Regulating Newton-Raphson's Method by Unconstrained Minimization of a Weighted Sum of Squared Residuals

Published

February 19, 2017

Author(s)

D M. Lorenzetti

Abstract

To control the Newton-Raphson solution of a nonlinear algebraic system, a descent method forces each step to reduce some norm of the residual errors. This prevents divergence, but risks stagnation in flat regions of the norm. Employing a family of cost functions -- here a weighted sum of squaredresiduals -- allows a solver, in principle, to avoid all minima except at the desired solution. Geometric arguments suggest several rules for choosing weights, and numerical experiments show these improve the global convergence of a representative method, provided the weights incorporate sufficient information from the residual models at each iteration. The technique extends to otherminimization based equation solvers, either directly, or through its interpretation as a means of automatically rescaling the residuals.

Citation

Siam Journal on Scientific Computing

Pub Type

Journals

Keywords

indefinite dogleg, Newton-Raphson, nonlinear system, scaling, trust region

Citation

Lorenzetti, D. (2017), Regulating Newton-Raphson's Method by Unconstrained Minimization of a Weighted Sum of Squared Residuals, Siam Journal on Scientific Computing (Accessed October 11, 2025)