Influence of the Condition Number on Interval Computations: Illustration on Some Examples

Abstract

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.