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Title: Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding (English)
Author: Rohn, Jiří
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 52
Issue: 2
Year: 2007
Pages: 105-115
Summary lang: English
Category: math
Summary: 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 (English)
Keyword: nonsingularity
Keyword: positive definiteness
Keyword: positive invertibility
Keyword: fixed-point rounding
MSC: 15A09
MSC: 15A12
MSC: 15A48
MSC: 65G40
MSC: 65G50
idZBL: Zbl 1164.15310
idMR: MR2305868
DOI: 10.1007/s10492-007-0005-6
Date available: 2009-09-22T18:28:45Z
Last updated: 2020-07-02
Stable URL:
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Reference: [9] J. Rohn: Computing the norm $\Vert {A}\Vert _{\infty ,1}$ is NP-hard.Linear Multilinear Algebra 47 (2000), 195–204. MR 1785027, 10.1080/03081080008818644


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