Converging interval methods for the iterative solution of a non-linear equation

The convergence criterion |x_i+1-x_i| < ε{lunate} is more reliable than |f(x_i| < ε{lunate}, particularly if at successive iterations the values of x_i are forced to alternate on both sides of the solution, by the iterative method. Two such methods are proposed. The first is a modification of the Interval Newton method and the second is a combination of the Newton and the Secant methods. These two methods require two starting points and numerical differentiation can be used with no loss of accuracy or rate of convergance. If numerical differentation is not required, these methods can be started with one initial point and converge faster.