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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
:
.
Other possibilities were considered by Axelsson and Eijkhout [15], Axelsson and Polman [20], and Concus, Golub and Meurant [55].
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 [62] for a general
proof, and Eijkhout and Polman [86] for a more detailed
analysis in the -matrix case.