Which vectorial norm should I use?

[RicardoMP]

I am to study how fast an iterative method for nonlinear system of equations converges to a certain root and found out that I can evaluate my rate of convergence by using the following formula: $r^{(k)}=\frac{||x^{(k+1)}-x^{(k)}||_V}{||x^{(k)}-x^{(k-1)}||_V}$. My question is which vectorial norm should I use? $||.||_{\infty}$? $||.||_1$? Etc... Is there any criteria in choosing them? And after getting my rate of convergence, how do I measure my order of convergence?

 

[RUber]

I think it is common to use the infinity norm. All the norms should be related by a constant multiple anyway, so when you talk about order, it should not matter too much.
The order of convergence is often displayed as the slope of your convergence plot on a log scale.

 

[Erland]

If you work in a normed vector space with finite dimension, all these norms are equivalent. They give rise to the same topology.