2 research outputs found
On the optimality and sharpness of Laguerre's lower bound on the smallest eigenvalue of a symmetric positive definite matrix
summary:Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on and have attracted attention recently, because they can be computed in operations when 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 and 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 and show that the gap becomes smallest when approaches 1 or . These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms
On the optimality and sharpness of Laguerre's lower bound on the smallest eigenvalue of a symmetric positive definite matrix
summary:Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on and have attracted attention recently, because they can be computed in operations when 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 and 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 and show that the gap becomes smallest when approaches 1 or . These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms