The nearest definite pair for the Hermitian generalized eigenvalue problem

The generalized eigenvalue problem Ax = lambda Bx has special properties when (A,B) is a Hermitian and definite pair. Given a general Hermitian pair (A, B) it is of interest to find the nearest definite pair having a specified Crawford number delta > 0. We solve the problem in terms of the inner numerical radius associated with the field of values of A + iB. We show that once the problem has been solved it is trivial to rotate the perturbed pair (A + Delta A, B + Delta B) to a pair ((A) over tilde, (B) over tilde) for which lambda(min),((B) over tilde) achieves its maximum value delta, 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. (C) 1999 Elsevier Science Inc. All rights reserved.