Numerical validation method for nonlinear equations using interval analysis and rational arithmetic

This paper proposes an algorithm with a guaranteed accuracy for the finite-dimensional nonlinear equation. First, an algorithm is shown which can determine the interval containing the true solution, based on the given approximate solution. The method is based on the interval analysis and considers the representation error of the equation. Then the interval iteration method is shown which can arbitrarily reduce the interval. The rational number arithmetic is used and the rounding of the rational number is utilized effectively. Finally, an experimental implementation of the algorithm on a computer is presented, together with some numerical examples.