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 Deteriorated HSS-like methods for the weighted Toeplitz least squares problem from image restoration Min-Li Zeng Keywords:Deteriorated HSS-like method, Matrix splitting iteration method, Convergence, Preconditioner, Krylov subspace method. Abstract: In this paper, we construct a deteriorated HSS-like (DHSS-like) iteration method for a class of large and sparse block two-by-two linear systems from image restoration. The detailed spectral properties and the quasi-optimal iteration parameters are investigated in detail. Because the DHSS-like iteration method naturally leads to a DHSS-like preconditioner, then we can use the circulant matrix to replace the Toeplitz matrix in the DHSS-like preconditioner approximately to obtain a circulant matrix-based DHSS-like (CDHSS-like) preconditioner. It is pointed out that the workload of the new preconditioned method is about $O(n\log n)$. Theoretically analysis shows that the eigenvalues of the CDHSS-like preconditioned matrix are clustered around 1. Implementations in linear systems from the image restoration problems are made to verify the correctness of the theoretical results and the efficiency of both the CDHSS-like iteration method and the CDHSS-like preconditioned method.