On Solving Nonlinear Equations with Simple Singularities or Nearly Singular Solutions

Abstract

The paper begins with a review of the convergence theory for Newton’s method near simple and other regular singularities, followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy. Then several ways to accelerate the slow convergence of Newton’s method are described in some detail. These are: Richardson extrapolation, overrelaxation, bordering by a singularity condition and appending the usual linear model by a quadratic term. The last two approaches require limited second derivative information which can be obtained analytically, by differencing or through secant updating. Numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy.