For the project I am working on I need to solve a very large system of sparse linear equations (3-4 Billion rows) describing flow along a network. Each edge of the network creates 4 matrix values with indexes with the form: (i,i) (i,j) (j,i) (j,j). Each row of the matrix, corresponds to a particular node of the network and each node can connect to 4 to 5 edges. I need to sum the diagonal elements from all the edges that connect to each node. My objects describing the edges of the network are large and can not be fit on a single node. My question is, does the cluster sparse solver sum up elements that are repeated for me, or do i need to perform that work on my own? As an alternative, does PARDISO perform this type of summation?
Second question, I read somewhere that the entire matrix must be able to fit onto each node. Is this still true?
Finally, I have a performance question. I have the ability to create my matrix such that it is Positive Definite (really simple if just flip the signs of the matrix and rhs side vector). Is solving a positive definite matrix fast, more precise, or both when compared to a sparse indefinite?