Full linear multistep methods as root-finders

Bart S. van Lith

doi:10.1016/j.amc.2017.09.003

Full linear multistep methods as root-finders

2018-03-01

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Bart S. van Lith *

J

J. H. M. ten Thije Boonkkamp

W

Wilbert L. IJzerman

DOI：10.1016/j.amc.2017.09.003

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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. (C) 2017 Elsevier Inc. All rights reserved.

Keyword:

Root-finder

Nonlinear equation

Linear multistep methods

Iterative methods

Convergence rate

Full linear multistep methods as root-finders

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