Verification of positive definiteness

Publikationstyp
Journal Article
Date Issued
2006-05-03
Sprache
English
Author(s)
Rump, Siegfried M.  orcid-logo
Institut
Zuverlässiges Rechnen E-19  
TORE-URI
http://hdl.handle.net/11420/8891
Journal
BIT  
Volume
46
Issue
2
Start Page
433
End Page
452
Citation
BIT Numerical Mathematics 46 (2): 433-452 (2006)
Publisher DOI
10.1007/s10543-006-0056-1
Scopus ID
2-s2.0-33745298703
Publisher
Springer Science + Business Media B.V.
We present a computational, simple and fast sufficient criterion to verify positive definiteness of a symmetric or Hermitian matrix. The criterion uses only standard floating-point operations in rounding to nearest, it is rigorous, it takes into account all possible computational and rounding errors, and is also valid in the presence of underflow. It is based on a floating-point Cholesky decomposition and improves a known result. Using the criterion an efficient algorithm to compute rigorous error bounds for the solution of linear systems with symmetric positive definite matrix follows. A computational criterion to verify that a given symmetric or Hermitian matrix is not positive definite is given as well. Computational examples demonstrate the effectiveness of our criteria.
Subjects
Cholesky decomposition
INTLAB
Positive definite
Rigorous error bounds
Self-validating methods
Semidefinite programming
Verification
DDC Class
510: Mathematik