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Imagine that you have a descritized matrix A which the points are $A_{i,j}$ with $i=1,...,n$ and $j=1,...,m$

Then we can label the points as $A_{k}$, where $k=i+n(m-j)$.

I had already seen some examples but I have never seen the proof of this.

Can someone explain me why this labeling $k=i+n(m-j)$ works? Or do you know where can I find the proof?

Thanks in advance to everyone

pipita
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  • Why we must have a proof of this? It is only one way to arrange the order of this points. If you want, you can arrange these points in a totally different way, whatever way you like – J. Yu Mar 05 '16 at 12:24
  • the arrangement you mentioned in main text, is to ensure the sparsity of the matrix, which is generated by the relation of neighboured nodes. – J. Yu Mar 05 '16 at 12:27
  • @J.Yu, Yes I know, but do you get to that arrangement? What I mean is, what is the thinking to get to this? Or does someone dream with equation $k=i+n(m-j)$ and it's done? – pipita Mar 05 '16 at 12:34
  • The thing is that with this ordering you get a nice sparse matrix for the linear system of equations, which can take advantage of the eigenvalues analysis for the 1D Poisson problem. Changing the ordering is like interchanging the lines of the nice sparse matrix that you have with your ordering. You can see the book Applied numerical linear algebra by James Demmel, p. 276, Lemma 6.3 and around it. – Svetoslav Mar 05 '16 at 12:39
  • Draw a picture. The matrix typically corresponds with starting at the upper left corner of the mesh and working across then down. The ordering you describe starts at the bottom left corner and works across and up. I think this is what you're asking based on your comment above. – postmortes Mar 05 '16 at 13:07

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