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

IF 0.8 Q2 MATHEMATICS

Special Matrices Pub Date : 2021-08-10 DOI:10.1515/spma-2021-0101

Yusaku Yamamoto

引用次数: 2

Abstract

Abstract In this short note, we define an s × s matrix Ks constructed from the Hilbert matrix Hs=(1i+j-1)i,j=1s{H_s} = \left( {{1 \over {i + j - 1}}} \right)_{i,j = 1}^s and prove that it has at least one pair of complex eigenvalues when s ≥ 2. Ks 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.

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关于保能/耗散连续阶段Runge-Kutta方法中矩阵的特征值

摘要本文定义了由希尔伯特矩阵Hs=(1i+j-1)i,j=1s{H_s} = \左({{1 \ / {i +j-1}}} \右)_{i,j =1}^s构造的s × s矩阵Ks，并证明了当s≥2时它至少有一对复特征值。k是一个与常微分方程的保能/耗散数值方法AVF配置法相关的矩阵，我们的结果从矩阵理论上证明了该方法在大于等于4阶时不具有大粒度并行性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Special Matrices MATHEMATICS-

CiteScore

1.10

自引率

20.00%

发文量

审稿时长

8 weeks

期刊介绍： Special Matrices publishes original articles of wide significance and originality in all areas of research involving structured matrices present in various branches of pure and applied mathematics and their noteworthy applications in physics, engineering, and other sciences. Special Matrices provides a hub for all researchers working across structured matrices to present their discoveries, and to be a forum for the discussion of the important issues in this vibrant area of matrix theory. Special Matrices brings together in one place major contributions to structured matrices and their applications. All the manuscripts are considered by originality, scientific importance and interest to a general mathematical audience. The journal also provides secure archiving by De Gruyter and the independent archiving service Portico.