On smoothness and invariance properties of the gauss-newton method

• Published in: Numerical functional analysis and optimization Vol. 14; no. 5-6; pp. 503 - 514
• Main Author: Beyn, W. -J.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Marcel Dekker, Inc 01.01.1993; Taylor & Francis
• Subjects: AMS(MOS) subject classification 65H10; Exact sciences and technology; foliations; Gauss-Newton method; invariant manifolds; Mathematics; Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; parametrized equations; Sciences and techniques of general use; Non linear equation; Invariant manifold; Foliation; Gauss Newton method; Equation system
• ISSN: 0163-0563; 1532-2467
• DOI: 10.1080/01630569308816536

We consider systems of m nonlinear equations in m + p unknowns which have p-dimensional solution manifolds. It is well-known that the Gauss-Newton method converges locally and quadratically to regular points on this manifold. We investigate in detail the mapping which transfers the starting point to its limit on the manifold. This mapping is shown to be smooth of one order less than the given system. Moreover, we find that the Gauss-Newton method induces a foliation of the neighborhood of the manifold into smooth submanifolds. These submanifolds are of dimension m, they are invariant under the Gauss-Newton iteration, and they have orthogonal intersections with the solution manifold.