Variants of existing sequential incomplete factorization preconditioners with a higher degree of parallelism have been devised, though they are perhaps less efficient in purely scalar terms than their ancestors. Some examples are: reorderings of the variables (see Duff and Meurant [79] and Eijkhout [85]), expansion of the factors in a truncated Neumann series (see Van der Vorst [201]), various block factorization methods (see Axelsson and Eijkhout [15] and Axelsson and Polman [21]), and multicolor preconditioners.
Multicolor preconditioners have optimal parallelism among incomplete factorization methods, since the minimal number of sequential steps equals the color number of the matrix graphs. For theory and applications to parallelism see Jones and Plassman [128][127].