An efficient algorithm for the Schrödinger-Poisson eigenvalue problem
Journal of Computational and Applied Mathematics
Convergence Analysis of Iterative Solvers in Inexact Rayleigh Quotient Iteration
SIAM Journal on Matrix Analysis and Applications
MINRES-QLP: A Krylov Subspace Method for Indefinite or Singular Symmetric Systems
SIAM Journal on Scientific Computing
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The SYMMLQ and MINRES methods for solving sparse, indefinite, symmetric systems of equations [SIAM J. Numer. Anal., 12 (1975), pp. 617--629] can be used inside the inverse or Rayleigh quotient iterations for computing the eigenpairs of large, sparse, generalized eigenproblems $A y = \lambda B y$. It is shown in this paper that the common opinion concerning the superiority of SYMMLQ over MINRES is not true in this context. Several results taken from quantum chemistry, acoustics, and structural mechanics are presented that show the superiority of MINRES and the new MINERR method over SYMMLQ, especially for ill-conditioned eigenproblems, where SYMMLQ may even diverge. Theoretical explanations of the observed phenomena are given. The use of MINRES or MINERR instead of SYMMLQ may thus lead to a significant increase in efficiency in large scale eigenvalue computations.