Abstract
In this paper, we construct a new algorithm which combines the conjugate gradient and Lanczos methods for solving nonlinear systems. The iterative direction can be obtained by solving a quadratic model via conjugate gradient and Lanczos methods. Using the backtracking line search, we will find an acceptable trial step size along this direction which makes the objective function nonmonotonically decreasing and makes the norm of the step size monotonically increasing. Global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, we present some numerical results to illustrate the effectiveness of the proposed algorithm.
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