Compact representation of the full Broyden class of quasi-Newton updates

In this paper, we present the compact representation for matrices belonging to the Broyden class of quasi-Newton updates, where each update may be either rank one or rank two. This work extends previous results solely for the restricted Broyden class of rank-two updates. In this article, it is not assumed that the same Broyden update is used in each iteration; rather, different members of the Broyden class may be used in each iteration. Numerical experiments suggest that a practical implementation of the compact representation is able to accurately represent matrices belonging to the Broyden class of updates. Furthermore, we demonstrate how to compute the compact representation for the inverse of these matrices and a practical algorithm for solving linear systems with members of the Broyden class of updates. We demonstrate through numerical experiments that the proposed linear solver is able to efficiently solve linear systems with members of the Broyden class of matrices with high accuracy. As an immediate consequence of this work, it is now possible to efficiently compute the eigenvalues of any limited-memory member of the Broyden class of matrices, allowing for the computation of condition numbers and the ability to perform sensitivity analysis.