Summary
Linear systems represent the computational kernel of many models that describe problems arising in the field of social, economic as well as technical and scientific disciplines. Therefore, much effort has been devoted to the development of methods, algorithms and software for the solution of linear systems. Finite precision computer arithmetics makes rounding error analysis and perturbation theory a fundamental issue in this framework (Higham 1996). Indeed, Interval Arithmetics was firstly introduced to deal with the solution of problems with computers (Moore 1979, Rump 1983), since a floating point number actually corresponds to an interval of real numbers. On the other hand, in many applications data are affected by uncertainty (Jerrell 1995, Marino & Palumbo 2002), that is, they are only known to lie within certain intervals. Thus, bounding the solution set of interval linear systems plays a crucial role in many problems. In this work, we focus on the state of the art of theory and methods for bounding the solution set of interval linear systems. We start from basic properties and main results obtained in the last years, then we give an overview on existing methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alefed, G. & Herzberger, J. (1983), Introduction to Interval Computation, Academic Press, New York.
Alefeld, G., Kreinovich, V. & Mayer, G. (1998), “The Shape of the Solution Set for Interval System with Dependent Coefficients’, Math. Nachr. 192, 23–36.
Alefeld, G. & Mayer, G. (2000), ‘Interval analysis: theory and applications’, Journal of Computational and Applied mathematics 121, 421–464.
Cormen, T.H., Leiserson, C.E., Rivest, R.L. & Stein, C. (2001), Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill.
Cryer, C.W. (1973), ‘The LU-factorization of totally positive matrices’, Linear Algebra and its Appl. 7, 83–92.
Frommer, A. & Mayer, G. (1993), ‘A New Criterion for Guarantee the Feasibility of the Interval Gaussian Algorithm’, SIAM J. Matrix Anal. Appl. 14(2), 408–419.
Funderlic, R.E., Neumann, M. & Plemmons, R.J. (1982), ‘LU decompositions of generalized diagonally dominat matrices’, Math. Comp. 40, 57–69.
Golub, G.H. & Van Loan, C.F. (1989), Matrix Computations, Academic Press, New York.
Hansen, E.R. (1992), ‘Bounding the solution of Interval Linear Equations’, SIAM J. Numer. Anal. 29(5), 1493–1503.
Hansen, E.R. (2005), ‘A Theorem on Regularity of Interval Matrices’, Reliable Computing 11(6), 495–497.
Higham, N.J. (1996), Accuracy and Stability of Numerical Algorithms, SIAM.
Jansson, C. (1997), ‘Calculation of Exact Bounds for the Solution Set of Linear Interval Systems’, Linear Algebra and its Appl. 251, 321–340.
Jansson, C. & Rohn, J. (1999), ‘An Algorithm for Checking Regularity of Interval Matrices’, SIAM J. Matrix Anal. Appl. 20(3), 756–776.
Jerrell, M.E. (1995), ‘Applications of Interval Computations to Economic Input-Output Models’, Estended Abstract of APIC’95, El Paso, Supplement to Reliable Computing, 102–104.
Kreinovich, V., Lakeyev, A., Rohn, J. & Kahla, R., (1998), Computational complexity and feasibility of data processing and interval computations, Kluwer Academic Publishers.
Kuttler, J. (1971), ‘A Fourth-Order Finite-Difference Approximation for the fixed Membrane Eigenproblem’, Math. Comp. 25, 237–256.
Marino, M. & Palumbo, F. (2002), ‘Interval arithmetic of imprecise data effetcs in least squares linear regression’, Stat. Appl. 14(3), 277–291.
Mayer, G. & Rohn, J. (1998), ‘On the Applicability of the Interval Gaussian Algorithm’, Reliable Computing 4, 205–222.
Moore, R.E. (1979), Methods and Applications of Interval Analysis, SIAM.
Neumaier, A. (1990), Interval methods for systems of equations, Cambridge University Press.
Ning, S. & Kearfott, R. B. (1997), ‘A Comparison of Some Methods for Solving Linear Interval Equations’, SIAM J. on Numerical Analysis 34(4), 1289–1305.
Oettli, W. & Prager, W. (1964), ‘Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-had sides’, Numer. Math. 6, 405–409.
Rex, G. & Rohn, J. (1998), ‘Sufficient Conditions for Regularity and Singularity of Interval Matrices’, SIAM J. Matrix Anal. Appl. 20 (2), 437–445.
Rohn, J. (1995), ‘Checking Bounds on Solutions of Linear Interval Equations is NP-Hard’, Linear Algebra and its Applications 223/224, 589–596.
rohn, J. (2003), ‘Solvability of Systems of Linear Interval Equations’, SIAM J. Matrix Anal. Appl. 25(1), 237–245.
Rohn, J. (2005), ‘How Strong Is Strong Regularity?’, Reliable Computing 11(6), 491–493.
Rump, M. (1983), ‘Solving algebraic problems with high accuracy’, in A New Approach to Scientific Computation, U. Kulisch and W.L. Miranker, eds., Academic Press, New York, 51–120.
Rump, S. M. (1997), ‘Bounds for the componentwise distance to the nearest singular matrix’, SIAM J. Matrix Anal. Appl. 18, 83–103.
Saad, Y. (2003), Iterative Methods for Sparse Linear Systems, SIAM.
Shary, S. P. (1995), ‘Solving the linear interval tolerance problem’, Mathematics and Computers in Simulation 39, 53–85.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Corsaro, S., Marino, M. Interval linear systems: the state of the art. Computational Statistics 21, 365–384 (2006). https://doi.org/10.1007/s00180-006-0268-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-006-0268-5