Influence of the Condition Number on Interval Computations: Illustration on Some Examples
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The condition number is a quantity that is well-known in “classical” numerical analysis, that is, where numerical computations are performed using floating-point numbers. This quantity appears much less frequently in interval numerical analysis, that is, where the computations are performed on intervals. The goal of this paper is twofold. On the one hand, it is stressed that the notion of condition number already appears in the literature on interval analysis, even if it does not bear that name. On the other hand, three small examples are used to illustrate experimentally the impact of the condition number on interval computations. As expected, problems with a larger condition number are more difficult to solve: this means either that the solution is not very accurate (for moderate condition numbers) or that the method fails to solve the problem, even inaccurately (for larger condition numbers). Different strategies to counteract the impact of the condition number are discussed and experimented: use of a higher precision, iterative refinement, bisection of the input. More strategies are discussed as a conclusion.
This work has been partially supported by the ANR project MetaLibm ANR-13-INSE-0007-04.
The author thanks H. D. Nguyen for his kind permission to reproduce the material of Sect. 4. She also thanks Olga Kosheleva for her kind invitation. Last but not least, she wishes to thank Vladik Kreinovich for his kindness and friendship during many years, and for his vitality that is contagious and beneficial to the community.
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