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How is it done?

Let's say I have a sparsely populated 1 gazillion by 1 gazillion square matrix in a formula like thisA*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.

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# Inverting big matrices. REALLY BIG

Can you offer guidance or do you also need help?

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