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### The special case of block tridiagonality

In many applications, a block tridiagonal structure can be found in the coefficient matrix. Examples are problems on a 2D regular grid if the blocks correspond to lines of grid points, and problems on a regular 3D grid, if the blocks correspond to planes of grid points. Even if such a block tridiagonal structure does not arise naturally, it can be imposed by renumbering the variables in a Cuthill-McKee ordering [57].

Such a matrix has incomplete block factorizations of a particularly simple nature: since no fill can occur outside the diagonal blocks , all properties follow from our treatment of the pivot blocks. The generating recurrence for the pivot blocks also takes a simple form. Let be the coefficient matrix, block indexed, and let be the sequence of pivots, then

The sequence consists of approximations to the inverses of the pivots in the manner outlined above.