Beyond Newton: a new root-finding fixed-point iteration for nonlinear equations

Abstract

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often leading to slow convergence or even divergence. In this paper, we present a new class of methods that improve upon the classical Newton's method. The key idea behind the new approach is to develop a relatively simple multiplicative transformation of the original equations, which leads to a significant reduction in nonlinearities, thereby alleviating the limitations of the Newton's method. Based on this idea, we propose two novel classes of methods and present their application to several mathematical functions (real, complex, and vector). Across all examples, our numerical experiments suggest that the new methods converge for a significantly wider range of initial guesses with minimal increase in computational cost. Given the ubiquity of Newton's method, an improvement in its applicability and convergence is a significant step forward, and will reduce computation times several-folds across many disciplines. Additionally, this multiplicative transformation may improve other techniques where a linear approximation is used.