• 대한수학회논문집
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• 제9권2호
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• pp.439-448
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• 1994

It is known that the steepest descent(SD) method and the conjugate gradient(CG) method [1, 2, 5, 6] converge when these methods are applied to solve linear systems of the form Ax = b, where A is symmetric and positive definite. For some finite difference discretizations of elliptic problems, one gets positive definite matrices that are almost symmetric. Practically, the SD method and the CG method work for these matrices. However, the convergence of these methods is not guaranteed theoretically. The SD method is also called Orthores(1) in iterative method papers. Elman [4] states that the convergence proof for Orthores($\kappa$), with $\kappa$ a positive integer, is not heard. In this paper, we prove that the SD method and the CG method converge when the $\iota$$^2$ matrix norm of the non-symmetric part of a positive definite matrix is less than some value related to the smallest and the largest eigenvalues of the symmetric part of the given matrix.(omitted)

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.839-850
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• 2008

In this paper, the symmetric accelerated overrelaxation (SAOR) method for solving $n{\times}n$ fuzzy linear system is discussed, then the convergence theorems in the special cases where matrix S in augmented system SX = Y is H-matrices or consistently ordered matrices and or symmetric positive definite matrices are also given out. Numerical examples are presented to illustrate the theory.

• Communications for Statistical Applications and Methods
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• 제22권4호
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• pp.321-331
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• 2015

The K-means clustering algorithm is a popular and widely used method for clustering. For covariance matrices, we consider a geodesic clustering algorithm based on the K-means clustering framework in consideration of symmetric positive definite matrices as a Riemannian (non-Euclidean) manifold. This paper considers a geodesic clustering algorithm for data consisting of symmetric positive definite (SPD) matrices, utilizing the Riemannian geometric structure for SPD matrices and the idea of a K-means clustering algorithm. A K-means clustering algorithm is divided into two main steps for which we need a dissimilarity measure between two matrix data points and a way of computing centroids for observations in clusters. In order to use the Riemannian structure, we adopt the geodesic distance and the intrinsic mean for symmetric positive definite matrices. We demonstrate our proposed method through simulations as well as application to real financial data.

• 대한수학회보
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• 제60권5호
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• pp.1181-1199
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• 2023

For solving complex symmetric positive definite linear systems, we propose a single step real-valued (SSR) iterative method, which does not involve the complex arithmetic. The upper bound on the spectral radius of the iteration matrix of the SSR method is given and its convergence properties are analyzed. In addition, the quasi-optimal parameter which minimizes the upper bound for the spectral radius of the proposed method is computed. Finally, numerical experiments are given to demonstrate the effectiveness and robustness of the propose methods.

• Journal of applied mathematics & informatics
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• 제28권3_4호
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• pp.569-582
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• 2010

The following "Periodic Jacobi Procrustes" problem is studied: find the Periodic Jacobi matrix X which minimizes the Frobenius (or Euclidean) norm of AX - B, with A and B as given rectangular matrices. The class of Procrustes problems has many application in the biological, physical and social sciences just as in the investigation of elastic structures. The different problems are obtained varying the structure of the matrices belonging to the feasible set. Higham has solved the orthogonal, the symmetric and the positive definite cases. Andersson and Elfving have studied the symmetric positive semidefinite case and the (symmetric) elementwise nonnegative case. In this contribution, we extend and develop these research, however, in a relatively simple way. Numerical difficulties are discussed and illustrated by examples.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.467-474
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• 2006

We present and study the stability and convergence of a deflation-preconditioned conjugate gradient(PCG) scheme for the interior generalized eigenvalue problem $Ax = {\lambda}Bx$, where A and B are large sparse symmetric positive definite matrices. Numerical experiments are also presented to support our theoretical results.

• Journal of applied mathematics & informatics
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• 제34권3_4호
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• pp.309-317
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• 2016

Let V be a Euclidean Jordan algebra with a symmetric cone K. We show that for a Z-transformation L with the additional property L(K) ⊆ K (which we will call ’cone-preserving’), GUS ⇔ strictly copositive on K ⇔ monotone + P. Specializing the result to the Stein transformation S_A(X) := X - AXA^T on the space of real symmetric matrices with the property $S_A(S^n_+){\subseteq}S^n_+$, we deduce that S_A GUS ⇔ I ± A positive definite.

• 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.

• 경영과학
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• 제19권2호
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• pp.155-168
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• 2002

In this paper, some efficient implementation techniques for the multifrontal method, which can be used to compute the Cholesky factor of a symmetric positive definite matrix, are presented. In order to use the cache effect in the cache-based computer architecture, a hybrid method for factorizing a frontal matrix is considered. This hybrid method uses the column Cholesky method and the submatrix Cholesky method alternatively. Experiments show that the hybrid method speeds up the performance of the supernodal multifrontal method by 5%～10%, and it is superior to the Cholesky method in some problems with dense columns or large frontal matrices.

• 대한수학회논문집
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• 제11권2호
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• pp.495-501
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• 1996

There is currently a regain of interest in ADI (Alternating Direction Implicit) method as a preconditioner for iterative Method for solving large sparse linear systems, because of its suitability for parallel computation. However the classical ADI is not applicable to FE(Finite Element) matrices. In this paper wer propose a Block-ADI method, which is applicable to Finite Element metrices. The new approach is a combination of classical ADI method and domain decompositi on. Also, we provide a partial proof of the convergence based on the results from the regular splittings, in case the discretization metrix is symmetric positive definite.