Full linear multistep methods as root-finders

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...

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Published in:arXiv.org 2017-09
Main Authors: van Lith, Bart S, Jan H M ten Thije Boonkkamp, IJzerman, Wilbert L
Format: Article
Language:English
Subjects:
Convergence
Interpolation
Mathematical analysis
Polynomials
Robustness (mathematics)
Solvers
Stability analysis
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Jan H M ten Thije Boonkkamp
IJzerman, Wilbert L
description 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.
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subjects Convergence
Interpolation
Mathematical analysis
Polynomials
Robustness (mathematics)
Solvers
Stability analysis
title Full linear multistep methods as root-finders
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