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Computing the Spectrum in One, Two and Three Limits

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Computing the eigenvalues of a matrix is a classic computational problem with obvious practical relevance. While calculating the eigenvalues of matrices has long had a solution, it is not without its complexities. We take these complexities as our starting point. Generalizing, we consider the Finite Section (Ritz) method for calculating the spectrum of an operator defined on Hilbert space. While this can provide a practical method for bounded self-adjoint operators, the computation introduces spectral pollution for some operators of physical significance. After introducing the notions of the Solvability Complexity Index (SCI) and Towers of Algorithms, we construct an algorithm that recovers the spectrum for any bounded operator on $l^2(\mathbb{N})$.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.

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