Spectral properties of flipped Toeplitz matrices and related preconditioning
Spectral properties of flipped Toeplitz matrices and related preconditioning
复制标题
DOI：
10.1007/s10543-018-0740-y
复制
发表时间：
2018-12
影响因子：
1.5
通讯作者：
M. Mazza;J. Pestana
中科院分区：
文献类型：
--
作者：
M. Mazza;J. Pestana
本文研究了“翻转”Toeplitz序列的谱，即其渐近谱行为，其中是由函数生成的实Toeplitz矩阵，是主反对角线上有1的交换矩阵。我们证明了矩阵值函数的特征值是渐近描述的，其特征值函数为。结果表明，大约有一半的特征值可以通过均匀采样的方式很好地逼近，而其余的特征值可以通过均匀采样的方式很好地逼近。当仅在一组度量0上消失时，这激发了频谱实际上是一半正一半负的。对相关预条件序列的谱分布也提供了一些见解。最后，大量的数值结果说明了我们的理论发现。
In this work, we investigate the spectra of “flipped” Toeplitz sequences, i.e., the asymptotic spectral behaviour of, whereis a real Toeplitz matrix generated by a function, andis the exchange matrix, with 1s on the main anti-diagonal. We show that the eigenvalues ofare asymptotically described by amatrix-valued function, whose eigenvalue functions are. It turns out that roughly half of the eigenvalues ofare well approximated by a uniform sampling of |f| over, while the remaining are well approximated by a uniform sampling ofover the same interval. Whenfvanishes only on a set of measure zero, this motivates that the spectrum is virtually half positive and half negative. Some insights on the spectral distribution of related preconditioned sequences are provided as well. Finally, a wide number of numerical results illustrate our theoretical findings.