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

Rohn, Jiří. "Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding." Applications of Mathematics 52.2 (2007): 105-115. <http://eudml.org/doc/33279>.

@article{Rohn2007,
abstract = {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},
author = {Rohn, Jiří},
journal = {Applications of Mathematics},
keywords = {nonsingularity; positive definiteness; positive invertibility; fixed-point rounding; nonsingularity; positive definiteness; positive invertibility; fixed-point rounding},
language = {eng},
number = {2},
pages = {105-115},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding},
url = {http://eudml.org/doc/33279},
volume = {52},
year = {2007},
}

TY - JOUR
AU - Rohn, Jiří
TI - Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding
JO - Applications of Mathematics
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 2
SP - 105
EP - 115
AB - 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
LA - eng
KW - nonsingularity; positive definiteness; positive invertibility; fixed-point rounding; nonsingularity; positive definiteness; positive invertibility; fixed-point rounding
UR - http://eudml.org/doc/33279
ER -