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Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding. (English) Zbl 1164.15310
Summary: For a real square matrix  \(A\) and an integer \(d\geq 0\), 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:
if \(d>\alpha (d)\), then nonsingularity (positive definiteness, positive invertibility) of  \(A_{(d)}\) implies the same property for  \(A\);
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 the computation of the NP-hard norm \(\| \cdot \| _{\infty ,1}\); for positive invertibility  \(\alpha (d)\) is given by an easily computable formula.
15A12 Conditioning of matrices
15B48 Positive matrices and their generalizations; cones of matrices
15A09 Theory of matrix inversion and generalized inverses
65G50 Roundoff error
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