Broyden’s Method for a Class of Problems Having Singular Jacobian at the Root

Abstract

For a class of nonlinear equations $F(x) = 0$, in $\mathbb{R}^n $, with the Jacobian $F'$ being singular at a root, $x^ * $, Broyden’s method is shown to yield a sequence that converges linearly to $x^ * $ if the initial guess is chosen in a special region. The asymptotic linear rate is ${{(\sqrt 5 - 1)} / 2}$.