When using Newton-Raphson to solve a system of equations of the form: $$\mathbf{r}(\mathbf{x})=\mathbf{0}, \quad \mathbf{r}=[r_1, r_2,...,r_N], \quad \mathbf{x}=[x_1, x_2,...,x_N]$$ The termination criterion is usually based on the euclidean norm of $\mathbf{r}$: $$ \sqrt{r_1^2+r_2^2+...+r_N^2} \le tolerance $$ As far as I understand, this criterion guarantees that the largest error for a single equation, $r_i$, will always be smaller than the desired tolerance. However, I feel like such a criterion is 'unfair' when the number of equations increases. An extreme example would be that of a system where $N=1$ (a single equation) vs a system where $N \gg 1$.
For $N=1$ we can assume that the value of $r_1$ at the point of termination will be close to the set tolerance.
For $N \gg 1$ my intuition is that each $r_i$ will be significantly smaller than the tolerance, since they are all added up to produce a value close to the tolerance. This means that more iterations are needed, although the set tolerance stays the same.
Then, my question would be: Is there a termination criterion that will provide solutions with the same order of accuracy, regardless of the number of the equations?
Yes. Replace the Euclidian norm with the infinity norm and terminate when $$\|\mathbf{r}\|_\infty \leq \tau$$ where $\tau$ is the tolerance.