推广的$q$ -离散Toda方程在计算Hessenberg完全非负矩阵特征值中的应用
Application of the extended $q$-discrete Toda equation to computing eigenvalues of Hessenberg totally nonnegative matrices
摘要：
Toda方程是最著名的可积系统之一，它的时间离散化就是计算三对角矩阵特征值的双差（qd）算法的递推公式。Toda方程的一个扩展是q-Toda方程，它是通过用所谓的q-导数代替标准导数得到的，q-导数涉及一个参数q，使得0 < q < 1。在我们以前的论文中，我们表明，离散化的q-Toda方程也适用于计算三对角特征值。在本文中，我们考虑的q-离散Toda方程的另一个扩展，并找到一个应用程序计算特征值的Hessenberg完全非负（TN）矩阵，这是矩阵的所有未成年人是非负的。我们的方法有两个关键组成部分。首先，我们考虑扩展的q-离散方程从移位LR变换的角度来看，类似于离散Toda和它的q-类似的情况。其次，我们澄清的渐近收敛的离散时间趋于无穷的q离散Toda方程集中在TN性质。我们还提出了两个例子来验证收敛到Hessenberg TN特征值数值。
英文摘要：
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.
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