I have a large square n x n matrix where n = 1,000,000. I need to calculate the inverse of this matrix. This would take a long time to do directly. However, the inverse matrix is very sparse. I know which elements of the inverse matrix are non-zero, and each row (and column) only has 100 non-zero entries.
So I thought I could directly calculate each of these 100,000,000 elements of the inverse matrix. Therefore I need MATLAB code to calculate individual elements of an inverse matrix in a much faster way than simply taking the inverse and checking the individual elements.
That is, given a matrix M, find element (i,j) of inv(M) quickly and directly using MATLAB. The answer must be much faster to calculate than simply taking the inverse and checking the appropriate elements.