Summary
Nonlinear stationary fixed point iterations inR n are considered. The Perron-Ostrowski theorem [23] guarantees convergence if the iteration functionG possesses an isolated fixed pointu. In this paper a sufficient condition for convergence is given ifG possesses a manifold of fixed points.
As an application, convergence of a nonlinear extension of the method of Kaczmarz is proved. This method is applicable to underdetermined equations; it is appropriate for the numerical treatment of large and possibly ill-conditioned problems with a sparse, nonsquare Jacobian matrix. A practical example of this type (nonlinear image reconstruction in ultrasound tomography) is included.
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Meyn, KH. Solution of underdetermined nonlinear equations by stationary iteration methods. Numer. Math. 42, 161–172 (1983). https://doi.org/10.1007/BF01395309
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DOI: https://doi.org/10.1007/BF01395309