In this paper, a Parallel Direct Eigensolver for Sequences of Hermitian Eigenvalue Problems with no tridiagonalization is proposed, denoted by \texttt{PDESHEP}, and it combines direct methods with iterative methods. \texttt{PDESHEP} first reduces a Hermitian matrix to its banded form, then applies a spectrum slicing algorithm to the banded matrix, and finally computes the eigenvectors of the original matrix via backtransform. Therefore, compared with conventional direct eigensolvers, \texttt{PDESHEP} avoids tridiagonalization, which consists of many memory-bounded operations. In this work, the iterative method in \texttt{PDESHEP} is based on the contour integral method implemented in FEAST. The combination of direct methods with iterative methods for banded matrices requires some efficient data redistribution algorithms both from 2D to 1D and from 1D to 2D data structures. Hence, some two-step data redistribution algorithms are proposed, which can be $10\times$ faster than ScaLAPACK routine \texttt{PXGEMR2D}. For the symmetric self-consistent field (SCF) eigenvalue problems, \texttt{PDESHEP} can be on average $1.25\times$ faster than the state-of-the-art direct solver in ELPA when using $4096$ processes. Numerical results are obtained for dense Hermitian matrices from real applications and large real sparse matrices from the SuiteSparse collection.
Structure preserving quaternion conjugate gradient-type methods for solving non-Hermitian quaternion linear systems
School of Mathematics and Statistics, Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Normal University（福建师范大学数学与统计学院，分析数学及应用教育部重点实验室）；School of Mathematics and Statistics, Hainan University（海南大学数学与统计学院）；School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）
This paper proposes an efficient method for computing selected generalized eigenpairs of a sparse Hermitian definite matrix pencil $(A,B)$. Based on Zolotarev's best rational function approximations of the signum function and conformal mapping techniques, we construct the best rational function approximation of a rectangular function supported on an arbitrary interval via function compositions with partial fraction representations. This new best rational function approximation can be applied to construct spectrum filters of $(A,B)$ with a smaller number of poles than a direct construction with…
This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues located inside a given disk, and the proposed algorithms approximate these eigenvalues and associated eigenvectors by harmonic Rayleigh-Ritz projections on subspaces built by computing range spaces of rational matrix functions through randomized range finders. These rational matrix functions are designed so that directions associated with non-sought eigenvalues are dampened to (approximately) zero. Var…
Recently a novel family of eigensolvers, called spectral indicator methods (SIMs), was proposed. Given a region on the complex plane, SIMs first compute an indicator by the spectral projection. The indicator is used to test if the region contains eigenvalue(s). Then the region containing eigenvalues(s) is subdivided and tested. The procedure is repeated until the eigenvalues are identified within a specified precision. In this paper, using Cayley transformation and Krylov subspaces, a memory efficient multilevel eigensolver is proposed. The method uses less memory compared with the early versi…
Improved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach
Aachen Institute for advanced study in Computational Engineering Science, RWTH Aachen University（亚琛高等计算工程科学研究院，亚琛工业大学）；Depto. de Ingeniería y Ciencia de Computadores, Universidad Jaume I（计算机科学与工程学院，海梅一世大学）
Convergence rates of block iterations for solving eigenvalue problems typically measure errors of Ritz values approximating eigenvalues. The errors of the Ritz values are commonly bounded in terms of principal angles between the initial or iterative subspace and the invariant subspace associated with the target eigenvalues. Such bounds thus cannot be applied repeatedly as needed for restarted block eigensolvers, since the left- and right-hand sides of the bounds use different terms. They must be combined with additional bounds which could cause an overestimation. Alternative repeatable bounds …
Convergence analysis of block iterative solvers for Hermitian eigenvalue problems and the closely related research on properties of matrix-based signal filters are challenging, and attract increasing attention due to their recent applications in spectral data clustering and graph-based signal processing. We combine majorization-based techniques pioneered for investigating the Rayleigh-Ritz method in [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1521-1537] with tools of classical analysis of the block power method by Rutishauser [Numer. Math., 13 (1969), pp. 4-13] to derive convergence rate bound…
We propose a verified computation method for partial eigenvalues of a Hermitian generalized eigenproblem. The block Sakurai-Sugiura Hankel method, a contour integral-type eigensolver, can reduce a given eigenproblem into a generalized eigenproblem of block Hankel matrices whose entries consist of complex moments. In this study, we evaluate all errors in computing the complex moments. We derive a truncation error bound of the quadrature. Then, we take numerical errors of the quadrature into account and rigorously enclose the entries of the block Hankel matrices. Each quadrature point gives rise…
Hermitian and unitary matrices are two representatives of the class of normal matrices whose full eigenvalue decomposition can be stably computed in quadratic computing com plexity. Recently, fast and reliable eigensolvers dealing with low rank perturbations of unitary and Hermitian matrices were proposed. These structured eigenvalue problems appear naturally when computing roots, via confederate linearizations, of polynomials expressed in, e.g., the monomial or Chebyshev basis. Often, however, it is not known beforehand whether or not a matrix can be written as the sum of an Hermitian or unit…
Randomized Block Davidson Eigensolvers for Plane-Wave Density-Functional Theory
Paul Scherrer Institute（保罗谢勒研究所）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Swiss National Supercomputing Centre (CSCS)（瑞士国家超级计算中心）；ETH Zurich（苏黎世联邦理工学院）
Non-Linear Least-Squares Optimization of Rational Filters for the Solution of Interior Eigenvalue Problems
RWTH Aachen University（亚琛工业大学）；Jülich Supercomputing Centre, Institute for Advanced Simulation, Forschungszentrum Jülich GmbH（于利希超级计算中心，先进模拟研究所，于利希研究中心有限公司）
symveig: Verified eigenvalue enclosures for symmetry-decomposed Hermitian matrices
National Forensic Sciences University（国家法证科学大学）；Department of Applied Science and Humanities, School of Engineering & Technology, Vivekananda Institute of Professional Studies - Technical Campus（维韦卡南达专业研究所技术校区工程与技术学院应用科学与人文系）
For dense Hermitian matrices with small off-diagonal (numerical) ranks and in a hierarchically semiseparable form, we give a stable divide-and-conquer eigendecomposition method with nearly linear complexity (called SuperDC) that significantly improves an earlier basic algorithm in [Vogel, Xia, et al., SIAM J. Sci. Comput., 38 (2016)]. We incorporate a sequence of key stability techniques and provide many improvements in the algorithm design. Various stability risks in the original algorithm are analyzed, including potential exponential norm growth, cancellations, loss of accuracy with clustere…
