• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.239-251
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• 2007

We show how the minimization can be used to solve the quadratic matrix equation and then compare two different types of conjugate gradient method which are Polak and Ribi$\acute{e}$re version and Fletcher and Reeves version. Finally, some results of the global and local convergence are shown.

• East Asian mathematical journal
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• v.29 no.3
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• pp.327-335
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• 2013

In this paper we consider the quadratic matrix equation which can be defined by $$Q(X)=AX^2+BX+C=0$$, where X is a $n{\times}n$ unknown complex matrix, and A, B and C are $n{\times}n$ given matrices with complex elements. We first introduce a couple of condition numbers of the equation Q(X) and present normwise condition numbers. Finally, we compare the results and some numerical experiments are given.

• Honam Mathematical Journal
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• v.35 no.3
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• pp.417-433
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• 2013

We consider the iterative solution of a quadratic matrix equation with special coefficient matrices which arises in the quasibirth and death problem. In this paper, we show that the elementwise minimal positive solvent of the quadratic matrix equations can be obtained using Newton's method if there exists a positive solvent and the convergence rate of the Newton iteration is quadratic if the Fr$\acute{e}$chet derivative at the elementwise minimal positive solvent is nonsingular. Although the Fr$\acute{e}$chet derivative is singular, the convergence rate is at least linear. Numerical experiments of the convergence rate are given.

• East Asian mathematical journal
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• v.38 no.1
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• pp.95-105
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• 2022

In this paper, we consider a quadratic matrix equation Q(X) = AX² + BX + C = 0 where A, B, C ∈ ℝ^n×n. A new approach is proposed to find solutions of Q(X), using the novel structure of the information processing system. We also present some numerical experimetns with Artificial Neural Network.

• East Asian mathematical journal
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• v.28 no.5
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• pp.587-595
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• 2012

In this paper we consider the quadratic matrix equation which can be defined be $$Q(X)=AX^2+BX+C=0$$, where X is a $n{\times}n$ unknown real matrix; A,B and C are $n{\times}n$ given matrices with real elements. Newton's method is considered to find the skew-symmetric solvent of the nonlinear matrix equations Q(X). We also show that the method converges the skew-symmetric solvent even if the Fr$\acute{e}$chet derivative is singular. Finally, we give some numerical examples.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.399-409
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• 2008

We consider the most generalized quadratic matrix equation, Q(X) = $A_7XA_6XA_5+A_4XA_3+A_2XA_1+A_0=0$, where X is m ${\times}$ n, $A_7$, $A_4$ and $A_2$ are p ${\times}$ m, $A_6$ is n ${\times}$ m, $A_5$, $A_3$ and $A_l$ are n ${\times}$ q and $A_0$ is p ${\times}$ q matrices with complex elements. The convergence of Newton's method for solving some different types of quadratic matrix equations are considered and we show that the elementwise minimal positive solvents can be found by Newton's method with the zero starting matrices. We finally give numerical results.

• The Pure and Applied Mathematics
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• v.23 no.3
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• pp.287-307
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• 2016

Using the fixed point method, we prove the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in matrix fuzzy normed spaces.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.199-214
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• 2017

We consider fixed-point iterations constructed by simple transforming from a quadratic matrix equation to equivalent fixed-point equations and assume that the iterations are well-defined at some solutions. In that case, we suggest real valued functions. These functions provide radii at the solution, which guarantee the local convergence and the uniqueness of the solutions. Moreover, these radii obtained by simple calculations of some constants. We get the constants by arbitrary matrix norm for coefficient matrices and solution. In numerical experiments, the examples show that the functions give suitable boundaries which guarantee the local convergence and the uniqueness of the solutions for the given equations.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.1
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• pp.19-25
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• 2012

The purpose of this paper is to derive powers $A^{n}$ using a system of recursive sequences for a given $2{\times}2$ matrix A. Introducing a recursive sequence we have a quadratic equation. Solutions to this quadratic equation are related with eigenvalues of A. By solving this quadratic equation we can easily obtain an explicit form of $A^{n}$. Our method holds when A is defined not only on the real field but also on the complex field.

• Journal of the Korean Institute of Telematics and Electronics
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• v.16 no.5
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• pp.18-26
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• 1979

This paper extends some well-known system theories on algebraic matrix Lyapunov and Riccati equations. These extended results contain two point boundary conditions in matrix differential equations and include conventional results as special cases. Necessary and sufficient conditions are derived under which linear systems are stabilizable with feedback gains derived from periodic two-point boundary matrix differential equations. An iterative computation method for two-point boundary differential Riccati equations is given with an initial guess method. The results in this paper are related to periodic feedback controls and also to the quadratic cost problem with a discrete state penalty.