# How does FindRoot decide if a solution has converged?

I am solving a 1D non-linear differential equation using the finite element method with NDSolve. From the documentation I understand that the equation is discretized and then solved with FindRoot, I am using the default option to use the AffineCovariantNewton method.

My main question is how does findroot decide that the iterative process has converged? What is the norm it computes, and how is this related to the accuracy and precisson goal? Ideally, I would like to know the convergence criteria as a function of the precission goal and accuracy goal to understand what is going on.

I find it hard to find these answers in the documentation.

• I am in a rush now, here are the relevant lit references: [1], Newton Methods for Nonlinear Problems,Affine Invariance and Adaptive Algorithms;Deufelhard, Peter; Springer, 2006 (Second Printing) ISBN 978-3-642-23898-7 (Softcover) [2], Reference implementations for [1], elib.zib.de/pub/elib/codelib/NewtonLib/index.html [3], A Family of Newton Codes for Systems of Highly Nonlinear Equations (1991), Nowak, U., Weimann, L.; ?, citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.3751 Commented Apr 1, 2022 at 14:15
• The method is purely error bases so it ignored accuracy goal. Commented Apr 1, 2022 at 14:16

Turning my comment into an answer.

I am in a rush now, Here are the relevant literature references for the affine covariant Newton solver that is implemented as a method of FindRoot:

[1], Newton Methods for Nonlinear Problems, Affine Invariance and Adaptive Algorithms; Deuflhard, Peter; Springer, 2006 (Second Printing) ISBN 978-3-642-23898-7 (Softcover)

[2], Reference implementations for [1], elib.zib.de/pub/elib/codelib/NewtonLib/index.html

[3], A Family of Newton Codes for Systems of Highly Nonlinear Equations (1991), Nowak, U., Weimann, L.; ?, citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.3751

The method is purely error based so it ignores the accuracy goal. In other words, only setting the PrecisionGoal has an effect.

The affine covariant Newton solver is not the default solver for FindRoot, but it is the default solver for nonlinear PDEs when the finite element method is used.

• Why are you always in a rush? It would be nice to take it easy once in a while, don't you think? Commented Jun 1, 2022 at 21:25
• @DanielLichtblau, actually, when I copied the comment into the answer, I was lazy to fix the wording of the comment. But I'll take the advice and try not to be in a rush. Commented Jun 2, 2022 at 4:52