The asymptotic spectrum of flipped multilevel Toeplitz matrices and of certain preconditionings
The asymptotic spectrum of flipped multilevel Toeplitz matrices and of certain preconditionings
复制标题
DOI：
10.1137/20m1379666
复制
发表时间：
2020-11
期刊：
影响因子：
--
通讯作者：
M. Mazza;J. Pestana
中科院分区：
文献类型：
--
作者：
M. Mazza;J. Pestana
在这项工作中，我们进行了翻转多层Toeplitz序列的谱分析，即我们研究了$\{Y_{\boldsymbol{n}} (f)\}_{\boldsymbol{n}} (f)$的渐近谱行为，其中$T_{\boldsymbol{n}}(f)$是由函数$f\在L^1([-\pi,\pi]^d)$中生成的实的、平方的多层Toeplitz矩阵，$Y_{\boldsymbol{n}}$是交换矩阵，在主反对角线上有$1$s。根据我们对单水平翻转Toeplitz矩阵序列的证明，渐近谱由一个$2\ × 2$矩阵值函数确定，其特征值为$\pm |f|$。在此基础上，对若干预条件翻转多层Toeplitz序列的特征值分布进行了刻画，并对多层Toeplitz序列和循环预条件序列进行了分析。最后，通过几个数值实验说明了我们所有的发现。
In this work, we perform a spectral analysis of flipped multilevel Toeplitz sequences, i.e., we study the asymptotic spectral behaviour of $\{Y_{\boldsymbol{n}}T_{\boldsymbol{n}}(f)\}_{\boldsymbol{n}}$, where $T_{\boldsymbol{n}}(f)$ is a real, square multilevel Toeplitz matrix generated by a function $f\in L^1([-\pi,\pi]^d)$ and $Y_{\boldsymbol{n}}$ is the exchange matrix, which has $1$s on the main anti-diagonal. In line with what we have shown for unilevel flipped Toeplitz matrix sequences, the asymptotic spectrum is determined by a $2\times 2$ matrix-valued function whose eigenvalues are $\pm |f|$. Furthermore, we characterize the eigenvalue distribution of certain preconditioned flipped multilevel Toeplitz sequences with an analysis that covers both multilevel Toeplitz and circulant preconditioners. Finally, all our findings are illustrated by several numerical experiments.