Small singular values can increase in lower precision

Petros Drineas, Christos Boutsikas, Ilse C. F. Ipsen

Abstract: We perturb a real matrix

A

of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, for two classes of perturbations: deterministic normwise absolute, and probabilistic componentwise relative. Both classes of bounds, which extend existing lower-order expressions, demonstrate a potential increase in the smallest singular values. Our perturbation results represent a qualitative model for the increase in the small singular values after a matrix has been demoted to a lower arithmetic precision. Numerical experiments confirm the qualitative validity of the model and its ability to predict singular values changes in the presence of decreased arithmetic precision.

Has companion code repository: https://github.com/cboutsikas/small_sigmas_increase

Mathematics Subject Classification ID

Factorization of matrices (15A23) Numerical computation of eigenvalues and eigenvectors of matrices (65F15) Inequalities involving eigenvalues and eigenvectors (15A42) Eigenvalues, singular values, and eigenvectors (15A18) Roundoff error (65G50)

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