Optimal spectral filtering in a finite subspace: a Lanczos-based implementation

Abstract

A new method is presented to achieve optimal spectral filtering within a finite Lanczos subspace, based on a criterion of minimising the error-norm about the designated reference filtering energy. The implementation of this approach is shown to be easily achieved by a straightforward extension of the minimum residual algorithm of Paige and Saunders [SIAMJ. Numer. Anal., 1975, 12, 617]. The convergence properties of the present optimal filtering (OF) approach are compared with other direct filtering methods and also diagonalization-based methods (Lanczos and filter diagonalization) using the benchmark HO₂ Hamiltonian as a test case. The OF method displays superior convergence properties to all of the other methods tested.

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