On nonlinear generalized conjugate gradient methods

where F (ξ) is a nonlinear operator from a real Euclidean space of dimension n or Hilbert space into itself. The Euclidean norm and corresponding inner product will be denoted by ‖·‖1 and (·, ·)1 respectively. A general different inner product with a weight function and the corresponding norm will be denoted by (·, ·)0 and ‖ · ‖ respectively. In the first part of this article (Sects. 2 and 3) we assume that the Jacobian of F (ξ) has symmetric parts uniformly positive definite. In the final part (Sect. 4) a method is presented where this assumption is not required. The Newton method coupled with direct linear system solvers is an efficient way to solve such nonlinear systems when the dimension of the Jacobian is small. When the Jacobian is large and sparse some kind of iterative method may be used. This can be a nonlinear iteration (for example functional iteration for contractive operators), or an inexact Newton method. In an inexact Newton the solution of the resulting linear systems is approximated by a linear iterative method. The following are typical steps in an inexact Newton method for solving this nonlinear system.