A treatment of breakdowns and near breakdowns in a reduction of a matrix to upper J -Hessenberg form and related topics - Archive ouverte HAL Access content directly
Journal Articles Journal of Computational and Applied Mathematics Year : 2019

A treatment of breakdowns and near breakdowns in a reduction of a matrix to upper J -Hessenberg form and related topics

Abstract

The reduction of a matrix to an upper J-Hessenberg form is a crucial step in the SR-algorithm (which is a QR-like algorithm), structure-preserving, for computing eigenvalues and vectors, for a class of structured matrices. This reduction may be handled via the algorithm JHESS or via the recent algorithm JHMSH and its variants. The main drawback of JHESS (or JHMSH) is that it may suffer from a fatal breakdown, causing a brutal stop of the computations and hence, the SR-algorithm does not run. JHESS may also encounter near-breakdowns, source of serious numerical instability. In this paper, we focus on these aspects. We first bring light on the necessary and sufficient condition for the existence of the SR-decomposition, which is intimately linked to J-Hessenberg reduction. Then we will derive a strategy for curing fatal breakdowns and also for treating near breakdowns. Hence, the J-Hessenberg form may be obtained. Numerical experiments are given, demonstrating the efficiency of our strategies to cure and treat break- downs or near breakdowns.
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Dates and versions

hal-03487006 , version 1 (20-12-2021)

Licence

Attribution - NonCommercial

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Ahmed Salam, Haithem Ben Kahla. A treatment of breakdowns and near breakdowns in a reduction of a matrix to upper J -Hessenberg form and related topics. Journal of Computational and Applied Mathematics, 2019, 359, pp.16 - 29. ⟨10.1016/j.cam.2019.03.011⟩. ⟨hal-03487006⟩
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