Spectral and singular value symbols are valuable tools to analyse the eigenvalue or singular value distributions of matrix-sequences in the Weyl sense. More recently, Generalized Locally Toeplitz (GLT) sequences of matrices have been introduced for the spectral/singular value study of the numerical approximations of differential operators in several contexts. As an example, such matrix-sequences stem from the large linear systems approximating Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), Integro Differential Equations (IDEs), using any discretization on reasonable grids via local methods, such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, Discontinuous Galerkin etc. Studying the asymptotic spectral behaviour of GLT sequences is useful in analysing classical techniques for the solution of the corresponding PDEs/FDEs/IDEs and in designing novel fast and efficient methods for the corresponding large linear systems or related large eigenvalue problems. The theory of GLT sequences, in combination with the results concerning the asymptotic spectral distribution of perturbed sequences of matrices, is one of the most powerful and successful tools for computing the spectral symbol . In this regard, it would be beneficial to design an automatic procedure to compute the spectral symbols of such matrix-sequences and Ahmed Ratnani partially pursued it. Here, in the case of one-dimensional and two-dimensional differential problems, we continue in this direction by proposing an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming that it is a GLT sequence satisfying mild conditions.

中文翻译：

---

GLT 序列和符号的自动计算

谱和奇异值符号是分析 Weyl 意义上的矩阵序列的特征值或奇异值分布的有用工具。最近，广义局部托普利茨 (GLT) 矩阵序列已被引入，用于多种情况下微分算子数值逼近的谱/奇异值研究。例如，此类矩阵序列源于近似偏微分方程 (PDE)、分数阶微分方程 (FDE)、积分微分方程 (IDE) 的大型线性系统，通过局部方法（例如有限差分）在合理网格上使用任何离散化、有限元、有限体积、等几何分析、不连续伽辽金等。研究 GLT 序列的渐近谱行为对于分析相应 PDE/FDE/IDE 求解的经典技术以及为相应的设计新的快速有效的方法非常有用大型线性系统或相关的大特征值问题。 GLT 序列理论与扰动矩阵序列的渐近谱分布结果相结合，是计算谱符号的最强大和最成功的工具之一。在这方面，设计一个自动程序来计算此类矩阵序列的频谱符号将是有益的，并且艾哈迈德·拉特纳尼（Ahmed Ratnani）部分追求了它。在这里，对于一维和二维微分问题，我们继续朝这个方向发展，提出了一个自动计算底层矩阵序列符号的过程，假设它是满足温和条件的 GLT 序列。