Well-conditioned eigenvalue problems that overflow

2020/05/11 by Carl Christian Kjelgaard Mikkelsen, Mikkelsen, Carl Christian Kjelgaard
Computer Science · Engineering · Mathematics · #65F15 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #G.1.3 #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2005.05356

openalex publication_date 2020/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we present a parameterized class of lower triangular matrices. The components of the eigenvectors grow rapidly and will exceed the representational range of any finite number system. The eigenvalues and the eigenvectors are well-conditioned with respect to componentwise relative perturbations of the matrix. This class of matrices is well suited for testing software for computing eigenvectors as these routines must be able to handle overflow successfully.

Related

• How well-conditioned can the eigenvalue problem be?
• A note on constructive treatment of eigenvectors
• Eigenvectors of random matrices: A survey
• Eigenvectors from eigenvalues revisited
• On condition numbers of polynomial eigenvalue problems with nonsingular leading coefficients