# How to describe function convergence and function tolerance for numerical root-finding?

I'm currently doing some practice problems on root-finding and am writing up some notes / comments on my code. In my solver code, if my function value is below the tolerance that I've set, should I write,

"the function converged to a root, since the function evaluation is below the function tolerance"

or

"the function converged to a root, since the function tolerance is achieved"?

Is there a better way to word it?

• What is your stopping criterion? @user37069 – Vefhug Sep 21 '20 at 20:37
• You could say the first option, which is what you did. – Vefhug Sep 21 '20 at 20:58
• @Vefhug Ok, will do -- thanks so much! – user37069 Sep 21 '20 at 21:00
• You're welcome. Maybe you could consider more precise stopping criteria. Indeed, it can happen that the residual is "large", but you're close to $0$ (when the derivative is high near the root). But also the converse can happen, for instance when the derivative is close to $0$ at the root you may find a small residual, but you're actually far from the solution. @user37096 – Vefhug Sep 21 '20 at 21:07
• The residual should decrease in a certain way with the iteration number, and monitoring this should tell more about the convergence than the residual size alone – Maxim Umansky Sep 22 '20 at 8:17