  Recently, two families of HSS-based iteration methods are constructed for
solving the system of absolute value equations (AVEs), which is a class of
non-differentiable NP-hard problems. In this study, we establish the
Picard-CSCS iteration method and the nonlinear CSCS-like iteration method for
AVEs involving the Toeplitz matrix. Then, we analyze the convergence of the
Picard-CSCS iteration method for solving AVEs. By using the theory about
nonsmooth analysis, we particularly prove the convergence of the nonlinear
CSCS-like iterationsolver for AVEs. The advantage of these methods is that they
do not require the storage of coefficient matrices at all, and the sub-system
of linear equations can be solved efficiently via the fast Fourier transforms
(FFTs). Therefore, computational cost and storage can be saved in practical
implementations. Numerical examples including numerical solutions of nonlinear
fractional diffusion equations are reported to show the effectiveness of the
proposed methods in comparison with some existing methods.
