• 호남수학학술지
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• 제35권3호
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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.

• 호남수학학술지
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• 제30권2호
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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.