We consider a regular splitting based on the Sherman-Morrison-Woodbury formula, which is especially effective with iterative methods for the numerical solution of large linear systems with matrices that are perturbations of circulant or block circulant matrices. Such linear systems occur usually in the numerical discretization of one-dimensional differential equations. We prove the convergence of the new iteration without any assumptions on the symmetry or diagonal-dominance of the matrix. An extension to 2-by-2 block matrices that occur in some saddle point problems is also presented. The new method converged very fast in all of the test cases we used. Due to its trivial implementation and complexity with nearly circulant matrices via the Fast Fourier Transform it can be useful in the numerical solution of various one-dimensional finite element and finite difference discretizations of differential equations. A comparison with Gauss-Seidel and GMRES methods is also presented.

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