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Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

Hanieh Tavakolipour 1 Fatemeh Shakeri 2
1 TROPICAL - TROPICAL
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : Let P(t) be an n × n (complex) exponentially parameterized pentadiagonal matrix. In this article, using a theorem of Akian, Bapat, and Gaubert, we present explicit formulas for asymptotics of the moduli of the eigenvalues of P(t) as t → ∞. Our approach is based on exploiting the relation with tropical algebra and the weighted digraphs of matrices. We prove that this asymptotics tends to a unique limit or two limits. Also, for n − 2 largest magnitude eigenvalues of P(t) we compute the asymptotics as n → ∞, in addition to t. When P(t) is also symmetric, these formulas allow us to compute the asymptotics of the 2‐norm condition number. The number of arithmetic operations involved, does not depend on n. We illustrate our results by some numerical tests.
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https://hal.inria.fr/hal-03088480
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Submitted on : Saturday, December 26, 2020 - 3:13:30 PM
Last modification on : Wednesday, November 3, 2021 - 9:54:46 AM

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Hanieh Tavakolipour, Fatemeh Shakeri. Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices. Numerical Linear Algebra with Applications, Wiley, 2020, 27 (6), ⟨10.1002/nla.2330⟩. ⟨hal-03088480⟩

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