On a shifted \(LR\) transformation derived from the discrete hungry Toda equation. (English) Zbl 1311.65036
Summary: The discrete hungry Toda (dhToda) equation is known as an integrable system which is derived from the study of the numbered box and ball system. In the authors’ paper [A. Fukuda et al., Phys. Lett., A 375, No. 3, 303–308 (2011; Zbl 1241.37010)], we associate the dhToda equation with a sequence of \(LR\) transformations for a totally nonnegative (TN) matrix, and then, in another paper [A. Fukuda et al., Ann. Mat. Pura Appl. (4) 192, No. 3, 423–445 (2013; Zbl 1349.37064)], based on the dhToda equation, we design an algorithm for computing the eigenvalues of the TN matrix. In this paper, in order to accelerate the convergence speed, we first introduce the shift of origin into the \(LR\) transformations associated with the dhToda equation, then derive a recursion formula for generating the shifted \(LR\) transformations. We next present a shift strategy for avoiding the breakdown of the shifted \(LR\) transformations. We finally clarify the asymptotic convergence and show that the sequence of TN matrices generated by the shifted \(LR\) transformations converges to a lower triangular matrix, exposing the eigenvalues of the original TN matrix. The asymptotic convergence is also verified through some numerical examples.
MSC:
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37N30 | Dynamical systems in numerical analysis |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
Keywords:
discrete hungry Toda equation; \(LR\) transformation; shift of origin; matrix eigenvalues; totally nonnegative matrix; convergence acceleration; numerical exampleReferences:
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