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A Simple Derivation of the Hansen-Bliek-Rohn-Ning-Kearfott Enclosure for Linear Interval Equations

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Reliable Computing

Abstract

Recently, Ning & Kearfott derived a formula for the interval enclosure of the solution set of linear systems of equations with uncertain data ranging in intervals, in the case when the coefficient matrix is an H-matrix. The enclosure is optimal when the midpoint matrix is diagonal, and when the midpoint is the identity, it reduces to the optimal method for enclosing preconditioned systems found by Hansen and Bliek and simplified by Rohn.

An elementary proof of this formula is given using only simple properties of H-matrices and Schur complements. The new proof gives additional insight into why the theorem is true. It is also shown how to preserve rigor in the enclosure when finite precision arithmetic is used.

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References

  1. Bliek, C.: Computer Methods for Design Automation, Ph.D. Thesis, Dept. of Ocean Engineering, Massachusetts Institute of Technology, 1992.

  2. Hansen, E.: Bounding the Solution of Interval Linear Equations, SIAM J. Numer. Anal. 29 (1992), pp. 1493–1503.

    Google Scholar 

  3. Neumaier, A.: Interval Methods for Systems of Equations, Cambridge Univ. Press, Cambridge, 1990.

    Google Scholar 

  4. Ning, S. and Kearfott, R. B.: A Comparison of Some Methods for Solving Linear Interval Equations, SIAM J. Numer. Anal. 34 (1997), pp. 1289–1305.

    Google Scholar 

  5. Rohn, J.: Cheap and Tight Bounds: The Recent Result by E. Hansen Can Be Made More Efficient, Interval Computations 4 (1993), pp. 13–21.

    Google Scholar 

  6. Rump, S. M.: INTLAB—INTerval LABoratory, Reliable Computing, to appear.

  7. Spellucci, P. and Krier, N.: Untersuchungen der Grenzgenauigkeit von Algorithmen zur Auflösung linearer Gleichungssysteme mit Fehlererfassung, in: Nickel, K. (ed.), Interval Mathematics, (Lecture Notes in Computer Science 29), Springer, Berlin, 1975, pp. 288–297.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090, Wien, Austria

    Arnold Neumaier

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  1. Arnold Neumaier
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Neumaier, A. A Simple Derivation of the Hansen-Bliek-Rohn-Ning-Kearfott Enclosure for Linear Interval Equations. Reliable Computing 5, 131–136 (1999). https://doi.org/10.1023/A:1009997221089

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  • DOI: https://doi.org/10.1023/A:1009997221089

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