A MODIFIED NEWTON METHOD WITH GUARANTEED ACCURACY BASED ON RATIONAL ARITHMETIC

In this paper, we are concerned with a problem of obtaining an approxi mate solution of a finite-dimensional nonlinear equation with guarante ed accuracy. Assuming that an approximate solution of a nonlinear equa tion is already calculated by a certain numerical method, we present c omputable conditions to validate whether there exists an exact solutio n in a neighborhood of this approximate solution or not. In order to c heck such conditions by computers, we present a method using rational arithmetic. In this method, both the effects of the truncation errors and the rounding errors of numerical computation are taken into consid eration. Moreover, based on rational arithmetic we propose a new modif ied Newton iteration to obtain an improved approximate solution with d esired accuracy.