Some tests of generalized bisection

This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically compares such generalized bisection algorithms to themselves, to continuation methods, and to hybrid steepest descent/quasi-Newton methods. A specific algorithm containing novel “expansion” and “exclusion” steps is fully described, and the effectiveness of these steps is evaluated. A test problem consisting of a small, high-degree polynomial system that is appropriate for generalized bisection, but very difficult for continuation methods, is presented. This problem forms part of a set of 17 test problems from published literature on the methods being compared; this test set is fully described here.

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