Adi Ben-Israel, Rutgers University

Abstract

The lecture has 3 sections:
(1) Given functions u,f:D → D ⊂ ℝ, if u(x)=1-f(x)/f'(x) for all x in D, we call f the inverse Newton transform of u, denoted f=N-1u. If 1/(x-u(x)) is integrable, then

(N-1u)(x)=C⋅exp{∫ dx/(x-u(x))}, C ≠0.

For such u, the iteration x+:=u(x) (away from its fixed points) is a Newton method on f, and the relations between (fixed points, monotonicity, of) u and (roots, convexity, of) f give a simple explanation of chaotic behavior, illustrated here for the logistic iteration (see citation in LaTeX document).
(2) A geometric interpretation of the complex Newton iteration, z+:=z-f(z)/f'(z), f analytic, (see citation in LaTeX document), allows extending the results of (1) to complex iterations. This is illustrated for the Mandelbrot set.
(3) An iterative method for minimizing a convex function f:ℝn→ℝ with an attained infimum, proceeds by bracketing the minimum value in nested, decreasing intervals. Each iteration consists of one Newton iteration, and the method has an advantage of fast convergence and a natural stopping criterion. This is illustrated for the Fermat--Weber location problem, (see citations in LaTeX document).