A Newton’s method characterization for real eigenvalue problems

Abstract

The current work is a continuation of Kim (An unconstrained global optimization framework for real symmetric eigenvalue problems, submitted), where an unconstrained optimization problem was proposed and a first order method was shown to converge to a global minimizer that is an eigenvector corresponding to the smallest eigenvalue with no eigenvalue estimation given. In this second part, we provide local and global convergence analyses of the Newton’s method for real symmetric matrices. Our proposed framework discovers a new eigenvalue update rule and shows that the errors in eigenvalue and eigenvector estimations are comparable, which extends to nonsymmetric diagonalizable matrices as well. At the end, we provide numerical experiments for generalized eigenvalue problems and for the trust region subproblem discussed in Adachi et al. (SIAM J Optim 27(1):269–291, 2017) to confirm efficiency and accuracy of our proposed method.