Authors

Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran

Abstract

It is proved that applying sufficient regularity conditions to the interval matrix $[A-|B|,A + |B|]$, we can create a new unique solvability condition for the absolute value equation $Ax + B|x|=b$, since regularity of interval matrices implies unique solvability of their corresponding absolute value equation. This condition is formulated in terms of positive de niteness of a certain point matrix. Special case $B=-I$ is veri ed too as an application.

Keywords

Main Subjects

###### ##### References

[1] H. Beeck, Zur Problematik der Hullenbestimmung von Inntervallgleichungssystemen, In: Interval Mathematics, Karlsruhe, 1975, Lecture Notes in Comput. Sci., 29, Springer, Berlin, (1975) 150-159.
[2] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge: Cambridge University Press 1985.
[3] R. Farhadsefat, T. Lot and J. Rohn, A note on regularity and positive de niteness of interval matrices, Cent. Eur. J. Math., 10(1) (2012) 322-328.
[4] O. L. Mangasarian, Absolute value programming, Compute Optim Applic, 36, (2007) 43-53.
[5] O. L. Mangasarian, R. R. Meyer, Absolute value equations, Linear Algebra Appl, 419(2-3), (2006) 359-367.
[6] O. Prokopyev, On equivalent reformulation for absolute value equations, Comput Optim Appl, 44, (2009) 363-372.
[7] G. Rex and J. Rohn, Sucient conditions for regularity and singularity of interval matrices, SIAM J. Matrix Anal. Appl., 20(2), (1999) 437-445.
[8] J. Rohn, An algorithm for solving the absolute value equation, Electronic Journal of Linear Algebra, 18, (2009) 589-599.
[9] J. Rohn, A Handbook of Results on Interval Linear Problems, Prague: Czech Academy of Sciences, 2005.

[10] J. Rohn, A theorem of the alternatives for the equation Ax + B|x|= b, Linear, Multilinear Algebra, 52(6) (2004) 421 426.
[11] J. Rohn, A theorem of the alternatives for the equation |Ax|- |B||x| = b, Optim Lett, 6, (2012) 585-591.
[12] J. Rohn, A residual existence theorem for linear equations, Optim Lett, 4(2), (2010) 287-292.
[13] J. Rohn, On unique solvability of the absolute value equation, Optim, Lett., 3, (2003) 603-606.
[14] J. Rohn, Checking properties of interval matrices, Technical Report 686, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, September 1996.
[15] J. Rohn, V. Hooshyarbakhsh and R. Farhadsefat, An iterative method for solving Absolute value equations and sufficient conditions for unique solvability, Optimization Letters, 2012.
[16] S. M. Rump, Veri cation methods for dence and sparse systems of equations, In: Topics in Validated Computations Oldenburg, 1993, Stud. Comput. Math., 5, North-Holland, Amsterdam, (1994) 63-135.