An Interval Step Control for Continuation Methods

arcs in the set Z Of (1.1). These applications divide naturally into two classes. In the rst class, H models a parametrized physical system, which we wish to study as the parameter varies, such as in [12]. In the second class, we introduce an articial parameter in order to solve (or nd all solutions to) a system of n nonlinear equations in n unknowns for which locally convergent methods are unsuccessful. For problems in the second class, we may have mathematically rigorous guarantees that we will nd all solutions to the original system Of equations, provided the continuation method reliably tracks the arcs in Z Of (1.1); see [15],