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A Note on Approximate Inverse Iteration

Different variants of approximate inverse iteration like the locally optimal block preconditioned conjugate gradient method became in recent years increasingly popular for the solution of the large matrix eigenvalue problems arising from the discretization of selfadjoint elliptic partial differential equations, in particular for the calculation of the minimum eigenvalue. We extend in this little note the classical convergence theory of D'yakonov and Orekhov [Math. Notes 27 (1980)] to the case of operators with an essential spectrum on infinite dimensional Hilbert spaces and allow for arbitrary, sufficiently small perturbations of the solutions of the equation that links the iterates. The note complements the much more elaborate convergence theory of Neymeyr and Knyazev and Neymeyr for the matrix case (see [Knyazev and Neymeyr, SIAM J. Matrix Anal. Appl. 31 (2009)] and the references therein) and is suitable for classroom presentation.