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Full linear multistep methods as root-finders. (English) Zbl 1426.65069
Summary: 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.
##### MSC:
 65H05 Numerical computation of solutions to single equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations
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