Sharp error bounds for Ritz vectors and approximate singular vectors

2018/10/05 by Yuji Nakatsukasa, Nakatsukasa, Yuji · 2 citations
Computer Science · Mathematics · #FOS: Mathematics #Numerical Analysis (math.NA) #cs.NA #math.NA

paper · pdf · doi:10.48550/arxiv.1810.02532

arxiv created 2019/12/29 · arxiv updated 2020/01/01

Abstract

We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue problems. Using information that is available or easy to estimate, our bounds improve the classical Davis-Kahan sinθ theorem by a factor that can be arbitrarily large, and can give nontrivial information even when the sinθ theorem suggests that a Ritz vector might have no accuracy at all. We also present extensions in three directions, deriving error bounds for invariant subspaces, singular vectors and subspaces computed by a (Petrov-Galerkin) projection SVD method, and eigenvectors of self-adjoint operators on a Hilbert space.

Cited by

• Sharp error bounds for approximate eigenvalues and singular values from subspace methods
• Local spectral approximation of unbounded operators: non-asymptotic and unified error quantification for subspace methods

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