Eigenvalue Problem For Symmetric Matrices Research Articles

On eigenvalues of real symmetric interval matrices: Sharp bounds and disjointness

In this paper, the eigenvalue problem of real symmetric interval matrices is studied. First, in the case of $2 \times 2$ real symmetric interval matrices, all the four endpoints of the two eigenvalue intervals are determined. These are not necessarily eigenvalues of vertex matrices, but it is shown that such a real symmetric interval matrix can be constructed from the original one. Then, necessary and sufficient conditions are provided for the disjointness of eigenvalue intervals. In the general $n\times n$ case, due to Hertz, a set of special vertex matrices determines the maximal eigenvalue and a similar statement holds for the minimal one. In a special case, namely if the right endpoints of the off-diagonal intervals are not smaller than the absolute value of the left ones, he concluded the vertex matrix of the right endpoints provides the maximal eigenvalue. Generalizing it, it is shown that in the case of real symmetric interval matrices with special sign pattern, a single vertex matrix determines one of the extremal bounds.

Nonlinear Random Matrix-Based Intelligent Management Model for Swimming Place Waters

In this paper, a nonlinear random matrix approach is used to analyze the management of swimming place waters, and in this way, an intelligent management model is designed and applied to the actual swimming place waters management. Firstly, some basic small deviation results of the random matrix are presented. Then, several types of small deviation inequalities are obtained for the maximum eigenvalues of independent stochastic semi-positive definite (PSD) matrices. These small deviation inequalities are independent of the matrix dimension, and the results apply to high-dimensional and even infinite-dimensional matrices. For the inverse eigenvalue problem of higher-order doubly random matrices, the conclusion that two symmetric doubly random matrices are combined into a higher-order symmetric doubly random matrix is proved. In other words, the method of constructing new symmetric random matrices using smaller matrices with known spectra is applied to the inverse eigenvalue problem of higher-order symmetric random matrices, and sufficient conditions for the existence of the solution of the inverse eigenvalue problem of higher-order random matrices are derived, avoiding the discussion of the parity of the number of eigenvalues. This paper focuses on realistic management, to explore the standardization of safety management of swimming venues, and is based on modern management theory, risk management theory and methods, using research integration method, fieldwork method, questionnaire survey method, expert interview method, and other research methods, detailing the methods and conclusions of risk identification in safety management of swimming venues, and further exploring the safety management standards of swimming venues on this basis. And, combined with risk management theory, depending on the probability of occurrence and the degree of harm of risk, four basic disposal methods such as risk avoidance, risk control, risk transfer, and risk self-retaining are proposed.

A Parallel Algorithm for Solving a Partial Eigenvalue Problem for Block-Diagonal Bordered Matrices

A hybrid algorithm of the iterative method for the solution subspace of a partial generalized eigenvalue problem for symmetric positive definite sparse matrices of block-diagonal structure with bordering on hybrid computers with graphic processors is proposed, efficiency coefficients of the algorithm are obtained, and the algorithm is tested against test and practical problems.

Eigenvalue Problems for Tensor-Block Matrices and Their Applications to Mechanics

In this paper, we state and examine the eigenvalue problem for symmetric tensor-block matrices of arbitrary even rank and arbitrary size m × m, m ≥ 1. We present certain definitions and theorems of the theory of tensor-block matrices. We obtain formulas that express classical invariants (that are involved in the characteristic equation) of a tensor-block matrix of arbitrary even rank and size 2×2 through the first invariants of powers of the same tensor-block matrix and also inverse formulas. A complete orthonormal system of tensor eigencolumns for a tensor-block matrix of arbitrary even rank and size 2 × 2 is constructed. The generalized eigenvalue problem for a tensor-block matrix is stated. As a particular case, the tensor-block matrix of tensors of elastic moduli is considered. We also present canonical representations of the specific energy of deformation and defining relations. We propose a classification of anisotropic micropolar linearly elastic media that do not possess a symmetry center.

Solution of symmetric positive semidefinite Procrustes problem

In this paper, the symmetric positive semidefinite Procrustes problem is considered. By using matrix inner product and matrix decomposition theory, an explicit expression of the solution is given. Based on the explicit expression given in this paper, it is easy to derive the explicit expression of the solution given by Woodgate [K.G. Woodgate. Least-squares solution of F = PG over positive semidefinite symmetric P . Linear Algebra Appl., 245:171–190, 1996.] and by Liao [A.P. Liao. On the least squares problem of a matrix equation. J. Comput. Math., 17:589–594, 1999.] for the Procrustes problem in some special cases. The explicit expression given in this paper also shows that the solution of the special inverse eigenvalue problem considered by Zhang [L. Zhang. A class of inverse eigenvalue problem for symmetric nonnegative definite matrices. J. Hunan Educational Inst., 2:11–17, 1995 (in Chinese).] can be computed exactly. Examples to illustrate the correctness of the theory results are given.

Solution of symmetric positive semidefinite Procrustes problem

In this paper, the symmetric positive semidefinite Procrustes problem is considered. By using matrix inner product and matrix decomposition theory, an explicit expression of the solution is given. Based on the explicit expression given in this paper, it is easy to derive the explicit expression of the solution given by Woodgate [K.G. Woodgate. Least-squares solution of F = PG over positive semidefinite symmetric P . Linear Algebra Appl., 245:171–190, 1996.] and by Liao [A.P. Liao. On the least squares problem of a matrix equation. J. Comput. Math., 17:589–594, 1999.] for the Procrustes problem in some special cases. The explicit expression given in this paper also shows that the solution of the special inverse eigenvalue problem considered by Zhang [L. Zhang. A class of inverse eigenvalue problem for symmetric nonnegative definite matrices. J. Hunan Educational Inst., 2:11–17, 1995 (in Chinese).] can be computed exactly. Examples to illustrate the correctness of the theory results are given.

Optimizations of the eigensolvers in the ELPA library

The solution of (generalized) eigenvalue problems for symmetric or Hermitian matrices is a common subtask of many numerical calculations in electronic structure theory or materials science. Depending on the scientific problem, solving the eigenvalue problem can easily amount to a sizeable fraction of the whole numerical calculation, and quite often is even the dominant part by far. For researchers in the field of computational materials science, an efficient and scalable solution of the eigenvalue problem is thus of major importance. The ELPA-library (Eigenvalue SoLvers for Petaflop-Applications) is a well-established dense direct eigenvalue solver library, which has proven to be very efficient and scalable up to very large core counts. It is in a wide-spread production use on a large variety of HPC systems worldwide, and is applied by many codes in the field of materials science. In this paper, we describe the latest optimizations of the ELPA-library for new HPC architectures of the Intel Skylake processor family with an AVX-512 SIMD instruction set, or for HPC systems accelerated with recent GPUs. Apart from those direct hardware-related optimizations, we also describe a complete redesign of the API in a modern modular way, which, apart from a much simpler and more flexible usability, leads to a new path to access system-specific performance optimizations. In order to ensure optimal performance for a particular scientific setting or a specific HPC system, the new API allows the user to influence in a straightforward way the internal details of the algorithms and of performance-critical parameters used in the ELPA-library. On top of that, we introduce an autotuning functionality, which allows for finding the best settings in a self-contained automated way, without the need of much user effort. In situations where many eigenvalue problems with similar settings have to be solved consecutively, the autotuning process of the ELPA-library can be done “on-the-fly”, without the need of preceding the simulation with an “artificial” autotuning step. Practical applications from materials science which rely on reaching a numerical convergence limit by so-called self-consistency iterations can profit from the on-the-fly autotuning. On some examples of scientific interest, simulated with the FHI-aims application, the advantages of the latest optimizations of the ELPA-library are demonstrated.

Inverse eigenvalue problems for checkerboard Toeplitz matrices

The inverse eigenvalue problem for real symmetric Toeplitz matrices motivates this investigation. The existence of solutions is known, but the proof, due to H. Landau, is not constructive. Thus a restriction, namely the required eigenvalues are to be equally spaced, is considered here. Two types of structured matrices arise, herein termed, “checkerboard” and “outer-banded”. Examples are presented. Properties of these structured matrices are explored and a full characterization of checkerboard matrices is given. The inverse eigenvalue problem is solved within the class of odd checkerboard matrices. In addition the “symmetric-spectrum” inverse eigenvalue problem is solved within a subclass of Hankel matrices. A regularity conjecture of H. Landau for the Toeplitz inverse eigenvalue problem is discussed and a similar conjecture for checkerboard Toeplitz matrices is given.

A class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth

In this paper, we propose and discuss a class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth. For an odd 2p+1 with a positive integer p, the problem is to construct an n×n real symmetric banded matrix with bandwidth 2p+1 whose m×m leading principal submatrix is a given m×m real symmetric banded matrix with bandwidth 2p+1 and spectrum is a given set of real numbers {λi}i=1n, where the number of distinct real numbers of {λi}i=1n is 2k when m=pk, or 2k+1 when pk<m<p(k+1), where m,n and k are positive integers and m<n. We point out that the well-known double dimensional (DD) problem is a special case of our proposed inverse eigenvalue problems. The necessary and sufficient condition for the solvability of the above inverse eigenvalue problem is derived, and the target real symmetric banded matrix can be constructed by the block Lanczos algorithms when the inverse eigenvalue problem is solvable. Several numerical examples show that our algorithms are feasible. Some concluding remarks are introduced.

Implementation of algebraic procedures on the GPU using CUDA architecture on the example of generalized eigenvalue problem

Abstract The ready to use set of functions to facilitate solving a generalized eigenvalue problem for symmetric matrices in order to efficiently calculate eigenvalues and eigenvectors, using Compute Unified Device Architecture (CUDA) technology from NVIDIA, is provided. An integral part of the CUDA is the high level programming environment enabling tracking both code executed on Central Processing Unit and on Graphics Processing Unit. The presented matrix structures allow for the analysis of the advantages of using graphics processors in such calculations.

Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices

We present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with high relative accuracy in O(n) operations. The algorithm is based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to the complex Hermitian case. The algorithm is similar to the algorithm for solving the eigenvalue problem for real symmetric arrowhead matrices from N. Jakovčević Stor et al. (2015) [16].

An algorithm for constructing nonnegative matrices with prescribed real eigenvalues

Provided with the real spectrum, this paper presents a numerical procedure based on the induction principle to solve two types of inverse eigenvalue problems, one for nonnegative matrices and another for symmetric nonnegative matrices. As an immediate application, our approach can offer not only the sufficient condition to solve inverse eigenvalue problems for nonnegative or symmetric nonnegative matrices, but also a numerical way to solve inverse eigenvalue problems for stochastic matrices. Numerical examples are presented to show the capacity of our method.

Barzilai-Borwein-like methods for the extreme eigenvalue problem

We consider numerical methods for the extreme eigenvalue problem of large scale symmetric positive definite matrices. By the variational principle, the extreme eigenvalue can be obtained by minimizing some unconstrained optimization problem. Firstly, we propose two adaptive nonmonotone Barzilai-Borwein-like methods for the unconstrained optimization problem. Secondly, we prove the global convergence of the two algorithms under some conditions. Thirdly, we compare our methods with eigsand the power method for the standard test problems from the UF Sparse Matrix Collection. The primary numerical experiments indicate that the two algorithms are promising.

Parallel Rayleigh Quotient Optimization with FSAI‐Based Preconditioning

The present paper describes a parallel preconditioned algorithm for the solution of partial eigenvalue problems for large sparse symmetric matrices, on parallel computers. Namely, we consider the Deflation‐Accelerated Conjugate Gradient (DACG) algorithm accelerated by factorized‐sparse‐approximate‐inverse‐ (FSAI‐) type preconditioners. We present an enhanced parallel implementation of the FSAI preconditioner and make use of the recently developed Block FSAI‐IC preconditioner, which combines the FSAI and the Block Jacobi‐IC preconditioners. Results onto matrices of large size arising from finite element discretization of geomechanical models reveal that DACG accelerated by these type of preconditioners is competitive with respect to the available public parallel hypre package, especially in the computation of a few of the leftmost eigenpairs. The parallel DACG code accelerated by FSAI is written in MPI‐Fortran 90 language and exhibits good scalability up to one thousand processors.

Convergence Analysis of Gradient Iterations for the Symmetric Eigenvalue Problem

Gradient iterations for the Rayleigh quotient are simple and robust solvers to determine a few of the smallest eigenvalues together with the associated eigenvectors of (generalized) matrix eigenvalue problems for symmetric matrices. Sharp convergence estimates for the Ritz values and Ritz vectors are derived for various steepest descent/ascent gradient iterations. The analysis shows that poorest convergence of the eigenvalue approximations is attained in a three-dimensional invariant subspace; explicit convergence estimates are then derived by means of a minidimensional analysis.

An overview on the eigenvalue computation for matrices

This paper sketches the research developments in the area of computational methods for solving the eigenvalue problems and how the methods developed relate to each other as they evolve over centuries. This is an attempt to write a complete overview on the research on computational aspects of eigenvalue problem, emphasize the history of methods that still play a role and some of those that no longer are considered to be on the main track but are somehow related to the present techniques in some smaller steps. This contribution brings out the state-of-the-art of the algorithms for solving large-scale eigenvalue problems for both symmetric and nonsymmetric matrices separately, thereby clearly drawing a comparison between the differences in the algorithms in practical use for the two. Some of the methods or modifications to the earlier methods that have been notable at the turn of the 21st century will also be covered through this paper under Modem Approaches. Also as the standard eigenvalue problem has become better understood, in a numerical sense, progress has been made in the generalized eigenvalue problem and this will briefly cover developments in solving these problems too.

A uniform object-oriented solution to the eigenvalue problem for real symmetric and Hermitian matrices

We present a system of classes, SHMatrix, to deal in a unified way with the computation of eigenvalues and eigenvectors in real symmetric and Hermitian matrices. Thus, two descendant classes, one for the real symmetric and other for the Hermitian cases, override the abstract methods defined in a base class. The use of the inheritance relationship and polymorphism allows handling objects of any descendant class using a single reference of the base class. The system of classes is intended to be the core element of more sophisticated methods to deal with large eigenvalue problems, as those arising in the variational treatment of realistic quantum mechanical problems. The present system of classes allows computing a subset of all the possible eigenvalues and, optionally, the corresponding eigenvectors. Comparison with well established solutions for analogous eigenvalue problems, as those included in LAPACK, shows that the present solution is competitive against them. Program summary Program title: SHMatrix Catalogue identifier: AEHZ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEHZ_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2616 No. of bytes in distributed program, including test data, etc.: 127 312 Distribution format: tar.gz Programming language: Standard ANSI C++. Computer: PCs and workstations. Operating system: Linux, Windows. Classification: 4.8. Nature of problem: The treatment of problems involving eigensystems is a central topic in the quantum mechanical field. Here, the use of the variational approach leads to the computation of eigenvalues and eigenvectors of real symmetric and Hermitian Hamiltonian matrices. Realistic models with several degrees of freedom leads to large (sometimes very large) matrices. Different techniques, such as divide and conquer, can be used to factorize the matrices in order to apply a parallel computing approach. However, it is still interesting to have a core procedure able to tackle the computation of eigenvalues and eigenvectors once the matrix has been factorized to pieces of enough small size. Several available software packages, such as LAPACK, tackled this problem under the traditional imperative programming paradigm. In order to ease the modelling of complex quantum mechanical models it could be interesting to apply an object-oriented approach to the treatment of the eigenproblem. This approach offers the advantage of a single, uniform treatment for the real symmetric and Hermitian cases. Solution method: To reach the above goals, we have developed a system of classes: SHMatrix. SHMatrix is composed by an abstract base class and two descendant classes, one for real symmetric matrices and the other for the Hermitian case. The object-oriented characteristics of inheritance and polymorphism allows handling both cases using a single reference of the base class. The basic computing strategy applied in SHMatrix allows computing subsets of eigenvalues and (optionally) eigenvectors. The tests performed show that SHMatrix is competitive, and more efficient for large matrices, than the equivalent routines of the LAPACK package. Running time: The examples included in the distribution take only a couple of seconds to run.

On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.

Explicit Solution of the Inverse Eigenvalue Problem of Real Symmetric Matrices and Its Application to Electrical Network Synthesis

A novel procedure for explicit construction of the entries of real symmetric matrices with assigned spectrum and the entries of the corresponding orthogonal modal matrices is presented. The inverse eigenvalue problem of symmetric matrices with some specific sign patterns (including hyperdominant one) is explicitly solved too. It has been shown to arise thereof a possibility of straightforward solving the inverse eigenvalue problem of symmetric hyperdominant matrices with assigned nonnegative spectrum. The results obtained are applied thereafter in synthesis of driving‐point immittance functions of transformerless, common‐ground, two‐element‐kind RLC networks and in generation of their equivalent realizations.

Numerical Methods for Solving Inverse Eigenvalue Problems for Nonnegative Matrices

Presented here are two related numerical methods, one for the inverse eigenvalue problem for nonnegative or stochastic matrices and another for the inverse eigenvalue problem for symmetric nonnegative matrices. The methods are iterative in nature and utilize alternating projection ideas. For the algorithm for the symmetric problem, the main computational component of each iteration is an eigenvalue-eigenvector decomposition, while for the other algorithm, it is a Schur matrix decomposition. Convergence properties of the algorithms are investigated and numerical results are also presented. While the paper deals with two specific types of inverse eigenvalue problems, the ideas presented here should be applicable to many other inverse eigenvalue problems, including those involving nonsymmetric matrices.