A block Chebyshev-Davidson method for linear response eigenvalue problems

We present a Chebyshev-Davidson method to compute a few smallest positive eigenvalues and corresponding eigenvectors of linear response eigenvalue problems. The method is applicable to more general linear response eigenvalue problems where some purely imaginary eigenvalues may exist. For the Chebyshev filter, a tight upper bound is obtained by a computable bound estimator that is provably correct under a reasonable condition. When the condition fails, the estimated upper bound may not be a true one. To overcome that, we develop an adaptive strategy for updating the estimated upper bound to guarantee the effectiveness of our new Chebyshev-Davidson method. We also obtain an estimate of the rate of convergence for the Ritz values by our algorithm. Finally, we present numerical results to demonstrate the performance of the proposed Chebyshev-Davidson method.