Real eigenvalue bounds of standard and generalized real interval eigenvalue problems

Abstract In this paper, we study the real eigenvalue bounds of standard and generalized real interval eigenvalue problems. Based on some known sufficient conditions for the regularity of interval matrices, a new algorithm for computing real eigenvalue bounds for real interval matrices is proposed. It can easily be extended for applying to generalized real interval eigenvalue problems. One advantage of the method is that the computation procedure takes less time than former methods we have proposed. Therefore it can be applied for solving larger interval eigenvalue problems. Finally, some numerical examples are presented which show the effectiveness of the method.

[1]  I. Elishakoff,et al.  Bounds of eigenvalues for structures with an interval description of uncertain-but-non-random parameters , 1996 .

[2]  Jiri Rohn,et al.  On the range of eigenvalues of an interval matrix , 2006, Computing.

[3]  J. Rohn.  Interval matrices: singularity and real eigenvalues , 1993 .

[4]  Milan Hladík,et al.  Characterizing and approximating eigenvalue sets of symmetric interval matrices , 2011, Comput. Math. Appl..

[5]  Milan Hladík,et al.  Bounds on Real Eigenvalues and Singular Values of Interval Matrices , 2010, SIAM J. Matrix Anal. Appl..

[6]  Zhiqing He,et al.  Computation of bounds for eigenvalues of structures with interval parameters , 2010, Appl. Math. Comput..

[7]  A. Deif,et al.  The Interval Eigenvalue Problem , 1991 .

[8]  Huinan Leng,et al.  Computing bounds to real eigenvalues of real‐interval matrices , 2008 .

[9]  Milan Hladík,et al.  A filtering method for the interval eigenvalue problem , 2011, Appl. Math. Comput..

[10]  D. Hertz.  The extreme eigenvalues and stability of real symmetric interval matrices , 1992 .

[11]  Lubomir V. Kolev.  Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies , 2006, Reliab. Comput..

[12]  Lubomir V. Kolev.  Eigenvalue range determination for interval and parametric matrices , 2010, Int. J. Circuit Theory Appl..

[13]  Robert L. Mullen,et al.  Natural Frequencies of a Structure with Bounded Uncertainty , 2006 .

[14]  Svatopluk Poljak,et al.  Checking robust nonsingularity is NP-hard , 1993, Math. Control. Signals Syst..

[15]  Sondipon Adhikari,et al.  Eigenvalue density of linear stochastic dynamical systems: A random matrix approach , 2012 .

[16]  Huinan Leng,et al.  Computing eigenvalue bounds of structures with uncertain-but-non-random parameters by a method based on perturbation theory , 2006 .

[17]  Peter C. Müller,et al.  An approximate method for the standard interval eigenvalue problem of real non‐symmetric interval matrices , 2001 .

[18]  A. S. Deif,et al.  On the invariance of the sign pattern of matrix eigenvectors under perturbation , 1994 .

[19]  Milan Hladík,et al.  Bounds on eigenvalues of real and complex interval matrices , 2013, Appl. Math. Comput..

[21]  Jiri Rohn,et al.  Sufficient Conditions for Regularity and Singularity of Interval Matrices , 1999, SIAM J. Matrix Anal. Appl..