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.

中文翻译：

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指数参数化五对角矩阵特征值的渐近性

令P（t）为n  ×  n（复数）指数参数化的五对角矩阵。在本文中，使用Akian，Bapat，和Gaubert的定理，我们提出明确的公式的本征值的模的渐近P（吨）作为吨 →交通 ∞。我们的方法基于利用与热带代数和矩阵加权有向图的关系。我们证明这种渐近趋向于一个唯一的极限或两个极限。此外，对于P（t）的n  -2个最大量级特征值，我们计算出的渐近性为n  →  ∞，除了t。当P（t）也对称时，这些公式使我们能够计算2范数条件数的渐近性。涉及的算术运算数不取决于n。我们通过一些数值测试来说明我们的结果。