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Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix $A\in {ℝ}^{m\times m}$ play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on $\mathrm{Tr}\left(A^{-1}\right)$ and $\mathrm{Tr}\left(A^{-2}\right)$ have attracted attention recently, because they can be computed in $O\left(m\right)$ 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...

We design shifted $LR$ transformations based on the integrable discrete hungry Toda equation to compute eigenvalues of totally nonnegative matrices of the banded Hessenberg form. The shifted $LR$ transformation can be regarded as an extension of the extension employed in the well-known dqds algorithm for the symmetric tridiagonal eigenvalue problem. In this paper, we propose a new and effective shift strategy for the sequence of shifted $LR$ transformations by considering the concept of the Newton shift....