| 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 |
| . |
