Componentwise enclosure for solutions of least squares problems and underdetermined systems

Abstract

Algorithms for enclosing solutions of least squares problems and underdetermined systems are proposed. The results obtained by these algorithms are “verified” in the sense that all the possible rounding errors have been taken into account. In order to develop these algorithms, theories for obtaining componentwise error bounds of numerical solutions are established. The error bounds by the proposed algorithms are vectors, as opposed that those by the fast algorithms proposed by Rump are scalars. The proposed algorithms require computational costs similar to those of Rumpʼs algorithms. It is moreover proved that each component of the error bounds by the proposed algorithms is equal or smaller than the error bounds by Rumpʼs algorithms. Numerical results show the properties of the algorithms.