关键词: 二次矩阵方程, 拟生灭过程, 最小非负解, 牛顿迭代, 收敛性

Abstract: Quadratic matrix equation is an important kind of equations in scientific and engineering computations, and it is a meaningful work to explore some effective numerical methods. A special class of quadratic matrix equations derived from quasi-birth-death processes is studied. The quasi-birth-death process has important applications in many fields such as stock price simulation, inventory control, queuing theory, etc. Under the assumption that the minimum non-negative solution exists and is unique, the Newton iteration method is proposed and its convergence is proved. When the initial matrix is zero matrix, the matrix sequence generated by Newton iteration method converges to the unique minimum non-negative solution. Finally, numerical examples are used to verify the effectiveness and feasibility of the algorithm.

Key words: quadratic matrix equation, the process of quasi-birth and death, minimum non-negative solution, Newton iteration, convergence