Local convergence of functional iterations for solving a quadratic matrix equation
Bull. Korean Math. Soc. 2017 Vol. 54, No. 1, 199-214
https://doi.org/10.4134/BKMS.b160022
Published online January 31, 2017
Hyun-Min Kim, Young-Jin Kim, and Jong-Hyeon Seo
Pusan National University, Pusan National University, Pusan National University
Abstract : We consider fixed-point iterations constructed by simple \linebreak 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.
Keywords : quadratic matrix equation (QME), functional iteration, fixed-point iterative method, contraction mapping theorem
MSC numbers : 15A24, 65F10, 65H10
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