Book Details

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This work is devoted to the fixed points of differentiable applications, to the zeros of non-linear systems and to Newton's method. It is aimed at masters students or preparing for the aggregation of mathematics and confirmed researchers. The first part is devoted to the method of successive approximations and confronts a “dynamical systems” point of view (Grobman-Hartman theorems, of the stable manifold) with examples resulting from numerical analysis. The second part of this work presents Newton's method and its most recent developments (Smale's alpha theory, under- or over-determined systems). It presents a new approach to this subject and a set of original results published for the first time in a French-language work. This is an advanced text on fixed points, zeros of nonlinear systems and the Newton method. Its first part, devoted to fixed points, includes the Grobman-Hartman and the stable manifold theorems. The second part describes the Newton method from a modern point of view: Smale's alpha theory, underdetermined and overdetermined systems of equations. These results are illustrated by various examples from numerical analysis.