Convergence acceleration of shifted $LR$ transformations for totally nonnegative Hessenberg matrices
Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura · Applications of Mathematics · 2020
summary: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-\epsilon $ for arbitrary $\epsilon >0$.