The nearest definite pair for the Hermitian generalized eigenvalue problem

The generalized eigenvalue problem Ax = lambda Bx has special properties wh en (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 specifie d Crawford number delta > 0. We solve the problem in terms of the inner num erical radius associated with the field of values of A + iB. We show that o nce 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 w hich lambda(min),((B) over tilde) achieves its maximum value delta, which i s a numerically desirable property when solving the eigenvalue problem by m ethods that convert to a standard eigenvalue problem by "inverting B". Nume rical examples are given to illustrate the analysis. (C) 1999 Elsevier Scie nce Inc. All rights reserved.