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?

  • $\begingroup$ 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. $\endgroup$ – vydesaster Feb 27 at 18:37

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