Global aspects of the continuous and discrete Newton method: A case study
In this paper and the related paper of \textit{D. Saupe} [ibid. 13, No.1/2, 59-80 (1988; reviewed above)] the qualitative convergence properties of Newton's method, some damped versions of Newton's method, and the continuous Newton's method are studied via the Julia set theory. In particular, the various forms of Newton's method are applied to complex polynomials, boundary value problems, and their corresponding discretizations. The present paper being to some extend experimental in nature consists of two lengthy chapters. Chapter 2 is concerned with the study of the Newton flow \((I): \dot x=[DG(x(t))]^{-1}G(x(t)),\) \(x(0)=x_ 0\) for general maps and for maps including a bifurcation parameter. Chapter 3 deals with the qualitative performance of Newton's method when G is a standard discretization of a boundary value problem. It deals with questions concerning the phase portraits, the role of the singular set, and Julia-like sets. Chapter 3 contains many insightful computer graphic diagrams.
scientific article; zbMATH DE number 4096692
scientific article