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### Special cases: central differences

Incomplete factorizations of type are particularly simple if the matrix is derived from central differences on a Cartesian product grid. As remarked above, we only have to calculate the pivots of the factorization; other elements in the triangular factors are equal to off-diagonal elements of .

In the following we will assume a natural, line-by-line, ordering of the grid points.

Letting , be coordinates in a regular 2D grid, it is easy to see that the pivot on grid point is only determined by pivots on points and . If there are n points on each of m grid lines, we get the following generating relations for the pivots:

Conversely, we can describe the factorization algorithmically as

In the above we have assumed that the variables in the problem are ordered according to the so-called ``natural ordering'': a sequential numbering of the grid lines and the points within each grid line. Below we will encounter different orderings of the variables.