cholesky

In the FETI method, one need the generalized inverse of a sparse matrix K.
In my case, K is a matrix arising from the finite element discretization on a tetrahedral mesh

of linear elasticity equations. I know how to compute the gereralized inverse from the LDL^T but I am not sure how to compute the LDL^T decomposition efficently. The algorithm use both rows and columns of the matrix...
Should I use a left or right looking algorihtm ?
If someone can provide me any hint to do that, I would appreciate a lot.
Thank you !