I have a large sparse matrix. Typical size is $299K$ by $300K$. I need to construct the null space of this matrix efficiently. The null space in this case is a $1000$ by $300K$ matrix which is in general dense.
My first question is what is the best way to compute this null space? I tried sparse QR factorization based method but the involved matrices are not so sparse and it is not as efficient as I would want it to be.
It seems that an LU factorization of a $300K$ by $300K$ can be computed quite fast so I was wondering if there is a way to achieve this using LU. I'm using Matlab so a Matlab function would be ideal.
Once I'll have the null space I actually need only several columns from it. I can simply extract these columns but it seems as a waste of time. Is there a way to compute only selected columns of the null space rather than all of it?
Have you looked at https://www.mathworks.com/help/matlab/ref/lu.html? It's the documentation for LU decomposition in matlab.