6.3 Additional Considerations

The algorithms presented in Section 6.1 and Section 6.2 are based on the work of Bennett [1]. The nomenclature is similar to that of Gill et al. [7]. These citations describe procedures for updating the factors of an LDU decomposition.

The procedure described by Bennett [1] is more general than the algorithms described in this section in that it applies to rank m changes to A. However, decomposing a rank m change into m rank one changes and applying the current algorithms has the same complexity as Bennett’s process and saves a little array space. Gill et al. [7] state that Bennett’s algorithm is theoretically unstable unless L = UT and y = z. In practice, Bennett’s algorithm has proven to be stable for many physical problems with reasonable values of α\alpha, y, and z. The algorithm rarely exhibits instability when it is applied to diagonally dominant matrices where pivoting is not required. Gill et al. [7] describe alternate algorithms for situations where stability problems arise.

Hager [12] provides a good overview of approaches to the problem of updating the inverse of a matrix and describes practical areas in which the problem arises. Chan and Brandwajn [2] examine applications in network analysis.