• Title/Summary/Keyword: symmetric positive definite matrix

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CONVERGENCE OF MULTISPLITTING METHOD FOR A SYMMETRIC POSITIVE DEFINITE MATRIX

  • YUN JAE HEON;OH SEYOUNG;KIM EUN HEUI
    • Journal of applied mathematics & informatics
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    • v.18 no.1_2
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    • pp.59-72
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    • 2005
  • We study convergence of symmetric multisplitting method associated with many different multisplittings for solving a linear system whose coefficient matrix is a symmetric positive definite matrix which is not an H-matrix.

MULTI SPLITTING PRECONDITIONERS FOR A SYMMETRIC POSITIVE DEFINITE MATRIX

  • Yun Jae-Heon;Kim Eun-Heui;Oh Se-Young
    • Journal of applied mathematics & informatics
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    • v.22 no.1_2
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    • pp.169-180
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    • 2006
  • We study convergence of multisplitting method associated with a block diagonal conformable multisplitting for solving a linear system whose coefficient matrix is a symmetric positive definite matrix which is not an H-matrix. Next, we study the validity of m-step multisplitting polynomial preconditioners which will be used in the preconditioned conjugate gradient method.

THE STEEPEST DESCENT METHOD AND THE CONJUGATE GRADIENT METHOD FOR SLIGHTLY NON-SYMMETRIC, POSITIVE DEFINITE MATRICES

  • Shin, Dong-Ho;Kim, Do-Hyun;Song, Man-Suk
    • Communications of the Korean Mathematical Society
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    • v.9 no.2
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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)

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MODIFLED INCOMPLETE CHOLESKY FACTORIZATION PRECONDITIONERS FOR A SYMMETRIC POSITIVE DEFINITE MATRIX

  • Yun, Jae-Heon;Han, Yu-Du
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.495-509
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    • 2002
  • We propose variants of the modified incomplete Cho1esky factorization preconditioner for a symmetric positive definite (SPD) matrix. Spectral properties of these preconditioners are discussed, and then numerical results of the preconditioned CG (PCG) method using these preconditioners are provided to see the effectiveness of the preconditioners.

SAOR METHOD FOR FUZZY LINEAR SYSTEM

  • Miao, Shu-Xin;Zheng, Bing
    • Journal of applied mathematics & informatics
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    • v.26 no.5_6
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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.

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SINGLE STEP REAL-VALUED ITERATIVE METHOD FOR LINEAR SYSTEM OF EQUATIONS WITH COMPLEX SYMMETRIC MATRICES

  • JingJing Cui;ZhengGe Huang;BeiBei Li;XiaoFeng Xie
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.5
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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.

ESOR METHOD WITH DIAGONAL PRECONDITIONERS FOR SPD LINEAR SYSTEMS

  • Oh, Seyoung;Yun, Jae Heon;Kim, Kyoum Sun
    • Journal of applied mathematics & informatics
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    • v.33 no.1_2
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    • pp.111-118
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    • 2015
  • In this paper, we propose an extended SOR (ESOR) method with diagonal preconditioners for solving symmetric positive definite linear systems, and then we provide convergence results of the ESOR method. Lastly, we provide numerical experiments to evaluate the performance of the ESOR method with diagonal preconditioners.

Geodesic Clustering for Covariance Matrices

  • Lee, Haesung;Ahn, Hyun-Jung;Kim, Kwang-Rae;Kim, Peter T.;Koo, Ja-Yong
    • Communications for Statistical Applications and Methods
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    • v.22 no.4
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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.

THE EXTREMAL RANKS AND INERTIAS OF THE LEAST SQUARES SOLUTIONS TO MATRIX EQUATION AX = B SUBJECT TO HERMITIAN CONSTRAINT

  • Dai, Lifang;Liang, Maolin
    • Journal of applied mathematics & informatics
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    • v.31 no.3_4
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    • pp.545-558
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    • 2013
  • In this paper, the formulas for calculating the extremal ranks and inertias of the Hermitian least squares solutions to matrix equation AX = B are established. In particular, the necessary and sufficient conditions for the existences of the positive and nonnegative definite solutions to this matrix equation are given. Meanwhile, the least squares problem of the above matrix equation with Hermitian R-symmetric and R-skew symmetric constraints are also investigated.

THE PERIODIC JACOBI MATRIX PROCRUSTES PROBLEM

  • Li, Jiao-Fen;Hu, Xi-Yan
    • Journal of applied mathematics & informatics
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    • v.28 no.3_4
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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.