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### Approximate inverses

In block factorizations a pivot block is generally forced to be sparse, typically of banded form, and that we need an approximation to its inverse that has a similar structure. Furthermore, this approximation should be easily computable, so we rule out the option of calculating the full inverse and taking a banded part of it.

The simplest approximation to is the diagonal matrix of the reciprocals of the diagonal of : . Figure: Algorithm for approximating the inverse of a banded matrix

Other possibilities were considered by Axelsson and Eijkhout , Axelsson and Polman , Concus, Golub and Meurant , Eijkhout and Vassilevski , Kolotilina and Yeremin , and Meurant . One particular example is given in figure . It has the attractive theoretical property that, if the original matrix is symmetric positive definite and a factorization with positive diagonal can be made, the approximation to the inverse is again symmetric positive definite.

Banded approximations to the inverse of banded matrices have a theoretical justification. In the context of partial differential equations the diagonal blocks of the coefficient matrix are usually strongly diagonally dominant. For such matrices, the elements of the inverse have a size that is exponentially decreasing in their distance from the main diagonal. See Demko, Moss and Smith  for a general proof, and Eijkhout and Polman  for a more detailed analysis in the -matrix case.

Jack Dongarra
Mon Nov 20 08:52:54 EST 1995