Fast accurate eigenvalue methods for graded positive definite matrices

Summary. Let where is a positive definite matrix and is diagonal and nonsingular. We show that if the condition number of is much less than that of then we can use algorithms based on the Cholesky factorization of to compute the eigenvalues of to high relative accuracy more efficiently than by Jacobi's method. The new methods are generally slower than tridiagonalization methods (which do not deliver the eigenvalues to maximal relative accuracy) but can be up to 4 times faster when the condition number of is very large. Received April 13, 1995