ONE- AND TWO-STAGE IMPLICIT INTERVAL METHODS OF RUNGE-KUTTA TYPE

Interval methods for solving the initial value problem are interesting due to interval-solutions obtained by such methods which contain their errors. Computer implementations of interval methods in floating-point interval arithmetic together with the representation of initial data in the form of minimal machine intervals, i. e. by intervals which ends are equal or neighboring machine numbers, yield interval solutions which contain all possible numerical errors. Explicit interval methods of Runge-Kutta type have been considered and analysed by Šokin [3,7]. In this paper we try to extend his approach for implicit methods. A reason to do this follows from a well-known fact concerning convential implicit Runge-Kutta methods higher orders of accuracy can be obtained than for explicit methods. This paper is dealt with oneand two-stage implicit interval methods of Runge-Kutta type, which are presented in sections 3 and 4. We prove that the exact solution of the initial value problem belongs to interval-solutions obtained by both kinds of these methods (section 5). In section 6 some approximations of the widths of interval-solution are given.