The present work is devoted to the eigenvalue asymptotic expansion of the Toeplitz matrix $T_{n}(a)$ whose generating function $a$ is complex valued and has a power singularity at one point. As a consequence, $T_{n}(a)$ is non-Hermitian and we know that the eigenvalue computation is a non-trivial task in the non-Hermitian setting for large sizes. We follow the work of Bogoya, B\"ottcher, Grudsky, and Maximenko and deduce a complete asymptotic expansion for the eigenvalues. After that, we apply matrix-less algorithms, in the spirit of the work by Ekstr\"om, Furci, Garoni, Serra-Capizzano et al, for computing those eigenvalues. Since the inner and extreme eigenvalues have different asymptotic behaviors, we worked on them independently, and combined the results to produce a high precision global numerical and matrix-less algorithm. The numerical results are very precise and the computational cost of the proposed algorithms is independent of the size of the considered matrices for each eigenvalue, which implies a linear cost when all the spectrum is computed. From the viewpoint of real world applications, we emphasize that the matrix class under consideration includes the matrices stemming from the numerical approximation of fractional diffusion equations. In the final conclusion section a concise discussion on the matter and few open problems are presented.
翻译:目前的工作致力于托普利茨矩阵 $T ⁇ n} (a) 美元,其产生功能美元的价值是复杂的,在某一点上具有超异性。 因此, $T ⁇ n} (a) 美元是非希腊文,我们知道,在非希腊文环境中,乙基值计算是一项非三重性任务,因为其规模很大。 我们跟踪博戈亚、博戈亚、博特切、格鲁德斯基、格鲁德斯基和马克西门科的工作,并推导出一个完全的乙基值的零星扩张。 之后,我们运用了无基值算法, 其精神是Ekstr\'om、 Furci、 Garoni、Serra- Capizzano et al等的工作精神, 计算这些乙基值。 由于内基值和极极值的最终行为不同, 我们独立地研究它们, 并将结果结合起来, 得出一个高精确的全球数字和无基数的等值扩展。 数字结果, 从每一类的直径分析角度看,, 算结果是精确和直径直径的矩阵的计算成本。