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The convergence criterion is $\| y^{k+1} - y^{k} \| < \varepsilon$ where $k$ is iteration number.

Question 1.

Is it possible to use simpler criterion $$ \max_j \left(\left | y_j^{k+1} - y_j^k \right |\right) < \varepsilon $$ where $j$ is equation number? If yes then

Question 2.

Probably for particular equations the condition $$ \left | y_j^{k+1} - y_{j}^k \right |< \varepsilon $$ will be fulfilled earlier than for others. Is it possible to exclude these equations from the following iterations?

Michael Hardy
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user2351152
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1 Answers1

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Q1. I guess yes, because norms are equivalent in finite dimension (but the $\varepsilon$ will not have the same value.

Q2. For each iteration, you have to check that for each $j$, $|y_j^{k+1}-y_j^k|<\varepsilon$. It could be that the max is obtained for one $j$ at one iteration, and for another one at another. So I don't think you can exclude some indexes $j$ in the criterion. Anyway, this criterion is computationally inexpensive so it shouldn't be a practical problem.

anderstood
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  • Q2. The question is not about computation of the criterion. The question is about computation of an equations. For example, if at the current iteration for equation j the criterion is fulfilled can I skip this j equation at all on subsequent iteration? It could be useful for large number of equations. – user2351152 May 28 '16 at 19:22
  • @user2351152 I believe I answered: a priori, I see no reason why $|y_j^{k+1}-y_j^k|<\varepsilon$ for some $k$ implies $|y_j^{k+2}-y_j^{k+1}|<\varepsilon$ (so I don't think you can skip this inequation). – anderstood May 28 '16 at 19:30