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.