On Generalized Newton Algorithms: Quadratic Convergence, Path-Following and Error Analysis

Newton iteration is known (under some precise conditions) to converge quadratically to zeros of non-degenerate systems of polynomials. This and other properties may be used to obtain theorems on the global complexity of solving systems of polynomial equations (See Shub and Smale in [6]), using a model of computability over the reals. However, it is not practical (and not desirable) to actually compute Newton iteration exactly. In this paper, approximate Newton iteration is investigated for several generalizations of the Newton operator. Quadratic covergence theorems and a robustness theorem are extended to approximate Newton iteration, generalizing some of the results in [6]. The results here can be used to prove complexity theorems on path following algorithms for solving systems of polynomial equations, using a model of computation over the integers (Malajovich [3])