The “global” convergence of Broyden-like methods with suitable line search

Abstract

Iterative methods for solving a square system of nonlinear equations g ( x ) = 0 often require that the sum of squares residual γ ( x ) ≡ ½∥ g ( x )∥ 2 be reduced at each step. Since the gradient of γ depends on the Jacobian ∇ g , this stabilization strategy is not easily implemented if only approximations B k to ∇ g are available. Therefore most quasi-Newton algorithms either include special updating steps or reset B k to a divided difference estimate of ∇ g whenever no satisfactory progress is made. Here the need for such back-up devices is avoided by a derivative-free line search in the range of g . Assuming that the B k are generated from an rbitrary B 0 by fixed scale updates, we establish superlinear convergence from within any compact level set of γ on which g has a differentiable inverse function g −1 .