I was reading this paper http://www.ersavas.com/bulut/projects/algorithms/smo_ge.pdf where it deals with solving linear equations with sparse matrices
So lets say we have an equations
Ax=b where A is a sparse matrix, then we can solve it using Gaussian elimination.
but first we can order A so that is has lesser band itself. So lets say I have a matrix
A = a11 0 a13 0 0 a22 a23 a24 a31 a32 a33 0 0 a42 0 a44
Then if I permute using $ \pi=\{1, 2, 3, 4\} $
The linear system after symmetric permutation will have
A = a11 a13 0 0 a31 a33 a32 0 0 a23 a22 a24 0 0 a42 a44
but if we actually change matrix A to the above form using permutation is the equation
Ax=b still valid
Also the paper mentions about reverse Cuthill-McKee Ordering, and states that it is efficient in comparision to just Cuthill-McKee Ordering for Gaussian elimination because of the way it is performed (left to right and down to bottom).
I didn't get what it meant by (left to right and down to bottom). Any insights?