Representing Symmetric Rank Two Updates / David M. Gay.

Format:

Book

Author/Creator:

Gay, David M.

Contributor:

National Bureau of Economic Research.

Series:

Working Paper Series (National Bureau of Economic Research) no. w0124.

NBER working paper series no. w0124

Language:

English

Subjects (All):

Matrix analytic methods.

Physical Description:

1 online resource: illustrations (black and white);

Place of Publication:

Cambridge, Mass. National Bureau of Economic Research 1976.

Cambridge, Mass. : National Bureau of Economic Research, 1976.

Summary:

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 [delta] as the sum of rank 1 matrices. We show that it is easy to compute rank 1 matrices [delta1] and [delta2] such that [delta] = [delta1] + [delta2] and [the norm of delta1]+ [the norm of delta2] is minimized, where ||.|| 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 improve efficiency over other representations. In the common case where [delta] is indefinite, a choice of the form [delta1] = [delta2 to the power of T] = [xy to the power of T] appears best. This case occurs for rank 2 quasi- Newton updates [delta] exactly when [delta] may be obtained by symmetrizing some rank 1 update; such popular updates as the DFP, BFGS, PSB, and Davidon's new optimally conditioned update fall into this category.

Notes:

Print version record

February 1976.