We design shifted LR transformations based on the integrable discrete hungry Toda equation to compute eigenvalues of totally nonnegative matrices of the banded Hessenberg form. The shifted LR transformation can be regarded as an extension of the extension employed in the well-known dqds algorithm for the symmetric tridiagonal eigenvalue problem. In this paper, we propose a new and effective shift strategy for the sequence of shifted LR transformations by considering the concept of the Newton shift. We show that the shifted LR transformations with the resulting shift strategy converge with order 2 − ε for arbitrary ε > 0.

中文翻译：

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完全非负 Hessenberg 矩阵移位 $LR$ 变换的收敛加速

我们设计了基于可积分离散饥饿 Toda 方程的移位 LR 变换来计算带状 Hessenberg 形式的完全非负矩阵的特征值。平移 LR 变换可以看作是著名的 dqds 算法中用于对称三对角特征值问题的扩展的扩展。在本文中，我们通过考虑牛顿移位的概念，为移位的 LR 变换序列提出了一种新的有效移位策略。我们表明，对于任意 ε > 0，带有结果移位策略的移位 LR 变换以 2 − ε 阶收敛。