# Article

 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: http://hdl.handle.net/10338.dmlcz/134666 . Reference: [1] G. H.  Golub, C. F.  van  Loan: Matrix Computations.The Johns Hopkins University Press, Baltimore, 1996. MR 1417720 Reference: [2] N. J.  Higham: Accuracy and Stability of Numerical Algorithms.SIAM, Philadelphia, 1996. Zbl 0847.65010, MR 1368629 Reference: [3] V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl: Computational Complexity and Feasibility of Data Processing and Interval Computations.Kluwer Academic Publishers, Dordrecht, 1998. MR 1491092 Reference: [4] S. Poljak, J. Rohn: Checking robust nonsingularity is NP-hard.Math. Control Signals Syst. 6 (1993), 1–9. MR 1358057, 10.1007/BF01213466 Reference: [5] J. Rohn: Inverse-positive interval matrices.Z. Angew. Math. Mech. 67 (1987), T492–T493. Zbl 0628.65027, MR 0907667 Reference: [6] J. Rohn: Systems of linear interval equations.Linear Algebra Appl. 126 (1989), 39–78. Zbl 0712.65029, MR 1040771 Reference: [7] J. Rohn: Nonsingularity under data rounding.Linear Algebra Appl. 139 (1990), 171–174. Zbl 0709.65036, MR 1071707 Reference: [8] J. Rohn: Positive definiteness and stability of interval matrices.SIAM J. Matrix Anal. Appl. 15 (1994), 175–184. Zbl 0796.65065, MR 1257627, 10.1137/S0895479891219216 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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