Simultaneous Global Convergence Failure of Newton's and Halley's Methods in Degree Four

We construct an explicit quartic polynomial for which Newton’s and Halley’s root-finding methods both fail to converge to a root on nonempty open sets of initial values. The example can be found through a reproducible computational search within a one-parameter family of quartics designed to exhibit a prescribed Newton two-cycle. We then prove analytically that Newton’s method has a superattracting two-cycle and that Halley’s method has a distinct attracting extraneous two-cycle, using a two-interval contraction argument. This fills a degree-four gap in the simultaneous Newton–Halley failure problem. The computational approach also identifies another quartic with numerical evidence of simultaneous failure of Newton’s, Halley’s, and Schröder’s methods.