Compact representation of the full Broyden class of quasi-Newton updates
This work provides a unified framework for representing and computing with the full Broyden class, enabling efficient eigenvalue computation and sensitivity analysis for limited-memory quasi-Newton methods.
This paper extends the compact representation of quasi-Newton updates to the full Broyden class, including both rank-one and rank-two updates, and demonstrates that a practical implementation can accurately represent these matrices and efficiently solve linear systems to high accuracy.
In this paper, we present the compact representation for matrices belonging to the 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 the same Broyden update is used each iteration; rather, different members of the Broyden class may be used 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, as well as 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 to 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 perform sensitivity analysis.