Compact quasi-Newton preconditioners for symmetric positive definite linear systems

In this paper, preconditioners for the conjugate gradient method are studied to solve the Newton system with symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of Symmetric Rank one (SR1) and Broyden-Fletcher-Goldfarb-Shanno (BFGS) low-rank updates. We develop conditions under which the SR1 update maintains the preconditioner symmetric positive definite. Spectral analysis of the SR1 preconditioned Jacobians shows an improved eigenvalue distribution as the Newton iteration proceeds. A compact matrix formulation of the preconditioner update is developed which reduces the cost of its application and is more suitable to parallel implementation. Some notes on the implementation of the corresponding Inexact Newton method are given and some numerical results on a number of model problems illustrate the efficiency of the proposed preconditioners.