Nonsingularity,positive definiteness,and positive invertibility under fixed-point data rounding

For a real square matrix A and an integer d ? 0, let A _(d) denote the matrix formed from A by rounding off all its coefficients to d decimal places. The main problem handled in this paper is the following: assuming that A _(d) has some property, under what additional condition(s) can we be sure that the original matrix A possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists a real number α(d), computed solely from A _(d) (not from A), such that the following alternative holdsif d > α(d), then nonsingularity (positive definiteness, positive invertibility) of A _(d) implies the same property for A if d < α(d) and A _(d) is nonsingular (positive definite, positive invertible), then there exists a matrix A′ with A′_(d) = A _(d) which does not have the respective property.For nonsingularity and positive definiteness the formula for α(d) is the same and involves computation of the NP-hard norm ‖ · ‖_∞,1; for positive invertibility α(d) is given by an easily computable formula.