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Newton’s method

  • Tobias Damm
Chapter
Part of the Lecture Notes in Control and Information Science book series (LNCIS, volume 297)

Abstract

This chapter contains some of our main results. It largely follows the presentation in [49]. Our object is to tackle rational matrix equations of the form (2.27), derived in Chapter 2, which we view, very generally, as nonlinear operator equations in an ordered vector space. A standard method to solve nonlinear equations is the Newton iteration. Its applicability to operator equations in normed spaces was first established by Kantorovich in [122]. In the operator-theoretic setup the method is therefore often referred to as the Newton-Kantorovich procedure. Kantorovich specified boundedness conditions on the first and second Fréchet derivatives of the nonlinear operator, which guarantee convergence of the Newton iteration starting in a small neighbourhood of the actual solution. These results can be simplified and generalized, if the underlying space is ordered and the sequence produced by the iteration can be shown to be monotonic and bounded; this can be the case, for instance, if the nonlinear operator satisfies certain convexity conditions (compare [196] and references therein).

Keywords

Riccati Equation Newton Iteration Quadratic Convergence Large Solution Algebraic Riccati Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Authors and Affiliations

  • Tobias Damm
    • 1
  1. 1.Technische Universität Braunschweig Institut für Angewandte Mathematik 38023 BraunschweigGermany

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