# Full linear multistep methods as root-finders

## TL;DR

This paper analyzes the use of full linear multistep methods as root-finders, demonstrating their convergence properties, limitations, and practical implementation with guaranteed convergence.

## Contribution

It provides a general theoretical analysis of LMM-based root-finders, establishes a fundamental convergence barrier, and offers a robust implementation with guaranteed convergence.

## Key findings

- Full LMM-based methods perform well in numerical tests.
- A fundamental barrier limits the convergence rate of LMM root-finders.
- A robust Brent-based implementation guarantees convergence.

## 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.