Introduction
Purpose
Given a sequence of vectors $\{s_k\}$ and $\{y_k\}$ \mbox{and scale factors} $\{\delta_k\}$, {s<sub>k</sub>} and {y<sub>k</sub>} and scalars {δ<sub>k</sub>}, {sk} and {yk} and scalars {deltak}, **obtain the product of a limited-memory secant approximation Hk$ (or its inverse) with a given vector**, using one of a variety of well-established formulae.
Currently, only the control and inform parameters are exposed; these are provided and used by other GALAHAD packages with C interfaces.
Authors
N. I. M. Gould, STFC-Rutherford Appleton Laboratory, England.
C interface, additionally J. Fowkes, STFC-Rutherford Appleton Laboratory.
Julia interface, additionally A. Montoison and D. Orban, Polytechnique Montréal.
Originally released
July 2014, C interface January 2022.
Method
Given a sequence of vectors $\{s_k\}$ and $\{y_k\}$ \mbox{and scale factors} $\{\delta_k\}$, {s<sub>k</sub>} and {y<sub>k</sub>} and scalars {δ<sub>k</sub>}, {sk} and {yk} and scalars {deltak}, a limited-memory secant approximation Hk$ is chosen so that $H_{\max(k-m,0)} = \delta_k I$, $H_{k-j} s_{k-j} = y_{k-j}$ and $\| H_{k-j+1} - H_{k-j}\|$ is “small” for $j = \min(k-1,m-1), \ldots, 0$. Different ways of quantifying “small” distinguish different methods, but the crucial observation is that it is possible to construct $H_k$ quickly from ${s_k}$, ${y_k}$ and $\delta_k$, and to apply it and its inverseto a given vector $v$. It is also possible to apply similar formulae to the “shifted” matrix $H_k + \lambda_k I$ that occurs in trust-region methods.
Reference
The basic methods are those given by
R. H. Byrd, J. Nocedal and R. B. Schnabel (1994) Representations of quasi-Newton matrices and their use in limited memory methods. Mathenatical Programming, 63(2) 129-156,
with obvious extensions.