On the solution of nonlinear equations in interval arithmetic

We consider the problem of finding a simple zero of a continuously differentiable functionf:R ⁿ R ⁿ . There is given an intervalvectorX ₀ ^I containing one zero off, and we will construct a contracting sequence of intervalvectors enclosing this zero. This can be done by Newton's method, which gives quadratic convergence, but requires inversion of an intervalmatrix at each step of the iteration. Alefeld and Herzberger, 1], give a modification of Newton's method, without the necessity of inversion, the convergence being superlinear. We give a slight modification of the latter method, with the property that the sequence of interval widths is dominated by a quadratically convergent sequence.