Acceleration of Adaptive normalized quasi-Newton algorithm with improved upper bounds of the condition number

Abstract

In 2013, Nguyen and Yamada proposed Adaptive normalized quasi-Newton algorithm and its adaptive step size for accurate and stable extraction of the first generalized eigenvector. The adaptive step size is determined by an upper bound of the condition number of a time-varying matrix. However, the employed upper bound is fairly tight only when the size of matrix is small, which degrades the performance of the algorithm for general case. In this paper, we propose new adaptive step sizes with aid of tighter upper bounds of the condition number. The proposed adaptive step sizes can be implemented efficiently, which are the same calculation order with the original adaptive step size. Numerical experiments show that the proposed adaptive step sizes succeed in extending the applicability of the algorithm.