One modification to the basic idea of incomplete factorizations is as follows: If the product is nonzero, and fill is not allowed in position , instead of simply discarding this fill quantity subtract it from the diagonal element . Such a factorization scheme is usually called a ``modified incomplete factorization''.

Mathematically this corresponds to forcing the preconditioner to have the same rowsums as the original matrix. One reason for considering modified incomplete factorizations is the behavior of the spectral condition number of the preconditioned system. It was mentioned above that for second order elliptic equations the condition number of the coefficient matrix is as a function of the discretization mesh width. This order of magnitude is preserved by simple incomplete factorizations, although usually a reduction by a large constant factor is obtained.

Modified factorizations are of interest because, in combination with small perturbations, the spectral condition number of the preconditioned system can be of a lower order. It was first proved by Dupont, Kendall and Rachford [81] that a modified incomplete factorization of gives for the central difference case. More general proofs are given by Gustafsson [112], Axelsson and Barker [.2]AxBa:febook, and Beauwens [32][31].

Instead of keeping row sums constant, one can also keep *column*
sums constant.
In computational fluid mechanics this idea is justified with the argument
that the material balance stays constant over all iterates.
(Equivalently, one wishes to avoid `artificial
diffusion'.)
Appleyard and Cheshire [4] observed that if and
have the same column sums, the choice

guarantees that the sum of the elements in (the material balance error) is zero, and that all further have elements summing to zero.

Modified incomplete factorizations can break down, especially when the variables are numbered other than in the natural row-by-row ordering. This was noted by Chan and Kuo [50], and a full analysis was given by Eijkhout [86] and Notay [161].

A slight variant of modified incomplete factorizations consists of the class of ``relaxed incomplete factorizations''. Here the fill is multiplied by a parameter before it is subtracted from the diagonal; see Ashcraft and Grimes [11], Axelsson and Lindskog [19][18], Chan [44], Eijkhout [86], Notay [162], Stone [194], and Van der Vorst [204]. For the dangers of MILU in the presence of rounding error, see Van der Vorst [206].

Mon Nov 20 08:52:54 EST 1995