# Full linear multistep methods as root-finders

### Uittreksel

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.

Originele taal-2 Engels 190-201 12 Applied Mathematics and Computation 320 https://doi.org/10.1016/j.amc.2017.09.003 Gepubliceerd - 1 mrt 2018

### Vingerafdruk

Linear multistep Methods
Interpolation
Roots
Convergence of numerical methods
Polynomials
Derivatives
Polynomial Interpolation
Nonlinear Function
Value Function
Rate of Convergence
Interpolate
Converge
Derivative
Numerical Examples
Zero
Estimate

### Citeer dit

@article{480938fbafe54243b339339a4700c209,
title = "Full linear multistep methods as root-finders",
abstract = "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.",
keywords = "Convergence rate, Iterative methods, Linear multistep methods, Nonlinear equation, Root-finder",
author = "{van Lith}, B.S. and {ten Thije Boonkkamp}, J.H.M. and W.L. IJzerman",
year = "2018",
month = "3",
day = "1",
doi = "10.1016/j.amc.2017.09.003",
language = "English",
volume = "320",
pages = "190--201",
journal = "Applied Mathematics and Computation",
issn = "0096-3003",
publisher = "Elsevier",

}

In: Applied Mathematics and Computation, Vol. 320, 01.03.2018, blz. 190-201.

TY - JOUR

T1 - Full linear multistep methods as root-finders

AU - van Lith, B.S.

AU - ten Thije Boonkkamp, J.H.M.

AU - IJzerman, W.L.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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.

AB - 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.

KW - Convergence rate

KW - Iterative methods

KW - Linear multistep methods

KW - Nonlinear equation

KW - Root-finder

UR - http://www.scopus.com/inward/record.url?scp=85030852642&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2017.09.003

DO - 10.1016/j.amc.2017.09.003

M3 - Article

AN - SCOPUS:85030852642

VL - 320

SP - 190

EP - 201

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

ER -