Geodesic
Convergence acceleration of shifted $LR$ transformations for totally nonnegative Hessenberg matrices
Applications of Mathematics, Tome 65 (2020) no. 5, pp. 677-702

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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 $\epsilon >0$.
DOI : 10.21136/AM.2020.0378-19
Classification : 34B16, 34C25
Keywords: $LR$ transformation; totally nonnegative matrix; Newton shift; convergence rate 
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     author = {Fukuda, Akiko and Yamamoto, Yusaku and Iwasaki, Masashi and Ishiwata, Emiko and Nakamura, Yoshimasa},
     title = {Convergence acceleration of shifted $LR$ transformations for totally nonnegative {Hessenberg} matrices},
     journal = {Applications of Mathematics},
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     publisher = {mathdoc},
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Fukuda, Akiko; Yamamoto, Yusaku; Iwasaki, Masashi; Ishiwata, Emiko; Nakamura, Yoshimasa. Convergence acceleration of shifted $LR$ transformations for totally nonnegative Hessenberg matrices. Applications of Mathematics, Tome 65 (2020) no. 5, pp. 677-702. doi: 10.21136/AM.2020.0378-19

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