论文标题
快速的非热toeplitz特征值计算，连接矩阵算法和FDE近似矩阵
Fast non-Hermitian Toeplitz eigenvalue computations, joining matrix-less algorithms and FDE approximation matrices
论文作者
论文摘要
目前的工作专门用于toeplitz矩阵$ t_ {n}（a）的特征值渐近扩展，其生成函数$ a $是复杂的，并且在某一时刻具有功率奇异性。结果，$ t_ {n}（a）$是非热的，我们知道特征值计算是大尺寸的非热门设置中的一项非琐事任务。我们遵循Bogoya，Böttcher，Grudsky和Maximenko的工作，并推断出特征值的完整渐近扩张。之后，我们将无基质算法应用于Ekström，Furci，Garoni，Serra-Capizzano等人的工作精神来计算这些特征值。由于内部和极端特征值具有不同的渐近行为，因此我们独立地对其进行了研究，并结合了结果以产生高精度的全局数值和无基质算法。 数值结果非常精确，并且所提出的算法的计算成本与每个特征值所考虑的矩阵的大小无关，这意味着当计算所有频谱时，都意味着线性成本。从现实世界应用的角度来看，我们强调的是，所考虑的矩阵类包括源自分数扩散方程的数值近似所产生的矩阵。在最后的结论部分中，关于此事的简要讨论，很少提出开放问题。
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, Böttcher, 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 Ekström, 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.
