On Some Properties of Interval Matrices

Authors: K. Ganesan

Abstract:

By using a new set of arithmetic operations on interval numbers, we discuss some arithmetic properties of interval matrices which intern helps us to compute the powers of interval matrices and to solve the system of interval linear equations.

Keywords: Interval arithmetic, Interval matrix, linear equations.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1070263

References:

[1] G. Alefeld and J. Herzberger, ''Introduction to Interval Computations'', Academic Press, New York, 1983.
[2] Atanu Sengupta, Tapan Kumar Pal, ''Theory and Methodology: On comparing interval numbers'', European Journal of Operational Research, vol. 127, pp. 28 - 43, 2000.
[3] K. Ganesan and P. Veeramani, ''On Arithmetic Operations of Interval Numbers'', International Journal of Unccertainty, Fuzziness and Knowledge - Based Systems, vol. 13, no. 6, pp. 619 - 631, 2005.
[4] E. R. Hansen and R. R. Smith, ''Interval arithmetic in matrix computations'', Part 2, SI AM. Journal of Numerical Analysis, vol. 4, pp. 1 - 9, 1967.
[5] E. R. Hansen, ''On the solution of linear algebraic equations with interval coefficients'', Linear Algebra Appl, vol. 2, pp. 153 - 165, 1969.
[6] E. R. Hansen, ''Global Optimization Using Interval Analysis'', Marcel Dekker, Inc., New York, 1992.
[7] E. R. Hansen, ''Bounding the solution of interval linear Equations'', SIAM. Journal of Numerical Analysis, vol. 29, no. 5, pp. 1493 - 1503, 1992.
[8] J. Kuttler, ''A Fourth-Order Finite-Difference Approximation for the Fixed Membrane Eigen- problem'', Math. Comp., vol. 25, pp. 237 - 256, 1971.
[9] Luc Jaulin, Michel Kieffer, Olivier Didrit and Eric Walter, ''Applied Interval Analysis'', Springer-Verlag, London, 2001.
[10] A. Neumaier, ''Interval Methods for Systems of Equations'', Cambridge University Press, Cambridge, 1990.
[11] S. Ning and R. B. Kearfott, ''A comparison of some methods for solving linear interval Equations'', SIAM. Journal of Numerical Analysis, vol. 34, pp. 1289 - 1305, 1997.
[12] J. Rohn, ''Interval matrices: singularity and real eigenvalues'', SIAM. Journal of Matrix Analysis and Applications, vol. 1, pp. 82 - 91, 1993.
[13] J. Rohn, ''Inverse interval matrix'', SIAM. Journal of Numerical Analysis, vol. 3, pp. 864 - 870, 1993.
[14] J. Rohn, ''Cheap and Tight Bounds: The recent result by E. Hansen can be made more efficient'', Interval Computations, vol. 4, pp. 13 - 21, 1993.