appeared in books of the same kind. As a matter of fact, the book, written in French, opens with an English ‘Preface’ by Steve Smale.

To the reviewer, the book is written in the more “classical” style of books treating a subject of Mathematical Analysis for students at an advanced level: after the statement of the proposition (or theorem), in most cases a proof is given, either in all extended form, or in a condensed form; in the other cases, the proof is left to the reader. Due to the treated matter, the number of drawings to get some better insight in some parts of the text is not very extended. The number of misprints seems to be rather modest, but as in any book they are present; for instance, on page 180 in the section about ‘Differential calculus on Hilbert spaces’ in chapter 6, on two different places a result is explained “…by use of the representation theorem of Riesz…”. But this does not harm the presentation by the Springer-Verlag.

Returning to the introductory Chapter 1, the author states that the book is intended for students doing higher level studies in mathematics, and for researchers. What is needed for reading it, he says, is a good knowledge of linear algebra, general topology and differential calculus, and some knowledge of functional analysis and of complex variables. The potential reader should also know that the two general frames in which the results are stated are either Banach spaces or Hilbert spaces (a few results are stated in complete metric spaces). In order to make the book in some sense “self-contained”, the author has added the appendices in Chapter 6, containing a.o. some definitions and results about differential calculus on Banach spaces, on Hilbert spaces and on Euclidean spaces.

The author describes the contents of the book as consisting of three parts. The first part is devoted to fixed points (chapter 2). By way of example, for differentiable maps on Banach spaces three categories of fixed points are considered: attractive, repulsive and hyperbolic fixed-points (connected to the theorem of Grobman-Hartman when the map is nonlinear); conntected to a fixed-point, we can consider stable and instable sets (going back to a theorem of Perron). Of those topics, the author describes results and gives proofs in a setting that may be more general than before; for this, he follows or adapts expositions from some of the more recent papers and books mentioned in the references. The second part consists of the chapters 3 and 4, and it treats the solution of systems of nonlinear equations by Newton’s method. In case we can speak of a “well-determined” system, the results are given for Banach spaces (chapter 3); on the other hand, chapter 4 treats the case of underdetermined systems using the notion of generalised inverse of a continuous linear operator (with closed range), and hence the presentation is given in Hilbert spaces. This last fact is also true for part three (chapter 5) of the paper, treating the case of overdetermined systems. The definition of generalised inverse and its immediate properties are given in Section 4.2; a continuation of properties for injective operateurs appears in Section 5.2.1. As for the contents of chapters 3,4 and 5 in which the so-called alpha-theory of S. Smale is presented the author states that at present this theory has for the most part not yet appeared in books of the same kind. As a matter of fact, the book, written in French, opens with an English ‘Preface’ by Steve Smale.

To the reviewer, the book is written in the more “classical” style of books treating a subject of Mathematical Analysis for students at an advanced level: after the statement of the proposition (or theorem), in most cases a proof is given, either in all extended form, or in a condensed form; in the other cases, the proof is left to the reader. Due to the treated matter, the number of drawings to get some better insight in some parts of the text is not very extended. The number of misprints seems to be rather modest, but as in any book they are present; for instance, on page 180 in the section about ‘Differential calculus on Hilbert spaces’ in chapter 6, on two different places a result is explained “…by use of the representation theorem of Riesz…”. But this does not harm the presentation by the Springer-Verlag.

Reviewer: Gilbert Crombez (Ghent)

### MSC:

65J15 | Numerical solutions to equations with nonlinear operators |

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

37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |

49M25 | Discrete approximations in optimal control |

58C15 | Implicit function theorems; global Newton methods on manifolds |

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

47H10 | Fixed-point theorems |

47J25 | Iterative procedures involving nonlinear operators |