On the Convergence of the Brent Method

Abstract

In this paper, we establish the semi-local convergence theorem of the Brent method with regional estimation. By an in-depth investigation in to the algorithm structure of the method, we convert the Brent method into an approximate Newton method with a special error term. Based on such equivalent variation, under a similar condition of the Newton-Kantorovich theorem of the Newton method, we establish a semi-local convergence theorem of the Brent method. This theorem provides a sufficient theoretical basis for initial choices of the Brent method.