On eigenvalues of a matrix arising in energy-preserving/dissipative continuous-stage Runge-Kutta methods

In this short note, we define an × matrix constructed from the Hilbert matrix and prove that it has at least one pair of complex eigenvalues when ≥ 2. is a matrix related to the AVF collocation method, which is an energy-preserving/dissipative numerical method for ordinary differential equations, an...

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Bibliographic Details
Published inSpecial matrices Vol. 10; no. 1; pp. 34 - 39
Main Author Yamamoto, Yusaku
Format Journal Article
LanguageEnglish
Published De Gruyter 01.01.2022
Subjects
continuous Runge-Kutta methods
eigenvalue
energy-preserving
Hilbert matrix
structure-preserving numerical methods
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Summary:In this short note, we define an × matrix constructed from the Hilbert matrix and prove that it has at least one pair of complex eigenvalues when ≥ 2. is a matrix related to the AVF collocation method, which is an energy-preserving/dissipative numerical method for ordinary differential equations, and our result gives a matrix-theoretical proof that the method does not have large-grain parallelism when its order is larger than or equal to 4.
ISSN:2300-7451
2300-7451
DOI:10.1515/spma-2021-0101