The well-known Asplund theorem states that the inverse of a (possibly one-sided) band matrix $A$ is a Green matrix. In accordance with quasiseparable theory, such a matrix admits a quasiseparable representation in its rank-structured part. Based on this idea, we derive algorithms that compute a quasiseparable representation of $A^{-1}$ with linear complexity. Many inversion algorithms for band matrices exist in the literature. However, algorithms based on a computation of the rank structure performed theoretically via the Asplund theorem appear for the first time in this paper. Numerical experiments confirm complexity estimates and offer insight into stability properties.

翻译：著名的Asplund定理指出，（可能为单侧的）带状矩阵$A$的逆矩阵为Green矩阵。根据拟可分理论，此类矩阵的秩结构部分具有拟可分表示。基于这一思想，我们推导出以线性复杂度计算$A^{-1}$拟可分表示的算法。文献中已存在多种带状矩阵求逆算法，但基于通过Asplund定理从理论上计算秩结构的算法首次出现在本文中。数值实验验证了复杂度估计，并揭示了稳定性特性。