A note on regularity and positive definiteness of interval matrices

Raena Farhadsefat; Taher Lotfi; Jiri Rohn

Open Mathematics (2012)

  • Volume: 10, Issue: 1, page 322-328
  • ISSN: 2391-5455

Abstract

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We present a sufficient regularity condition for interval matrices which generalizes two previously known ones. It is formulated in terms of positive definiteness of a certain point matrix, and can also be used for checking positive definiteness of interval matrices. Comparing it with Beeck’s strong regularity condition, we show by counterexamples that none of the two conditions is more general than the other one.

How to cite

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Raena Farhadsefat, Taher Lotfi, and Jiri Rohn. "A note on regularity and positive definiteness of interval matrices." Open Mathematics 10.1 (2012): 322-328. <http://eudml.org/doc/269157>.

@article{RaenaFarhadsefat2012,
abstract = {We present a sufficient regularity condition for interval matrices which generalizes two previously known ones. It is formulated in terms of positive definiteness of a certain point matrix, and can also be used for checking positive definiteness of interval matrices. Comparing it with Beeck’s strong regularity condition, we show by counterexamples that none of the two conditions is more general than the other one.},
author = {Raena Farhadsefat, Taher Lotfi, Jiri Rohn},
journal = {Open Mathematics},
keywords = {Interval matrix; Regularity condition; Positive definiteness; interval matrices; regularity condition; positive definiteness; regular interval matrices; singular interval matrices; strong regularity; symmetric interval matrices; NP-hard; NP-complete},
language = {eng},
number = {1},
pages = {322-328},
title = {A note on regularity and positive definiteness of interval matrices},
url = {http://eudml.org/doc/269157},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Raena Farhadsefat
AU - Taher Lotfi
AU - Jiri Rohn
TI - A note on regularity and positive definiteness of interval matrices
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 322
EP - 328
AB - We present a sufficient regularity condition for interval matrices which generalizes two previously known ones. It is formulated in terms of positive definiteness of a certain point matrix, and can also be used for checking positive definiteness of interval matrices. Comparing it with Beeck’s strong regularity condition, we show by counterexamples that none of the two conditions is more general than the other one.
LA - eng
KW - Interval matrix; Regularity condition; Positive definiteness; interval matrices; regularity condition; positive definiteness; regular interval matrices; singular interval matrices; strong regularity; symmetric interval matrices; NP-hard; NP-complete
UR - http://eudml.org/doc/269157
ER -

References

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  1. [1] Beeck H., Zur Problematik der Hüllenbestimmung von Intervallgleichungssystemen, In: Interval Mathematics, Karlsruhe, 1975, Lecture Notes in Comput. Sci., 29, Springer, Berlin, 1975, 150–159 Zbl0303.65025
  2. [2] Fiedler M., Nedoma J., Ramík J., Rohn J., Zimmermann K., Linear Optimization Problems with Inexact Data, Springer, New York, 2006 Zbl1106.90051
  3. [3] Garey M.R., Johnson D.S., Computers and Intractability, Ser. Books Math. Sci., W.H. Freeman and Company, San Francisco, 1979 
  4. [4] Horn R.A., Johnson C.R., Matrix Analysis, Cambridge University Press, Cambridge, 1985 Zbl0576.15001
  5. [5] Oettli W., Prager W., Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides, Numer. Math., 1964, 6, 405–409 http://dx.doi.org/10.1007/BF01386090 Zbl0133.08603
  6. [6] Poljak S., Rohn J., Checking robust nonsingularity is NP-hard, Math. Control Signals Systems, 1993, 6(1), 1–9 http://dx.doi.org/10.1007/BF01213466 Zbl0780.93027
  7. [7] Rex G., Rohn J., Sufficient conditions for regularity and singularity of interval matrices, SIAM J. Matrix Anal. Appl., 1999, 20(2), 437–445 http://dx.doi.org/10.1137/S0895479896310743 Zbl0924.15003
  8. [8] Rohn J., Positive definiteness and stability of interval matrices, SIAM J. Matrix Anal. Appl., 1994, 15(1), 175–184 http://dx.doi.org/10.1137/S0895479891219216 Zbl0796.65065
  9. [9] Rohn J., Forty necessary and sufficient conditions for regularity of interval matrices: a survey, Electron. J. Linear Algebra, 2009, 18, 500–512 Zbl1189.65088
  10. [10] Rump S.M., Verification methods for dense and sparse systems of equations, In: Topics in Validated Computations, Oldenburg, 1993, Stud. Comput. Math., 5, North-Holland, Amsterdam, 1994, 63–135 

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