We will begin our discussion of pivoting by identifying a condition in which pivoting is unnecessary. The matrix A is row diagonally dominant when the following inequality holds.
| $$\left|{a}_{ii}\right|>\sum_{j\neq i}\left|{a}_{ij}\right|,\mbox{ where }i=1,\cdots,n$$ | (52) |
The matrix A is column diagonally dominant when the following inequality holds.
| $$\left|{a}_{jj}\right|>\sum_{j\neq i}\left|{a}_{ij}\right|,\mbox{ where }j=1,\cdots,n$$ | (53) |
If either of these conditions apply, the LU decomposition algorithms discussed in this document are numerically stable without pivoting.