• 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)

• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.101-109
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• 2002

We extend two inequalities involving Hadamard Products of Positive definite Hermitian matrices to positive semi-definite Hermitian matrices. Simultaneously, we also show the sufficient conditions for equalities to hold. Moreover, some other matrix inequalities are also obtained. Our results and methods we different from those which are obtained by S. Liu in [J. Math. Anal. Appl. 243:458-463(2000)] and B.-Y Wang et al in [Lin. Alg. Appl. 302-303: 163-172(1999)] .

• 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.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.365-376
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• 2004

The purpose of this study is to show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met. In particular, as a basis function, we are interested in using a conditionally positive definite function $\Phi$ whose generalized Fourier transform is of the form $\Phi(\theta)\;=\;F(\theta)$\mid$\theta$\mid$^{-2m}$ with a bounded function F > 0.

• 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.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1131-1141
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• 2010

We develop an algorithm for computing a sparse approximate inverse for a nonsymmetric positive definite matrix based upon the FFAPINV algorithm. The sparse approximate inverse is computed in the factored form and used to work with some Krylov subspace methods. The preconditioner is breakdown free and, when used in conjunction with Krylov-subspace-based iterative solvers such as the GMRES algorithm, results in reliable solvers. Some numerical experiments are given to show the efficiency of the preconditioner.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.917-925
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• 2020

It has been shown that the geometric mean A#B of positive definite Hermitian matrices A and B is the maximal element X of Hermitian matrices such that $$\(\array{A&X\\X&B}\)$$ is positive semi-definite. As an extension of this result for the 2 × 2 block matrix, we consider in this article the block matrix [[A#w_ijB]] whose (i, j) block is given by the Riemannian geodesics of positive definite Hermitian matrices A and B, where w_ij ∈ ℝ for all 1 ≤ i, j ≤ m. Under certain assumption of the Loewner order for A and B, we establish the equivalent condition for the parameter matrix ω = [w_ij] such that the block matrix [[A#w_ijB]] is positive semi-definite.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.739-763
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• 2014

In this paper, the nonlinear matrix equation $$X+\sum_{i=1}^{m}A_i^*X^{-q}A_i=Q(0<q{\leq}1)$$ is investigated. Some necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are derived. Two iterative methods for the maximal positive definite solution are proposed. A perturbation estimate and an explicit expression for the condition number of the maximal positive definite solution are obtained. The theoretical results are illustrated by numerical examples.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.421-431
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• 2006

In this paper, we establish a matrix inequality and its equivalent form. Applying the results, some matrix inequalities involving Khatri-Rao products of positive semi-definite matrices are generalized.

• East Asian mathematical journal
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• v.26 no.5
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• pp.649-654
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• 2010

A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find some finiteness theorems which ensure 2-universality of Hermitian lattices over several imaginary quadratic number fields.