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For a given subspace, the Rayleigh-Ritz method projects the large quadraticeigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similarto the Rayleigh-Ritz method for the linear eigenvalue problem, theRayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEPwith respect to the projection subspace. We analyze the convergence of themethod when the angle between the subspace and the desired eigenvectorconverges to zero. We prove that there is a Ritz value that converges to thedesired eigenvalue unconditionally but the Ritz vector converges conditionallyand may fail to converge. To remedy the drawback of possible non-convergence ofthe Ritz vector, we propose a refined Ritz vector that is mathematicallydifferent from the Ritz vector and is proved to converge unconditionally. Weconstruct examples to illustrate our theory.

On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems