Abstract
In this paper we present a linear time algorithm for checking whether a tridiagonal matrix will become singular under certain perturbations of its coefficients. The problem is known to be NP-hard for general matrices. Our algorithm can be used to get perturbation bounds on the solutions to tridiagonal systems.
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References
I. Bar-On and M. Leoncini,Reliable sequential algorithms for solving bidiagonal systems of linear equations, submitted.
A. Deif,Sensitivity Analysis in Linear Systems, Springer-Verlag, 1986.
G. Alefeld and J. Herzberger,Introduction to Interval Computations, Academic Press, New York, 1983.
G. H. Golub and C. F. V. Loan,Matrix Computations 2nd ed., The Johns Hopkins University Press, Baltimore, 1989.
J. W. Demmel,On Condition Numbers and the distance to the nearest ill-posed problem, Numer. Math., 51 (1987), pp. 251–289.
J. W. Demmel,The Componentwise distance to the nearest singular matrix, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 10–19.
N. J. Higham,Bounding the error in Gaussian elimination for tridiagonal systems, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 521–530.
N. J. Higham,Algorithm 694: A collection of test matrices in MATLAB, ACM Trans. Math. Softw., 17 (1991), pp. 289–305.
W. Kahan,Numerical Linear Algebra, Canadian Math. Bull., 9 (1966), pp. 757–801.
J. C. Doyle,Analysis of feedback systems with structured uncertainties, in Proc. IEEE 129 (1982), pp. 242–250.
K. Knopp,Elements of the Theory of Functions, Dover, New York, 1952.
M. Mansour,Robust stability of interval matrices, in Proceedings of the 28th Conference on Decision and Control, Tampa, FL, 1989, pp. 46–51.
MATLAB,The Math Works, Inc., Natick, MA., 1994.
A. Nemirovskii,Several NP-Hard problems arising in robust stability analysis, Mathematics of Control, Signals, and Systems, 6 (1993), pp. 99–105.
A. Neumaier,Interval Methods for Systems of Equations, Cambridge University Press, 1990.
W. Oettli and W. Prager,Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides, Numer. Math., 6 (1964), pp. 405–409.
S. Poljak and J. Rohn,Checking robust nonsingularity is NP-Hard, Mathematics of Control, Signals, and Systems, 6 (1993), pp. 1–9.
J. Rohn and V. Kreinovich,Computing exact componentwise bounds for the solution of linear systems with interval data is NP-Hard, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 415–420.
S. M. Rump,Bounds for the componentwise distance to the nearest singular matrix, Berichte 95.3, Technische Universität Hamburg, Germany, May 1995.
R. Skeel,Scaling for numerical stability in Gaussian elimination, J. Assoc. Comput. Mach., 26 (1979), pp. 494–526.
G. Stewart and J. Sun,Matrix Perturbation Theory, Academic Press, 1990.
J. H. Wilkinson,The Algebraic Eigenvalue Problem, Oxford University Press, 1965. Reprinted in Oxford Science Publications, 1988.
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Bar-On, I., Codenotti, B. & Leoncini, M. Checking robust nonsingularity of tridiagonal matrices in linear time. Bit Numer Math 36, 206–220 (1996). https://doi.org/10.1007/BF01731979
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DOI: https://doi.org/10.1007/BF01731979