论文标题
计算半侵犯的准特征矩阵的特征值
Computing eigenvalues of semi-infinite quasi-Toeplitz matrices
论文作者
论文摘要
准toeplitz（QT）矩阵是$ a = t（a）+$ $ a = t（a）$的半偶然矩阵，其中$ t（a）$是带有条目$（t（a））_ {t（a））_ {i，j} = a_ ___ {j-i} $的toeplitz矩阵1 $，而$ e $是代表$ \ ell^2 $的紧凑型操作员的矩阵。如果$ a_k = 0 $ for $ k <-m $，而对于$ k> n $（给定$ m，n> 0 $，并且如果$ e $具有有限的nonzero条目，则矩阵$ a $是有限表示的。研究了有限代表QT矩阵的数值计算本特征的问题，即对$（λ，{\ bf v}）$，以至于$ a {\ bf v} =λ{\ bf v} $，带有$λ{\ bf v} $ v} =（v_j）_ {j \ in \ mathbb z^+} $，$ {\ bf v} \ ne 0 $，和$ \ sum_ {j = 1}^\ infty | v_j | v_j |^2 <\ infty $。结果表明，该问题被简化为有限的非线性特征值问题，即$ WU（λ）{\pmbβ} = 0 $，其中$ w $是常数矩阵，$ u $取决于$λ$，并且可以以A Vandermonde Matrix或canspanion Matrix或a vandermonde矩阵。分析了依赖于方程式$ \ det Wu（λ）= 0 $的牛顿方法的算法。数值实验显示了这种方法的有效性。该算法已包含在CQT-Toolbox [Numer。算法81（2019），第1期。 2，741--769]。
A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form $A=T(a)+E$ where $T(a)$ is the Toeplitz matrix with entries $(T(a))_{i,j}=a_{j-i}$, for $a_{j-i}\in\mathbb C$, $i,j\ge 1$, while $E$ is a matrix representing a compact operator in $\ell^2$. The matrix $A$ is finitely representable if $a_k=0$ for $k<-m$ and for $k>n$, given $m,n>0$, and if $E$ has a finite number of nonzero entries. The problem of numerically computing eigenpairs of a finitely representable QT matrix is investigated, i.e., pairs $(λ,{\bf v})$ such that $A{\bf v}=λ{\bf v}$, with $λ\in\mathbb C$, ${\bf v}=(v_j)_{j\in\mathbb Z^+}$, ${\bf v}\ne 0$, and $\sum_{j=1}^\infty |v_j|^2<\infty$. It is shown that the problem is reduced to a finite nonlinear eigenvalue problem of the kind $ WU(λ){\pmb β}=0$, where $W$ is a constant matrix and $U$ depends on $λ$ and can be given in terms of either a Vandermonde matrix or a companion matrix. Algorithms relying on Newton's method applied to the equation $\det WU(λ)=0$ are analyzed. Numerical experiments show the effectiveness of this approach. The algorithms have been included in the CQT-Toolbox [Numer. Algorithms 81 (2019), no. 2, 741--769].
