In this paper, we first show that the second-order cone linear complementarity problem (SOCLCP) can be solved by finding a positive zero s * ∈ R of a particular rational function h(s), and we then propose a Krylov subspace method to reduce h(s) to h (s) as in the model reduction. The zero s * of h(s) can be accurately approximated by that of h (s) = 0, which itself can be cast as a small eigenvalue problem. The new method is made possible by a complete characterization of the curve of h(s), and it has several advantages over the bisection-Newton (BN) iteration recently proposed by [L.-H. Zhang and W. H. Yang, Math. Comp., 83 (2013), pp. 1701-1720] and shown to be very efficient for small-to medium-size problems. The method is tested and compared against the BN iteration and two other state-of-the-art packages: SDPT3 and SeDuMi. Our numerical results show that the method is very efficient for both small to medium dense problems and large-scale ones.

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Large scale eigenvalue computation is about approximating certain invariant subspaces associated with the interested part of the spectrum, and the interested eigenvalues are then extracted from projecting the problem by approximate invariant subspaces into a much smaller eigenvalue problem. In the case of the linear response eigenvalue problem (aka the random phase eigenvalue problem), it is the pair of deflating subspaces associated with the first few smallest positive eigenvalues that needs to be computed. This paper is concerned with approximation accuracy relationships between a pair of approximate deflating subspaces and approximate eigenvalues extracted by the pair. Lower and upper bounds on eigenvalue approximation errors are obtained in terms of canonical angles between exact and computed pair of deflating subspaces. These bounds can also be interpreted as lower/upper bounds on the canonical angles in terms of eigenvalue approximation errors. They are useful in analyzing numerical solutions to linear response eigenvalue problems.

The numerical solution of a large scale linear response eigenvalue problem is often accomplished by computing a pair of deflating subspaces associated with the interesting part of the spectrum. This paper is concerned with the backward perturbation analysis for a given pair of approximate deflating subspaces or an approximate eigenquadruple. Various optimal backward perturbation bounds are obtained, as well as bounds for approximate eigenvalues computed through the pair of approximate deflating subspaces or approximate eigenquadruple. These results are reminiscent of many existing classical ones for the standard eigenvalue problem.