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Harmonic Ritz and Lehmann bounds. (English) Zbl 0918.65027

The author surveys a selection of results relating to optimal bounds for matrix eigenvalues, including both known and new results. The paper concentrates on obtaining information about matrix eigenvalues in the interior of its spectrum from its action on vectors in a given subspace.
Initially the Ritz values, the simplest estimates, are discussed, together with the variants harmonic Ritz, which gives best bounds to the upper eigenvalues of \(Kx=\lambda Mx\), and the dual harmonic Ritz which gives best bounds to the lower eigenvalues.
This section is followed by discussion of right-definite and left-definite variants of Lehmanns’s optimal bounds, after which two new convenient computational reformulations are introduced, together with a discussion as to why these should be preferable to the right-definite bounds.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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