Relative perturbation theory for diagonally dominant matrices Articles uri icon

publication date

  • October 2014

start page

  • 1303

end page

  • 1328

issue

  • 4

volume

  • 35

International Standard Serial Number (ISSN)

  • 0895-4798

Electronic International Standard Serial Number (EISSN)

  • 1095-7162

abstract

  • In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.

keywords

  • accurate computations; diagonally dominant matrices; diagonally dominant parts; eigenvalues; inverses; linear systems; relative perturbation theory; singular values; eigenvalues and eigenfunctions; inverse problems; linear systems; matrix algebra; number theory; accurate computations; diagonally dominant; eigenvalues; inverses; relative perturbation; singular values; perturbation techniques