Iterative methods for nonlinear operator equations

摘要

A nonlinear conjugate gradient method has been introduced and analyzed by J.W. Daniel. This method applies to nonlinear operators with symmetric Jacobians. The conjugate gradient method applied to the normal equations can be used to approximate the solution of general nonsymmetric linear systems of equations if the condition of the coefficient matrix is small. In this article, we obtain nonlinear generalizations of this method which apply directly to nonlinear operator equations. Under conditions on the Hessian and the Jacobian of the operators, we prove that these methods converge to a unique solution. Error bounds and local convergence results are also obtained.