Cholesky Factor Updating Techniques for Rank-Two Matrix Modifications.

Gill, Golub, Murray and Saunders have described 5 methods by which the Cholesky factors of a positive-definite matrix may be updated when the matrix is subjected to a symmetric rank- one modification. In many minimization algorithms symmetric rank two modification are found. We show how each of the rank- one methods gives rise to a single-application rank-two method. For some of the methods this involves a new Householder transformation technique designed to eliminate elements of two vectors at once using a rank 1 correction of the identity matrix. On parallel and vector machines it is more economical to perform rank-two updates rather than two rank-one updates.