摘要
对阻尼牛顿算法作了适当的改进,证明了新算法的收敛性.基于新算法,运用计算机代数系统Matlab,研究了迭代次数k,参数对(μ,λ)与初值x0三者间的依赖关系,研究了病态问题在新算法下趋于稳定的渐变(瞬变)过程.数值结果表明:(1)阻尼牛顿迭代中,参数对(μ,λ)与迭代次数k间存在特有的非线性关系;(2)适当的参数对(μ,λ)与阻尼因子α的共同作用能够在迭代中大幅度地降低病态问题的Jacobi阵的条件数,使病态问题逐渐趋于稳定,从而改变原问题的收敛性与收敛速度.
In this paper,the damped Newton method is improved suitably and the convergence for new method is proved.Based on the new algorithm,a program is proposed and fulfilled by numerical and symbolic computations in Matlab,we study the relation among the iteration degrees k,parameters（μ,λ） and initial value x0.We also study the gradual（transient） process of the ill-conditioned systems nonlinear equations tending stable.Numerical results show that there is a special nonlinear relation between the parameters（μ,λ） and the iteration degrees k for damped Newton method and that the suitable parameters（μ,λ） and damping coeffcient α can greatly decrease condition number of Jacobi matrix of ill-conditioned systems nonlinear equations.The ill-conditioned problems can gradually become stable and thus the convergence and the convergence speed of the ill-conditioned systems nonlinear equations can be changed.
出处
《纯粹数学与应用数学》
CSCD
2012年第4期433-439,共7页
Pure and Applied Mathematics
基金
四川省教育厅2011年重点科研项目(10ZA073)
关键词
阻尼牛顿法
雅可比矩阵的条件数
病态问题
参数对(μ
λ)
damped Newton method； condition number of the Jacobi matrix； ill-conditioned systems equations； parameters（μ； λ）
