The Roles of Condition Number for the Interval Hull Solution Set in Least Squares Equation

Abstract

The paper considers nearness to singularity for which the Co-variance matrix in the least squares equation is well known. A synchronization of the condition number with ill-conditioning is highlighted which relates the quality of approximate solution to the described system. Various theoretical lower and upper bounds to the perturbed least squares problems have been described for which, reach ability theory has strong representation. In particular, a theorem due to Rump as exemplified by Popova was revisited and examined in detail; a slight modification was made to the theorem by neglecting the second term appearing in the equation. This was found to have strong favourable appeals on the interval least squares problem. As a comparison to the computed results, a procedure described in Kramer/Rohn was used to crop the corner points solution of the linear interval system which is obtained from least squares equation based on the appropriate choice of the orthant (where there are possibilities). This leads to solving systems of linear inequalities for the interval Hull of solution set. Furthermore, the Rump/Krawczyk method was used to narrow, the computed corner point solution in order to obtain tighter approximate solution bounds of the interval Hull which may be applicable to both nonparametric and parametric interval linear equations. The loss function for the computed result obtained from Rump method for the set of data points is reported.