On the optimality and sharpness of Laguerre’s lower bound on the smallest eigenvalue of a symmetric positive definite matrix

• Published in: Applications of mathematics (Prague) Vol. 62; no. 4; pp. 319 - 331
• Main Author: Yamamoto, Yusaku
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2017; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Classical and Continuum Physics; Eigenvalues; Iterative methods; Lower bounds; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Matrix methods; Optimization; Sharpness; Theoretical; 15A42; symmetric positive definite matrix; dqds algorithm; eigenvalue bound; singular value computation; Laguerre bound; 15A18
• ISSN: 0862-7940; 1572-9109
• DOI: 10.21136/AM.2017.0022-17

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ R m × m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr( A −1 ) and Tr( A −2 ) have attracted attention recently, because they can be computed in O ( m ) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr( A −1 ) and Tr( A −2 ) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr( A −1 )} 2 /Tr( A −2 ) approaches 1 or m . These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.