A Generalized Univariate Newton Method Motivated by Proximal Regularization

We devise a new generalized univariate Newton method for solving nonlinear equations, motivated by Bregman distances and proximal regularization of optimization problems. We prove quadratic convergence of the new method, a special instance of which is the classical Newton’s method. We illustrate the possible benefits of the new method over classical Newton’s method by means of test problems involving the Lambert W function, Kullback-Leibler distance and a polynomial. These test problems provide insight as to which instance of the generalized method could be chosen for a given nonlinear equation. Finally, we derive a closed-form expression for the asymptotic error constant of the generalized method and make further comparisons involving this constant.