On Modifications to the Spectral Dichotomy Algorithm

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We introduce modifications of an existing spectral dichotomy algorithm [6] with respect to a circle for general regular matrix pencils. The modifications result in a cost-effective algorithm, mathematically equivalent to the original one, but six times less expensive. The algorithm approximates iteratively the spectral projector onto the right deflating subspace associated with the eigenvalues inside/outside the circle as well as a criterion to ascertain its numerical quality. We also show how to approximate the left projector from the algorithm without any extra work. Effective stopping criteria for the algorithm are discussed. Numerical examples which illustrate the theory are presented.