Unconstrained variational principles for eigenvalues of real symmetric matrices
SIAM Journal on Mathematical Analysis
On the limited memory BFGS method for large scale optimization
Mathematical Programming: Series A and B
Globally and rapidly convergent algorithms for symmetric eigenproblems
SIAM Journal on Matrix Analysis and Applications
Representations of quasi-Newton matrices and their use in limited memory methods
Mathematical Programming: Series A and B
Applied numerical linear algebra
Applied numerical linear algebra
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
The Quasi-Cauchy Relation and Diagonal Updating
SIAM Journal on Optimization
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
An implicit riemannian trust-region method for the symmetric generalized eigenproblem
ICCS'06 Proceedings of the 6th international conference on Computational Science - Volume Part I
A DC programming approach for solving the symmetric Eigenvalue Complementarity Problem
Computational Optimization and Applications
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In certain circumstances, it is advantageous to use an optimization approach in order to solve the generalized eigenproblem, Ax = λBx, where A and B are real symmetric matrices and B is positive definite. In particular, this is the case when the matrices A and B are very large and the computational cost, prohibitive, of solving, with high accuracy, systems of equations involving these matrices. Usually, the optimization approach involves optimizing the Rayleigh quotient.We first propose alternative objective functions to solve the (generalized) eigenproblem via (unconstrained) optimization, and we describe the variational properties of these functions.We then introduce some optimization algorithms (based on one of these formulations) designed to compute the largest eigenpair. According to preliminary numerical experiments, this work could lead the way to practical methods for computing the largest eigenpair of a (very) large symmetric matrix (pair).