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Two CSCS-based iteration methods for solving absolute value equations
Keywords:Absolute value equation, CSCS-based iteration, Toeplitz matrix, Nonsmooth analysis, Fast Fourier transform.
      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 iteration solver 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 experiments including the solution of nonlinear fractional diffusion equations are presented to illustrate the effectiveness of the proposed methods in comparison with some existing methods.