论文标题
扩展$ q $ -discrete TODA方程的应用到Hessenberg的计算特征值完全非负矩阵
Application of the extended $q$-discrete Toda equation to computing eigenvalues of Hessenberg totally nonnegative matrices
论文作者
论文摘要
TODA方程是最著名的集成系统之一，其时间差不多只是用于计算Tridiagonal矩阵特征值的商差异（QD）算法的递归公式。 TODA方程的扩展是Q-TODA方程，它是通过用所谓的Q衍生物替换涉及参数Q的Q衍生物来得出的，因此在上一篇论文中，我们表明Q-TODA方程的离散化也适用于计算Tridiaiaogonal eigenal eigeNal eigeNal的计算。在本文中，我们考虑了Q-Discrete TODA方程的另一个扩展，并找到了计算Hessenberg完全非负（TN）矩阵的应用程序，这些矩阵是所有未成年人都不存在的矩阵。我们的方法有两个关键组成部分。首先，我们从移动的LR转换的角度考虑了扩展的Q-污点方程，类似于离散的TODA及其Q-Analogue案例。其次，我们通过关注TN特性来阐明渐近收敛为离散时间在QDISCRETE TODA方程中的无穷大。我们还提出了两个示例，以数字验证收敛到Hessenberg TN特征值。
The Toda equation is one of the most famous integrable systems, and its time-discretization is simply the recursion formula of the quotient-difference (qd) algorithm for computing eigenvalues of tridiagonal matrices. An extension of the Toda equation is the q-Toda equation, which is derived by replacing standard derivatives with the so-called q-derivatives involving a parameter q such that 0 < q < 1. In our previous paper, we showed that a discretization of the q-Toda equation is shown to be also applicable to computing tridiagonal eigenvalues. In this paper, we consider another extension of the q-discrete Toda equation and find an application to computing eigenvalues of Hessenberg totally nonnegative (TN) matrices, which are matrices where all minors are nonnegative. There are two key components to our approach. First, we consider the extended q-discrete equation from the perspective of shifted LR transformations, similarly to the discrete Toda and its q-analogue cases. Second, we clarify asymptotic convergence as discrete-time goes to infinity in the qdiscrete Toda equation by focusing on TN properties. We also present two examples to numerically verify convergence to Hessenberg TN eigenvalues numerically.
