Inverting big matrices. REALLY BIG

[Okefenokee]

Inverting big matrices. REALLY BIG!

How is it done?

Let's say I have a sparsely populated 1 gazillion by 1 gazillion square matrix in a formula like this A*x = b. What sort of efficient methods exist to do the following?: find the rank, invert it if it has full rank, find the null vectors if it does not have full rank.

Also, it would be great if the technique lends itself to parallel programing. It's for a pet project that I'm playing with. I can think of a couple ways to do this but I'm sure my solution would be sloppy compared to what is already out there. One of my ideas was to package the inner matrices into manageable blocks and do some Gaussian elimination in stages.

 

[fresh_42]

It all depends on we can use further properties of the matrix. In general I would look for an efficient algorithm for matrix multiplication. I'm not sure where the current record holder for the matrix exponent lies, but I guess numerical stability is more of a problem than to implement an algorithm which only works for really big matrices, the more as the I/O operations outperform calculation times by far.