Interval Newton/generalized bisection when there are singularities near roots

Interval Newton methods in conjunction with generalized bisection are important elemetns of algorithms which find theglobal optimum within a specified box X ⁿ of an objective function whose critical points are solutions to the system of nonlinear equationsF(X)=0with mathematical certainty, even in finite presision arithmetic. The overall efficiency of such a scheme depends on the power of the interval Newton method to reduce the widths of the coordinate intervals of the box. Thus, though the generalized bisection method will still converge in a box which contains a critical point at which the Jacobian matrix is singular, the process is much more costly in that case. Here, we propose modifications which make the generalized bisection method isolate singular solutions more efficiently. These modifications are based on an observation about the verification property of interval Newton methods and on techniques for detecting the singularity and removing the region containing it. The modifications assume no special structure forF. Additionally, one of the observations should also make the algorithm more efficient when finding nonsingular solutions. We present results of computational experiments.