Small singular values can increase in lower precision

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.

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