open method
A root-finding method (fixed-point iteration, Newton's method, the secant method) that tracks a single running guess instead of a shrinking bracket -- and so is never guaranteed to converge.
An open method produces a sequence of approximations x_0, x_1, x_2, \ldots from a starting guess (or two), without maintaining an interval known to contain the root. Unlike a bracketing method, nothing forces the sequence toward a root at all -- an open method can converge quickly when it works, but can also diverge or leave the region entirely, trading the bracketing method's guarantee for speed.
Examples
Newton's method started at x_0=0 for f(x)=x^3-2x+2 never settles down -- it cycles 0\to1\to0\to\cdots forever, exactly the risk an open method carries that bisection never does.
See also