# Can't understand how to get coarsened matrix in classic algebraic multigrid method [closed]

I'm trying to write code for CAMG using information from this book(https://www.wias-berlin.de/people/john/LEHRE/MULTIGRID/multigrid.pdf, 57-59 pages).

Example of CAMG in this book consists of solving 3 diagonal matrix.
But immediately after he said that matrix is 3 diagonal he says "Consider a 5×5 patch and choose some node as C-node"(page 57, at the end of the page).

Matrix, that he is solving:

5x5 patch

How did we get that 5x5 patch, why is it 5x5?

What is the meaning of this patch, is it a graph, if it's a graph, then by what rule we define existence of connection between nodes?

Prior to this author says that nodes represent unknowns, which means if his patch has 25 nodes, than our original matrix is 25x25 or what?

And here is another problem why every node of his patch is connected to one another, if it's a three diagonal matrix than the graph of this matrix would be chain, and every node would have had only two neighbors at most, but his nodes has 3 and even 4 neighbors?

Expected graph form

If someone would give simple example on certain matrix, and steps(with this certain matrix) that will lead to getting the approximate solution of matrix equation I would be very grateful.

## closed as too broad by Eevee Trainer, John Omielan, Shailesh, YiFan, mrtaurhoMar 15 at 6:45

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