Exact Formula for Solving a Degenerate System of Quadratic Equations

Abstract

The paper is devoted to the solution of a nonlinear system of equations \(F(x{{) = 0}_{n}}\), where \(F\) is a quadratic mapping acting from \({{\mathbb{R}}^{n}}\) to \({{\mathbb{R}}^{n}}\). The derivative \(F{\kern 1pt} '\) is assumed to be degenerate at the solution point, which is a major characteristic property of nonlinearity of the mapping. Based on constructions of the p-regularity theory, a 2-factor method is proposed for solving the system of equations, which converges at a quadratic rate. Moreover, an exact formula is obtained for solving this quadratic system of equations in the case of a 2-regular mapping \(F(x)\).