TITLE: I have a very large sparse matrix, 'A', in Ax = b. What work in advance of getting 'b' can be done to reduce solving time?
QUESTION [9 upvotes]: This question borders between a programming and math question (more math). I have a little matrix knowledge but this is past my ability, so any help is very much appreciated.
Question
I have a very large sparse matrix (say one million rows square). I'd like to solve this for many different sets of constant values (to clarify, by constant values, I mean the 1xn matrix 'b' in Ax = b). Performance is absolutely crucial - so it would make sense to do whatever simplification I can in advance of introducing each set of constants.
Goal
I'd like to get the time complexity for solving each constant set down to O(n)... I may be dreaming however.
Other Info

The matrix really is a million (maybe even 100 million) rows
It's very sparse
It has a bandwidth ranging between 2 and as much as a few hundred
Even for larger bandwidths, each row has max. 7 or 8 non zero entries
It takes an (extremely rough) diagonal form
Each matrix 'b' in Ax = b is actually very sparse itself
Due to the nature of the problem, a solution should always be available 
All nonzero values are real, floating point numbers. They can be negative however.

REPLY [2 votes]: There is a huge amount of research on this problem --- it turns out that solving huge linear systems is pretty ubiquitous in applied maths, because linear algebra is one of the few things that we can do efficiently for large problems.
Sparse LU decompositions have already been mentioned; the other main approach is Krylov subspace methods (such as GMRES) with a vast choice of preconditioners --- there is a whole lot of strategies, and many of them work only for specific problems.
I would suggest contacting someone in person; see who works in numerical analysis at your local institution. $10^6$ to $10^8$ is already quite a large problem, and it is likely that some tricks of the trade are needed to get things working. All this especially if you say that "performance is crucial".
Also, in any case try to use an existing program or library (Matlab, for some quick experimentation if the problem fits in memory for it, or Scipy, PetSC, Trilinos, ...). If you are not an expert, rewriting things is unlikely to match state-of-the-art code in terms of performance.