Newton's method for solving a quadratic matrix equation with special coefficient matrices

2nd International Conference on Mathematical Modeling in Physical Sciences 2013

物理学;数学

Seo, Sang-Hyup^1 ; Seo, Jong Hyun^1 ; Kim, Hyun-Min^1

Department of Mathematics, Pusan National University, 609-735 Busan, Korea, Republic of^1

关键词: Coefficient matrix;    Convergence rates;    Matrix equations;    Newton iterations;    Newton's methods;    Nonsingular;    Numerical experiments;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/490/1/012001/pdf
DOI  :  10.1088/1742-6596/490/1/012001

来源: IOP

【 摘 要 】

We consider the iterative method for solving a quadratic matrix equation with special coefficient matrices which arises in the quasi-birth-death problem. In this paper, we show that the elementwise minimal positive solvents to quadratic matrix equations can be obtained using Newton's method. We also prove that the convergence rate of the Newton iteration is quadratic if the Fre´chet derivative at the elementwise minimal positive solvent is nonsingular. However, if the Fre´chet derivative is singular, the convergence rate is at least linear. Numerical experiments of the convergence rate are given.(This is summarized a paper which is to appear in Honam Mathematical Journal.)