# 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?

• In general it is possible to perform pivoting for ILU. I am not sure if ILU is used for solving linear equation systems. I have only seen it as a preconditioner. – vydesaster Feb 27 '19 at 18:37