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 $A\in \mathbb {R}^{m\times m}$ play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on ${\rm Tr}(A^{-1})$ and ${\rm Tr}(A^{-2})$ 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 ${\rm Tr}(A^{-1})$ and ${\rm Tr}(A^{-2})$ 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 $\{{\rm Tr}(A^{-1})\}^2/{\rm Tr}(A^{-2})$ approaches 1 or $m$. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms

Error Analysis of the Cholesky QR-Based Block Orthogonalization Process for the One-Sided Block Jacobi SVD Algorithm

The one-sided block Jacobi method (OSBJ) has attracted attention as a fast and accurate algorithm for the singular value decomposition (SVD). The computational kernel of OSBJ is orthogonalization of a column block pair, which amounts to computing the SVD of this block pair. Hari proposes three methods for this partial SVD, and we found through numerical experiments that the variant named "V2", which is based on the Cholesky QR method, is the fastest variant and achieves satisfactory accuracy. While it is a good news from a practical viewpoint, it seems strange considering the well-known instability of the Cholesky QR method. In this paper, we perform a detailed error analysis of the V2 variant and explain why and when it can be used to compute the partial SVD accurately. Thus, our results provide a theoretical support for using the V2 variant safely in the OSBJ method

A fast and accurate computation method for reflective diffraction simulations

We present a new computation method for simulating reflection high-energy electron diffraction and the total-reflection high-energy positron diffraction experiments. The two experiments are used commonly for the structural analysis of material surface. The present paper improves the conventional numerical method, the multi-slice method, for faster computation, since the present method avoids the matrix-eigenvalue solver for the computation of matrix exponentials and can adopt higher-order ordinary differential equation solvers. Moreover, we propose a high-performance implementation based on multi-thread parallelization and cache-reusable subroutines. In our tests, this new method performs up to 2,000 times faster than the conventional method

Roundoff error analysis of the double exponential formula-based method for the matrix sign function

In this paper, we perform a roundoff error analysis of an integration-based method for computing the matrix sign function recently proposed by Nakaya and Tanaka. The method expresses the matrix sign function using an integral representation and computes the integral numerically by the double-exponential formula. While the method has large-grain parallelism and works well for well-conditioned matrices, its accuracy deteriorates when the input matrix is ill-conditioned or highly nonnormal. We investigate the reason for this phenomenon by a detailed roundoff error analysis.Comment: 6 pages, 1 figur

Identification of multiple actin-binding sites in cofilin-phosphatase Slingshot-1L

AbstractSlingshot-1L (SSH1L) is a phosphatase that specifically dephosphorylates and activates cofilin, an actin-severing and -depolymerizing protein. SSH1L binds to and is activated by F-actin in vitro, and co-localizes with F-actin in cultured cells. We examined the F-actin-binding activity, F-actin-mediated phosphatase activation, and subcellular distribution of various mutants of SSH1L. We identified three sites involved in F-actin binding of SSH1L: Trp-458 close to the C-terminus of the phosphatase domain, an LHK motif in the N-terminal region, and an LKR motif in the C-terminal region. These sites play unique roles in the control of subcellular localization and F-actin-mediated activation of SSH1L