Comparison Results for the Brown Method with Convexity Hypotheses
by
J.P. Milaszewicz
Instituto Argentino de Matemática, Saavedra 15, 1083 Buenos Aires, Argentina

For a continuously differentiable function F, consider the associated system F(x)=0. If F is nonlinear, Newton's method is often the basic tool to solve it. In order to find an approximate solution, Brown has proposed a combination of the one-dimensional Newton method with a Gauss-Seidel-like extension to the nonlinear case of the Gaussian elimination process. Although quadratic convergence for Brown's analytic method was established, its original implementation showed a greater cost of algebraic operations per iteration, when comparing it with Newton's. Subsequently the same cost as Newton's was made available with another implementation of the method.

With the new implementation of the method, in the context of the Monotone Newton Theorem (MNT), Frommer has proved both a monotone Brown theorem, and that Brown's analytic method converges componentwise at least as fast as Newton's analytic method. He also extended these results for the corresponding Fourier iterations in both methods. Since comparison theorems hold for Newton's method in the MNT context, it is thus interesting to obtain analogous results for Brown's method, which is what we present here. A typical result reads that if two convenient initial points y and z are chosen so that z is bracketed coordinatewise between the root y* and y, then the Brown iterates generated by z are bracketed between y* and the corresponding Brown iterates generated by y. Similar results hold for the corresponding Fourier iterates.

Date received: May 24, 2001