• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제3권2호
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• pp.5-16
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• 1999

We introduce a new projector for accelerating convergence of a symmetric eigenvalue problem Ax = x, and devise a power/Lanczos hybrid algorithm. Acceleration can be achieved by removing the hard-to-annihilate nonsolution eigencomponents corresponding to the widespread eigenvalues with modulus close to 1, by estimating them accurately using the Lanczos method. However, the additional Lanczos results can be obtained without expensive matrix-vector multiplications but a very small amount of extra work, by utilizing simple power-Lanczos interconversion algorithms suggested. Numerical experiments are given at the end.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제5권2호
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• pp.63-69
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• 2001

In this study, we shall be concerned with computing in parallel a few of the smallest eigenvalues and their corresponding eigenvectors of the eigenvalue problem, $Ax={\lambda}Bx$, where A is symmetric, and B is symmetric positive definite. Both A and B are large and sparse. Recently iterative algorithms based on the optimization of the Rayleigh quotient have been developed, and CG scheme for the optimization of the Rayleigh quotient has been proven a very attractive and promising technique for large sparse eigenproblems for small extreme eigenvalues. As in the case of a system of linear equations, successful application of the CG scheme to eigenproblems depends also upon the preconditioning techniques. A proper choice of the preconditioner significantly improves the convergence of the CG scheme. The idea underlying the present work is a parallel computation of the Multi-Color Block SSOR preconditioning for the CG optimization of the Rayleigh quotient together with deflation techniques. Multi-Coloring is a simple technique to obatin the parallelism of order n, where n is the dimension of the matrix. Block SSOR is a symmetric preconditioner which is expected to minimize the interprocessor communication due to the blocking. We implemented the results on the CRAY-T3E with 128 nodes. The MPI(Message Passing Interface) library was adopted for the interprocessor communications. The test problems were drawn from the discretizations of partial differential equations by finite difference methods.

• Journal of applied mathematics & informatics
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• 제38권1_2호
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• pp.175-187
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• 2020

Recently Salkuyeh and Rahimian in (Comput. Math. Appl. 74 (2017) 2940-2949) proposed a modification of the generalized shift-splitting (MGSS) method for solving singular saddle point problems. In this paper, we present the spectral analysis of the MGSS preconditioner when it is applied to precondition the singular saddle point problems with the (1, 1) block being symmetric. Some eigenvalue bounds for the spectrum of the preconditioned matrix are given. We show that all the real eigenvalues of the preconditioned matrix are in a positive interval and all nonzero eigenvalues having nonzero imaginary part are contained in an intersection of two circles.

• Journal of applied mathematics & informatics
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• 제25권1_2호
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• pp.435-443
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• 2007

Computing the interior spectrum of large sparse generalized eigenvalue problems $Ax\;=\;{\lambda}Bx$, where A and b are large sparse and SPD(Symmetric Positive Definite), is often required in areas such as structural mechanics and quantum chemistry, to name a few. Recently, CG-type methods have been found useful and hence, very amenable to parallel computation for very large problems. Also, as in the case of linear systems proper choice of preconditioning is known to accelerate the rate of convergence. After the smallest eigenpair is found we use the orthogonal deflation technique to find the next m-1 eigenvalues, which is also suitable for parallelization. This offers advantages over Jacobi-Davidson methods with partial shifts, which requires re-computation of preconditioner matrx with new shifts. We consider as preconditioners Incomplete LU(ILU)(0) in two variants, ever-relaxation(SOR), and Point-symmetric SOR(SSOR). We set m to be 5. We conducted our experiments on matrices from discretizations of partial differential equations by finite difference method. The generated matrices has dimensions up to 4 million and total number of processors are 32. MPI(Message Passing Interface) library was used for interprocessor communications. Our results show that in general the Multi-Color ILU(0) gives the best performance.

• Journal of the Korean Data and Information Science Society
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• 제14권3호
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• pp.623-630
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• 2003

Recently, Tanaka(1988) derived two influence functions related to an eigenvalue problem $(A-\lambda_sI)\upsilon_s=0$ of real symmetric matrix A and used them for sensitivity analysis in principal component analysis. In this paper, we deal with the perturbation expansions up to quadratic terms of the same functions and discuss the application to sensitivity analysis in principal component regression analysis(PCRA). Numerical example is given to show how the approximation improves with the quadratic term.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제4권2호
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• pp.99-110
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• 2000

This paper is devoted to the parallelism of a numerical matrix eigenvalue problem. The eigenproblem arises in a variety of applications, including engineering, statistics, and economics. Especially we try to approach the industrial techniques from mathematical modeling. This paper has developed a parallel algorithm to find all eigenvalues. It is contributed to solve a specific practical problem, a vibration problem in the industry. Also we compare the runtime between the serial algorithm and the parallel algorithm for the given problems.

• 한국인터넷방송통신학회논문지
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• 제19권3호
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• pp.173-178
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• 2019

Blazed 격자구조에 의한 광 신호의 회절 특성을 분석하기 위하여 처음으로 격자구조의 Toeplitz 유전율 tensor를 2D spatial Fourier 급수로 정의하고 공식화하였다. 그때 각 층에서의 필드들은 고유치 문제에 기초하여 표현하였으며, 완전한 해는 적절한 경계 값 문제에 의존하는 모드 전송선로 이론 (MTLT)을 사용하여 정확하게 유도하였다. 비대칭형 blazed 격자구조의 Toeplitz 유전율 tensor에 기초하여 대칭형과 톱니형 격자구조의 Toeplitz 행렬을 정의하고 각 격자구조에 대한 회절특성을 수치해석 하였다. 수치해석 결과, 비대칭형과 대칭형 구조는 무반사 (anti-reflection) GMR 필터 특성을 나타내었으며, 대칭형 구조가 비대칭형 구조보다 광대역 필터특성을 보였다. 이에 반하여 톱니형 격자 구조는 무반사보다 무투과 (anti-transmission) 필터의 특성이 더욱 강하게 나타났다.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 1994년도 Proceedings of the Korea Automatic Control Conference, 9th (KACC) ; Taejeon, Korea; 17-20 Oct. 1994
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• pp.364-368
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• 1994

Algebraic matrix Riccati equations of the form, FP+PF$^{T}$ -PRP+Q=0. are analyzed with reference to the stability of closed-loop system F-PR. Here F, R and Q are n * n real matrices with R=R$^{T}$ and Q=Q$^{T}$ .geq.0 (nonnegative-definite). Such equations have been playing key roles in optimal control and filtering problems with R .geq. 0. and also in the solutions of in H$_{\infty}$ control problems with R taking the form R=H$_{1}$$^{T}$ H$_{1}$-H$_{2}$$^{T}$ H$_{2}$. In both cases an existence of stabilizing solution, i.e. the solution yielding asymptotically stable closed-loop system, is an important problem. First, we briefly review the typical results when R is of definite form, namely either R .geq. 0 as in LQG problems or R .leq. 0. They constitute two extrence cases of Riccati to the cases H$_{2}$=0 and H$_{1}$=0. Necessary and sufficient conditions are shown for the existence of nonnegative-definite or positive-definite stabilizing solution. Secondly, we focus our attention on more general case where R is only assumed to be symmetric, which obviously includes the case for H$_{\infty}$ control problems. Here, necessary conditions are established for the existence of nonnegative-definite or positive-definite stabilizing solutions. The results are established by employing consistently the so-called algebraic method based on an eigenvalue problem of a Hamiltonian matrix.x.ix.x.

• 한국전산구조공학회:학술대회논문집
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• 한국전산구조공학회 1998년도 봄 학술발표회 논문집
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• pp.283-290
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• 1998

A solution method is presented to solve the eigenproblem arising in tile dynamic analysis of non-proportional damping systems with symmetric matrices. The method is based on tile use of Lanczos method to generate a Krylov subspace of trial vectors, witch is then used to reduce a large eigenvalue problem to a much smaller one. The method retains the η order quadratic eigenproblem, without the need to the method of matrix augmentation traditionally used to cast the problem as a linear eigenproblem of order 2n. In the process, the method preserves tile sparseness and symmetry of the system matrices and does not invoke complex arithmetics, therefore, making it very economical for use in solving large problems. Numerical results are presented to demonstrate the efficiency and accuracy of the method.

• 대한기계학회논문집A
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• 제27권1호
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• pp.66-76
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• 2003

The arbitrary V-notched crack problem is considered. The general expressions for the stress components on this problem are obtained as explicit series forms composed of independent unknown coefficients which are denoted by coefficients of eigenvector. For this results eigenvalue equation is performed first through introducing complex stress functions and applying the traction free boundary conditions. Next solving this equation, eigenvalues and corresponding eigenvectors are obtained respectively, and finally inserting these results into stress components, the general equations are obtained. These results are also shown to be applicable to the symmetric V-notched crack or straight crack. It can be shown that this solutions are composed of the linear combination of Mode I and Mode II solutions which are obtained from different characteristic equations, respectively. Through performing asymptotic analysis for stresses, the stress intensity factor is given as a closed form equipped with the unknown coefficients of eigenvector. In order to calculate the unknown coefficients. based on these general explicit equations, numerical programming using the overdetermined boundary collocation method which is algorithmed originally by Carpenter is also worked out. As this programming requires the input data, the commercial FE analysis for stresses is performed. From this study, for some V-notched problems, unknown coefficients can be calculated numerically and also fracture parameters are determined.