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

Rohn, Jiří

Applications of Mathematics, Tome 52 (2007) no. 2, pp. 105-115.

Résumé

For a real square matrix $A$ and an integer $d\ge 0$, 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 $\alpha (d)$, computed solely from $A_{(d)}$ (not from $A$), such that the following alternative holds: $\bullet $ if $d>\alpha (d)$, then nonsingularity (positive definiteness, positive invertibility) of $A_{(d)}$ implies the same property for $A$; $\bullet $ if $d\alpha (d)$ and $A_{(d)}$ is nonsingular (positive definite, positive invertible), then there exists a matrix $A^{\prime }$ with $A^{\prime }_{(d)}=A_{(d)}$ 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 $\Vert \cdot \Vert _{\infty ,1}$; for positive invertibility $\alpha (d)$ is given by an easily computable formula. 0178.57901 1013.81007 0635.58034 1022.81062 0372.43005 1058.81037 0986.81031 0521.33001 0865.65009 0847.65010 0945.68077 0780.93027 0628.65027 0712.65029 0709.65036 0796.65065 0964.65049

Détail

DOI : 10.1007/s10492-007-0005-6

Classification : 15A09, 15A12, 15A48, 65G40, 65G50
Keywords: nonsingularity; positive definiteness; positive invertibility; fixed-point rounding