An efficient refinement algorithm is proposed for symmetric eigenvalue
problems. The algorithm is simple, primarily comprising matrix
multiplications. It constructs arbitrarily accurate eigenvalue decomposition
by iteration, up to the limit of computational precision. We show that since
the proposed algorithm is based on Newton's method, it quadratically converges
for sufficiently distinct simple eigenvalues. Numerical results demonstrate
excellent performance of the proposed algorithm in terms of convergence rate
and overall computational cost,and show that it is considerably faster than a
straightforward approach using multiple-precision arithmetic. Unfortunately,
the algorithm is not applicable to a matrix having nearly multiple eigenvalues
unless a sufficiently accurate initial guess is provided. However, it is worth
noting that the proposed algorithm works for exactly multiple eigenvalues.