一类二次矩阵方程的牛顿迭代法及其收敛性  

Newton Iterative Methods for a Class of Quadratic Matrix Equations and Its Convergence

作　　者：刘兰冬[1] 刘铭 LIU Landong;LIU Ming(School of Science,China University of Mining and Technology(Beijing),Beijing 100083)

机构地区：[1]中国矿业大学(北京)理学院,北京100083

出　　处：《工程数学学报》2024年第3期587-594,共8页Chinese Journal of Engineering Mathematics

基　　金：中国矿业大学(北京)心桥工程项目;中国矿业大学(北京)“课程思政”示范课程建设项目(62911008).

摘　　要：二次矩阵方程是科学与工程计算中一类重要的方程,探讨有效的数值方法是一项有意义的工作,拟生灭过程在股价模拟、库存控制、排队论等很多领域都有着重要的应用,对一类来源于拟生灭过程的特殊的二次矩阵方程进行了研究。在最小非负解存在且唯一的假设条件下,提出了牛顿迭代法并证明其收敛性。当初始矩阵取零矩阵时,牛顿迭代法产生的矩阵列收敛到方程的唯一最小非负解。最后通过数值例子验证算法的有效性与可行性。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.

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 

分 类 号：O241.7[理学—计算数学]