On solving systems of equations using interval arithmetic

Introduction. In this paper, we consider the problem of applying interval arithmetic to bound a solution of a system of nonlinear equations. Moore [1, Section 7.3] has discussed the same problem. His approach, as well as ours, is to extend the multidimensional Newton method and implement it in interval arithmetic. In Section 2, it is shown that a particular detail of Moore's method can be modified to improve convergence and yield sharper bounds. In extreme cases, the modification can yield convergence where the original method fails. To illustrate this procedure, we consider (in Section 3) the problem of bounding complex roots of polynomials. Previous literature on the use of interval arithmetic to bound polynomial roots was restricted to the case of real polynomials with real roots. We use the obvious expedient of separating a polynomial equation into real and imaginary parts. This yields two real equations in two real variables to be solved by the method of Section 2. In Section 4, we consider the matrix eigenvalue-vector problem. Bounds for the solution of this problem are obtained by a method which is essentially that of Section 2. We show that our method is directly related to Wielandt inverse iteration.