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Journal Articles Numerical Linear Algebra with Applications Year : 2020

Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

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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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Dates and versions

hal-03088480 , version 1 (26-12-2020)

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