Rank-one update of Cholesky factorization

Symmetric positive definite matrices appear in most methods for Unconstrained Optimization. The method widely adopted for factoring these matrices is Cholesky Factorization. Furthermore, in Quassi-Newton methods for unconstrained optimization these matrices are continually updated and factorized. Here we consider factoring an n x n symmetric positive definite matrix of the form: A' = A + CXZZT , where A is symmetric positive definite, a is a scalar and z is a vector of length n. We assume that A has already been factorized by Cholesky factorization. The adopted methods are due to Gill et. al. [GGS75, GM72]