Pivot tightening for some direct methods for solving systems of linear interval equations

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2011
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JABER, Khalid, ed.. Proceedings of the 3rd Conference of Mathematical Sciences : CMS 2011, 27-28 April 2011, Zarqa, Jordanien. Zarqa: Zarqa University, 2011, pp. 2137-2156. Available under: doi: 10.1007/s00607-011-0159-7
Zusammenfassung

The paper considers systems of linear interval equations, i. e., linear systems where the coefficients of the matrix and the right hand side vary between given bounds. We consider direct methods for the enclosure of the solution set of such a system. The algorithms are obtained from the ordinary elimination proce-dures by replacing the real numbers by the related intervals and the real operations by the respective interval operations. We report on methods by which the breakdown of the interval variants of the Gaussian elimination and the Neville elimination caused by division of an interval containing zero can be avoided for some classes of matrices with identically signed inverses. These include the non-singular totally nonnegative matrices. The approach consists of a tightening of the interval pivot by determining the exact range of the pivot over the matrix interval. By means of the interval Cholesky method an enclo-sure of the solution set for symmetric matrices can be found. We present a method by which the diagonal entries of the interval Cholesky factor can be tightened for positive definite interval matrices, such that a breakdown of the algorithm can be prevented. In the case of positive definite symmetric Toeplitz matrices, a further tightening of the interval pivots and other entries of the Cholesky factor is possible.

Zusammenfassung in einer weiteren Sprache
Fachgebiet (DDC)
510 Mathematik
Schlagwörter
Interval Neville elimination, interval Cholesky method, totally nonnegative matrix, positive definite matrix, Toeplitz matrix
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3rd Conference of Mathematical Sciences, 27. Apr. 2011 - 28. Apr. 2011, Zarqa, Jordanien
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ISO 690GARLOFF, Jürgen, 2011. Pivot tightening for some direct methods for solving systems of linear interval equations. 3rd Conference of Mathematical Sciences. Zarqa, Jordanien, 27. Apr. 2011 - 28. Apr. 2011. In: JABER, Khalid, ed.. Proceedings of the 3rd Conference of Mathematical Sciences : CMS 2011, 27-28 April 2011, Zarqa, Jordanien. Zarqa: Zarqa University, 2011, pp. 2137-2156. Available under: doi: 10.1007/s00607-011-0159-7
BibTex
@inproceedings{Garloff2011Pivot-18703,
  year={2011},
  doi={10.1007/s00607-011-0159-7},
  title={Pivot tightening for some direct methods for solving systems of linear interval equations},
  publisher={Zarqa University},
  address={Zarqa},
  booktitle={Proceedings of the 3rd Conference of Mathematical Sciences : CMS 2011, 27-28 April 2011, Zarqa, Jordanien},
  pages={2137--2156},
  editor={Jaber, Khalid},
  author={Garloff, Jürgen}
}
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