This paper discusses the explicit inverse of a class of seven-diagonal (near) Toeplitz matrices, which arises in the numerical solutions of nonlinear fourth-order differential equation with a finite difference method. A non-recurrence explicit inverse formula is derived using the Sherman-Morrison formula. Related to the fixed-point iteration used to solve the differential equation, we show the positivity of the inverse matrix and construct an upper bound for the norms of the inverse matrix, which can be used to predict the convergence of the method.

翻译：本文件讨论七对角矩阵(近)的明显反向,它产生于非线性四级差分方程的数字解决方案中,使用一定的差分法。使用谢尔曼-莫里森公式得出了不重复的明显反向公式。与用于解决差分方程的固定点迭代有关,我们显示了反向矩阵的假设性,为反向矩阵的规范构造了一个上方界限,可用于预测方法的趋同性。