Small Singular Values Can Increase in Lower Precision
Christos Boutsikas, Petros Drineas, Ilse C. F. Ipsen · SIAM Journal on Matrix Analysis and Applications · 2024
Abstract. We perturb a real matrix [Formula: see text] of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions, demonstrate the potential increase in the smallest singular values and represent a qualitative model for the increase in the small singular values after a matrix has been downcast to a lower arithmetic precision. Numerical experiments confirm the qualitative validity of this model and its ability to predict singular values changes in the presence of decreased arithmetic precision.