Added a bandwidth feature

[HenryWu2101]

#13565
Added a bandwidth feature that gives the bandwidth of a matrix. The bandwidth of a matrix measures how far from the main diagonal the nonzero entries extend. Mostly used for storage efficiency / reordering to speed up solvers etc. For now it is just a new feature to add.

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cc @tscrim

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[dcoudert]

The order of operations could be improved by visiting the matrix by diagonals and stop as soon as a non zero value is found. it is best to visit simultaneously the upper and the lower diagonals.
You can use #39963 here.

I'm not sure that the method should be called bandwidth. May be get_bandwidth ? When you ask for the bandwidth, you usually expect to get an ordering of the rows/columns that minimizes the bandwidth.
See for instance method bandwidth in graphs https://doc.sagemath.org/html/en/reference/graphs/sage/graphs/graph_decompositions/bandwidth.html

[HenryWu2101]

@dcoudert For the namesake, I wasn't careful to see that there is a method with the same name but different behavior implemented elsewhere. In that case, naming it the same would not be appropriate, will change to get_bandwidth. And additionally, I'm not sure if it would be necessary to propose implement bandwidth in matrix2.pyx as a follow-up issue.

As for using diagonal to improve efficiency, it was exactly what I thought when I created the diagonal PR and then got to this issue when looking for something similar in the issue pool. But as I understand it, the bandwidth of a matrix is the smallest distance for all non-zero entries to lie from the main diagonal. At that time, I felt iterating from main diagonal outward would not be an improvement (since it still needs to iterate to the end, in case a non-zero entry lies at the top corner).

So to clarify what you proposed, you're saying if I go through every diagonal starting from the corner to the main diagonal via diagonal, and I stop as soon as I reach a non-zero entry (and visit from both directions simultaneously), I would have an improvement with only $O(n^2)$ as the worst case scenario.

[dcoudert]

@dcoudert For the namesake, I wasn't careful to see that there is a method with the same name but different behavior implemented elsewhere. In that case, naming it the same would not be appropriate, will change to get_bandwidth. And additionally, I'm not sure if it would be necessary to propose implement bandwidth in matrix2.pyx as a follow-up issue.

I don't think it's necessary.

So to clarify what you proposed, you're saying if I go through every diagonal starting from the corner to the main diagonal via diagonal, and I stop as soon as I reach a non-zero entry (and visit from both directions simultaneously), I would have an improvement with only $O(n^2)$ as the worst case scenario.

Precisely and this is what was proposed in the original patch #13565. The idea is to stop computation as soon as possible. This should be an improvement for large matrix.