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

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