  Root-finders based on full linear multistep methods (LMMs) use previous
function values, derivatives and root estimates to iteratively find a root of a
nonlinear function. As ODE solvers, full LMMs are typically not zero-stable.
However, used as root-finders, the interpolation points are convergent so that
such stability issues are circumvented. A general analysis is provided based on
inverse polynomial interpolation, which is used to prove a fundamental barrier
on the convergence rate of any LMM-based method. We show, using numerical
examples, that full LMM-based methods perform excellently. Finally, we also
provide a robust implementation based on Brent's method that is guaranteed to
converge.
