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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.
##### MSC:
 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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