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Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices
Author(s) -
Tavakolipour Hanieh,
Shakeri Fatemeh
Publication year - 2020
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2330
Subject(s) - mathematics , parameterized complexity , eigenvalues and eigenvectors , matrix (chemical analysis) , exponential growth , norm (philosophy) , moduli , combinatorics , pure mathematics , mathematical analysis , physics , materials science , quantum mechanics , political science , law , composite material
Summary 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.