In this paper, we have considered fast solutions of the linear system arising from the fractional Sturm–Liouville problem, whose coefficient matrix contains the product of Toeplitz‐like matrices. Based on suitable circulant approximations of the related coefficient matrix, we establish a matching preconditioner of matrix‐free form. In theory, the spectrum of the preconditioned matrix is shown to cluster around [1/2, 1), which suggests the fast convergence speed of the proposed preconditioner within Krylov subspace acceleration. In addition, to reduce the computation time and storage, we consider an all‐at‐once discretized system and explore its low‐rank tensor structure and alternating iterative tensor algorithms. Numerical experiments are given to show the effectiveness of the proposed solution techniques compared with some existing methods.

中文翻译：

---

分数Sturm-Liouville方程的快速预处理迭代方法

在本文中，我们考虑了由分数Sturm–Liouville问题引起的线性系统的快速解，其系数矩阵包含类Toeplitz矩阵的乘积。基于相关系数矩阵的适当循环近似，我们建立了一个无矩阵形式的匹配前置条件。从理论上讲，预处理矩阵的光谱显示为聚集在[1/2，1）附近，这表明拟议的预处理器在Krylov子空间加速中的快速收敛速度。此外，为了减少计算时间和存储量，我们考虑了一次全离散化系统，并探讨了其低秩张量结构和交替迭代张量算法。数值实验表明，与现有方法相比，所提出的求解技术是有效的。