Abstract. We present several variations on the harmonic Rayleigh–Ritz method. First, we introduce a relative harmonic approach for the standard, generalized, and polynomial eigenproblem. Second, a harmonic extraction method is studied for rightmost eigenvalues of generalized eigenvalue problems. Third, we propose harmonic extraction methods for large eigenvalues of generalized and polynomial eigenproblems, where we also discuss avoidance of infinite eigenvalues when the finite eigenvalues are of interest. We give an oversight of the different methods with their relations and several typical numerical examples. AMS subject classifications. 65F15, 65F50. Key words. Rayleigh–Ritz, harmonic Rayleigh–Ritz, refined Rayleigh–Ritz, relative harmonic Rayleigh–Ritz, rational harmonic Rayleigh–Ritz, subspace method, subspace extraction, large eigenvalues, interior eigenvalues, rightmost eigenvalues, generalized eigenvalue problem, polynomial eigenvalue problem. 1. Introduction. The harmonic Rayleigh–Ritz subspace extraction, mainly due to early work by Morgan  and its formal introduction by Paige, Parlett, and Van der Vorst  is a very helpful technique to approximate eigenvalues in the interior o
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