Journal of the
Korean Mathematical Society JKMS

In this paper, we propose a new rank two quasi-Newton method based on adjoint Broyden updates for solving symmetric nonlinear equations, which can be seen as a class of adjoint BFGS method. The new rank two quasi-Newton update not only can guarantee that $B_{k+1}$ approximates Jacobian $F'(x_{k+1})$ along direction $s_k$ exactly, but also shares some nice properties such as positive definiteness and least change property with BFGS method. Under suitable conditions, the proposed method converges globally and superlinearly. Some preliminary numerical results are reported to show that the proposed method is effective and competitive.

Keywords: adjoint Broyden update, quasi-Newton method, symmetric nonlinear equations, global convergence, superlinear convergence

MSC numbers: Primary 65K05, 65H10, 90C53