Fast Generation of Real-Symmetric Matrices and their Exact Eigenpairs

• Published in: Journal of Advanced Simulation in Science and Engineering Vol. 12; no. 1; pp. 44 - 60
• Main Authors: Ozaki, Katsuhisa, Imamura, Toshiyuki, Uchino, Yuki, Terao, Takeshi
• Format: Journal Article
• Language: English
• Published: Japan Society for Simulation Technology 2025; 一般社団法人 日本シミュレーション学会
• Subjects: Communication-avoiding; Floating-point arithmetic; Symmetric eigenvalue problems; Test matrices
• ISSN: 2188-5303; 2188-5303
• DOI: 10.15748/jasse.12.44

We propose a method to rapidly generate matrices for real-symmetric eigenproblems. The proposed method produces a reproducible matrix with explicit eigenpairs, where the distribution of the eigenvalues can be controlled by a user. All elements of the generated matrix are rigorous floating-point numbers and can be represented in simple expressions involving the exact eigenvalues. The exact eigenpairs of the generated matrix are known in advance; thus, the proposed method contributes to the validation of errors in approximate eigenpairs. Several constraints on the matrix generated by the proposed method, were produced theoretically.