The nearest definite pair for the Hermitian generalized eigenvalue problem

✍ Scribed by Sheung Hun; Nicholas J.

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 199 KB
• Volume: 302-303
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(99)00026-9

✦ Synopsis

The generalized eigenvalue problem ex kfx has special properties when eY f is a Hermitian and de®nite pair. Given a general Hermitian pair eY f it is of interest to ®nd the nearest de®nite pair having a speci®ed Crawford number d b 0. We solve the problem in terms of the inner numerical radius associated with the ®eld of values of e if. We show that once the problem has been solved it is trivial to rotate the perturbed pair e DeY f Df to a pair ẽY f for which k min e f achieves its maximum value d, which is a numerically desirable property when solving the eigenvalue problem by methods that convert to a standard eigenvalue problem by ``inverting B''. Numerical examples are given to illustrate the analysis.