Fast Accurate Eigenvalue Computations Using the Cholesky Factorization

Let H = DAD where A is a positive definite matrix and D is diagonal and nonsingular. We show that if the condition number of a is much less than that of D then we can use algorithms based on the Cholesky factorization of H to compute the eigenvalues of H 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 D is very large