New results on verified inclusions

Abstract

The computational results of traditional numerical algorithms on computers are usually good approximations to the solution of a given problem. However, no verification is provided for some bound on the maximum relative error of the approximation. As can be demonstrated by ill-conditioned examples, those approximations may be drastically wrong. The algorithms based on the inclusion theory (cf. [38]) do have an automatic verification process. Rather than approximations to the solution an inclusion of the solution is computed and the correctness of the bounds together with the existence and uniqueness of the solution within the bounds is automatically verified by the computer without any effort on the part of the user. The computing time of these algorithms is of the order of a comparable, standard floating-point algorithm (such as Gaussian elimination in case of general linear systems). In the following some new results complementing the inclusion theory are given. One of the main results is that the inclusion sets need not to be convex. Therefore other types of inclusion sets such as torus-sectors can be used. Another main observation is that the new and old theorems can be proved without using fixed point theorems. Moreover improvements of existing theorems of the inclusion theory by means of weaker assumptions are presented. Another fundamental observation is the following. It is well-known that a real iteration in IRn with affine iteration function converges if and only if the spectral radius of the iteration matrix is less than one. It can be shown that a similar result holds for our inclusion algorithm: An inclusion will be achieved if and only if the spectral radius of the iteration matrix is less then one. This result is best possible. It is demonstrated by means of theorems and examples that even for extremely ill-conditioned examples very sharp inclusions of the solution are computed. The inclusions are almost always of least significant bit accuracy, i.e. the left and right bounds of the inclusion are adjacent floating-point numbers.