J. Rohn, Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodarenskou vezi 2, 182 07 Prague 8, Czech Republic, e-mail: rohn@cs.cas.cz
Abstract: For a real square matrix $A$ and an integer $d\geq0$, 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 $\alpha(d)$, computed solely from $A_{(d)}$ (not from $A$), such that the following alternative holds: \item{$\bullet$} if $d>\alpha(d)$, then nonsingularity (positive definiteness, positive invertibility) of $A_{(d)}$ implies the same property for $A$; \item{$\bullet$} if $d<\alpha(d)$ 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 $\alpha(d)$ is the same and involves computation of the NP-hard norm $\|\cdot\|_{\infty,1}$; for positive invertibility $\alpha(d)$ is given by an easily computable formula.
Keywords: nonsingularity, positive definiteness, positive invertibility, fixed-point rounding
Classification (MSC 2000): 15A12, 15A48, 65G40, 65G50
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