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Analyse numérique. Avec la collaboration de Claude Brezinski, Claude Carasso, Jean-Marc Chassery, Françoise Chatelin, Jean-François Maitre, Jean Roux, Gerhard Wanner. (French) Zbl 0757.65001

Collection Enseignement des Sciences 38. Paris: Hermann (ISBN 2-7056-6093-2/pbk). iii, 561 p. (1991).
This book – intended for second year students (in a French system of university education) – gives a survey of classical and more recent methods of numerical mathematics together with an insight into its philosophy.
The first part, due to J. Baranger, deals with the fundamental methods: basic methods of numerical algebra in an \(n\)-dimensional space (solution of linear and nonlinear systems), methods for the numerical solution of functional equations (this means problems for differential equations, also in variational formulation, up to parabolic equations of evolution), and methods to approximate functions (interpolation and norm approximation, including numerical integration).
A chapter on the difficulties arising in writing machine programs contains such items as machine arithmetics, stability, condition, errors and their propagation, all explained for concrete methods – an impressive exposition. For proofs frequently is referred to a former, smaller book of the author with the same title (1977; Zbl 0363.65001).
The second part, due to seven specialists, repeats these subjects and gives a deeper insight into remedies for special, more complicated situations: large linear and nonlinear systems ( gradient methods with preconditioning, too, resp. quasi-Newton methods), more issues of eigenvalues and eigenvectors (due to F. Chatelin), in the reviewer’s opinion a presentation with a highly interesting frame), approximation by splines and Padé approximations, fast Fourier transforms, and stiff differential equations.
Everywhere exercises and problems for programming are added. References (not everywhere up to date) and useful comments on them are found. Hence, this is a textbook sufficient to give an appropriate, authentic knowledge of numerical mathematics, especially (caused by the reasonably low amount of functional analysis) for those who are interested in applications (as students of science or engineering, too).
(However, what about the influence of computer algebra making headway on numerical mathematics, especially in applier’s point of view? There (on p. 142) is Forsythe’s famous example to solve \(ax^ 2+bx+c=0\) for extreme combinations of \(a\), \(b\), \(c\). The answer is: Do it with computer algebra – without problems.).

MSC:

65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65Fxx Numerical linear algebra
65Jxx Numerical analysis in abstract spaces
65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65Hxx Nonlinear algebraic or transcendental equations
65Nxx Numerical methods for partial differential equations, boundary value problems
65Dxx Numerical approximation and computational geometry (primarily algorithms)
65Lxx Numerical methods for ordinary differential equations
65Gxx Error analysis and interval analysis
65T50 Numerical methods for discrete and fast Fourier transforms
65Y05 Parallel numerical computation

Citations:

Zbl 0363.65001
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