Iterative refinement for symmetric eigenvalue decomposition

An efficient refinement algorithm is proposed for symmetric eigenvalue problems. The structure of the algorithm is straightforward, primarily comprising matrix multiplications. We show that the proposed algorithm converges quadratically if a modestly accurate initial guess is given, including the case of multiple eigenvalues. Our convergence analysis can be extended to Hermitian matrices. Numerical results demonstrate excellent performance of the proposed algorithm in terms of convergence rate and overall computational cost, and show that the proposed algorithm is considerably faster than a standard approach using multiple-precision arithmetic.