# Numerical analysis, pivoting and incomplete LU decomposition

When doing LU decomposition, the algorithm will break down if any of the diagonal element $$x_{ii}$$ is zero. Therefore, we can use pivoting on the matrix such that $$x_{ii}$$ is no longer zero. That is, instead of looking at the $$x_{ii}$$ we look at another element $$x$$ in the matrix.

One problem with LU decomposition is that when our matrix $$A$$ is sparse, we would like to keep the sparsity pattern of $$A$$, something that will not happen when we do LU decomposition. Therefore, we can use incomplete LU decomposition, where if $$x_{ij}$$ of $$A$$ is zero, we also skip $$x_{ij}$$ of $$L$$ or $$U$$ depending on where $$x_{ij}$$ is in $$A$$.

My question is: will we do pivoting for incomplete LU decomposition? If we do, how do we do it? If we don’t, how do we make sure that our algorithm doesn't break?

• This may be rather a question for Scientific Computing SE, scicomp.stackexchange.com – Joce Feb 14 at 16:14
• I am not a specialist but I would say yes. You would perform it the same way as for a dense matrix, choosing the pivot among the nonzero elements and performing elimination only on the nonzero elements. It can probably break, but not with a worse probability than without pivoting (just a guess). – Yves Daoust Feb 14 at 16:24