Interval Newton method: Hansen-Greenberg approach—some procedural improvements

摘要

E.R. Hansen and R.I. Greenberg have presented an interval Newton method to solve a system of n nonlinear equations in n real variables. Following R.E. Moore, they have linearized the system using a mean value expansion, and used preconditioning technique due to Hansen and R.R. Smith to modify the system. The modified system is subjected to a Hansen-Sengupta step to obtain an updated interval containing the solution. Hansen and Greenberg have in fact used the subalgorithms of preconditioning and Hansen-Sengupta step witha real (local, noninterval) iteration and an elimination procedure to provide an algorithm of greater efficiency. In this paper, we indicate procedures which will further improve the efficiency of the Hansen-Greenberg algorithm. Firstly, a better approximating matrix for the identity is obtained. Secondly, a successive overrelaxation (SOR) technique is introduced to replace the Hansen-Sengupta step if necessary. Finally, an interval iteration is suggested to provide an alternative to the real (noninterval) inner iteration introduced by Hansen and Greenberg. Examples are included to show improvement in results and/or the forms the new procedures would take. the additional procedures provide more efficient results besides improving the techniques.