Relations Between Several Path Following Algorithms and Local and Global Newton Methods

This paper discusses certain common features of several algorithms for finding approximations to a zero of a mapping F:Rⁿ → Rⁿ. In particular, a recent simplicial pivot algorithm of the authors, Merrill’s algorithm, Newton’s method, and a “global Newton method” presented by Smale can be described in terms of the differential equations (or their discretizations) which determine the paths “followed by the algorithms.” This paper surveys the relations between these differential equations and presents simple proofs of several new results.