An extension of the theory of secant preconditioners

A theory of inexact Newton methods with secant preconditioners for solving large nonlinear systems of equations has been developed recently by Martيnez (Math. Comput., 1993). According to this theory, local and superlinear convergence with bounded work per iteration of the inexact Newton method is obtained if the first trial increment at each iteration is a suitable quasi-Newton step computed using least-change secant-update procedures. The Jacobian approximation is interpreted as a preconditioner of the iterative linear method. In this paper, we extend the theory in two ways. On the one hand, since in many iterative methods the true residual is not computed but the preconditioned residual is, we show how to stop the linear iteration using the preconditioned residual instead of the original one. On the other hand, we introduce damping parameters that modify the usual unitary secant step. Two natural damping parameters are introduced, one of them tries to reduce the true residual and the other one tries to reduce the preconditioned residual. We prove that the main results of the theory of secant preconditioners hold under these modifications.