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I am trying to solve 5 coupled 1st order PDES on a 100 by 70 grid. So I end up with 35000, eqs. The method I am attempting to solve these eqs with is Newton-Raphson. Which then leads me to compute a Jacobian that is 35000 X 35000. As in Newton-Raphson I have to solve J = v F[x1,x1...x35000]. Where F is the vector of equations and v is the new solution. Then I add this v to my initial guess, and iterate.

Anyways the biggest problem I'm having now is trying to get the solution for this matrix equation J = v F[x1,x1...x35000], given that I have such a large matrix. Now the good thing is the the matrix is sparse. The bad thing is that I don't really know how to deal with that. I've read online about lots of libraries and packages that will solve Sparse matrices for me. But I would like to understand just what they are doing.

So far I understand the basic idea of compressed-column-formatting (CCF) , and how this reduces the storage of the full matrix. What I don't understand is how do you now operate on this CCF matrix, and get the solution to your original matrix equation.

You don't have to go into a long discussion of it if you don't want, I totally understand. Just please point me to a good reference. I've been trying to find such a reference for the past couple of days and its driving me nuts. Everyone just explains the compression formats and then says "here is the following library/software" that solves sparse matrix. But HOW? Please help.

Thanks in advanced.