论文标题
翻转多级toeplitz矩阵和某些预处理的渐近光谱
The asymptotic spectrum of flipped multilevel Toeplitz matrices and of certain preconditionings
论文作者
论文摘要
在这项工作中，我们对翻转的多级toeplitz序列进行光谱分析，即，我们研究$ \ {y _ _ {y _ {\ boldsymbol {\ boldsymbol {n}} t _ {\ boldsymbol $ t _ {\ boldsymbol {n}}（f）$是一个真实的，平方的多级toeplitz矩阵，该矩阵由l^1 in l^1的函数$ f \生成（[ - π，π]^d）$和$ y _ {\ boldsymbol {n}} $是$ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $。与我们为Unilevel翻转Toeplitz矩阵序列所显示的内容一致，渐近光谱由$ 2 \ times 2 $矩阵值函数确定，其特征值为$ \ pm | f | $。此外，我们通过涵盖多级toeplitz和循环预调节器的分析来表征某些预处理的多级toeplitz序列的特征值分布。最后，我们所有的发现都通过几个数值实验来说明。
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 $f\in L^1([-π,π]^d)$ 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.
