Representing Symmetric Rank Two Updates
David T Gay · National Bureau of Economic Research · 1976
Various quasi-Newton methods periodically add a symmetric "correction" matrix of rank at most 2 to a matrix approximating some quantity A of interest (such as the Hessian of an objective function).In this paper we examine several ways to express a symmetric rank 2 matrix as the sum of rank 1 matrices.We show that it is easy to compute rank 1 matrices and such that + 2 and I + 2I I is minimized, where H 1 I is any inner product norm.Such a representation recommends itself for use in those computer programs that maintain A explicitly, since it should reduce cancellation errors and/or iiiprove efficiency over other representations.In the common case where is indefinite, a choice of the form appears best.This case occurs for rank 2 quasi-Newton updates L exactly when L may be obtained by syrmnetrizing some rank 1 update; such popular updates as the IJFP, BFGS, PSB, and Davidon's new optimally conditioned update fall into this category..