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.

翻译：我们考虑基于Sherman-Morrison-Woodbury公式的正则分裂，该方法特别适用于求解大型线性系统的迭代方法，此类系统的矩阵是循环矩阵或块循环矩阵的扰动形式。这类线性系统通常出现在一维微分方程的数值离散中。我们证明了新迭代在矩阵无需对称或对角占优假设下的收敛性。该方法还扩展至某些鞍点问题中出现的2×2块矩阵。在所有测试案例中，新方法均实现了快速收敛。由于其实现简单，且可通过快速傅里叶变换处理近似循环矩阵，该方法可用于求解多种一维微分方程的有限元与有限差分数值离散问题。文中还给出了与Gauss-Seidel和GMRES方法的比较。