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# On the Solution of Interval Linear Systems

Citation Link: https://doi.org/10.15480/882.295

Publikationstyp

Journal Article

Publikationsdatum

1992

Sprache

English

Author

Institut

Citation

Computing 47:337-353, 1992

In the literature efficient algorithms have been described for calculating guaranteed inclusions for the solution of a number of standard numerical problems [3], [4], [8], [11], [12], [13]. The inclusions are given by means of a set containing the solution. In [12], [13] this set is calculated using an affine iteration which is stopped when a nonempty and compact set is mapped into itself. For exactly given input data (point data) it has been shown that this iteration stops if and only if the iteration matrix is convergent (cf. [13]).

In this paper we give a necessary and sufficient stopping criterion for the above mentioned iteration for interval input data and interval operations. Stopping is equivalent to the fact that the algorithm presented in [12] for solving interval linear systems computes an inclusion of the solution. An algorithm given by Neumaier is discussed and an algorithm is proposed combining the advantages of our algorithm and a modification of Neumaier's. The combined algorithm yields tight bounds for input intervals of small and large diameter.

Using a paper by Jansson [6], [7] we give a quite different geometrical interpretation of inclusion methods. It can be shown that our inclusion methods are optimal in a specified geometrical sense. For another class of sets, for standard simplices, we give some interesting examples.

In this paper we give a necessary and sufficient stopping criterion for the above mentioned iteration for interval input data and interval operations. Stopping is equivalent to the fact that the algorithm presented in [12] for solving interval linear systems computes an inclusion of the solution. An algorithm given by Neumaier is discussed and an algorithm is proposed combining the advantages of our algorithm and a modification of Neumaier's. The combined algorithm yields tight bounds for input intervals of small and large diameter.

Using a paper by Jansson [6], [7] we give a quite different geometrical interpretation of inclusion methods. It can be shown that our inclusion methods are optimal in a specified geometrical sense. For another class of sets, for standard simplices, we give some interesting examples.

Schlagworte

linear system

iteration

inclusion method

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