求解大规模非线性方程组的分层多元谱梯度算法  

HIERARCHICAL-MULTIVARIATE SPECTRAL GRADIENT ALGORITHM FOR LARGE-SCALE NONLINEAR SYSTEMS

作　　者：韩乐[1,2] 喻高航[3] 关履泰[1] 

机构地区：[1]中山大学科学计算与计算机应用系,广州510275 [2]华南理工大学理学院数学系,广州510640 [3]赣南师范学院数学与计算机科学学院,赣州341000

出　　处：《高等学校计算数学学报》2011年第1期28-38,共11页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金(10572154);广东省自然科学基金(7003624)

摘　　要：1引言考虑非线性方程组问题:F(x)=0,x∈R^n(1)其中,F:R^n→R^n为连续可微的非线性映射.我们讨论大规模情形,并假设F(x)的Jacobian矩阵无法获取,或存储量太大无法承受.仅在极其特殊的情况下,求解非线性方程组(1)才可能有直接解法,对大部分问题要依赖迭代法.已有的迭代方法有多种[1],常用方法大多是基于Newton法的,这些方法理论上有很多好的性质,如它们具有局部超线性收敛性,但在实际计算过程中。Hierarchical-multivariate spectral gradient algorithm is proposed in this paper for large-scale nonlinear systems. The search direction is determined by a diagonal matrix and the nonlinear mapping according to the structure of Jacobian matrix. The layers are reduced along with the increase in the number of iterations and the decrease in the difference of diagonal elements in the diagonal matrix. In order to avoid calculating the Jacobian matrix and solving linear equations, the line search in each iteration step is taken in a systematic way. At the same time the non-monotone line search guarantcs the global convergence of the algorithm. Numerical experimental results show good.

关 键 词：求解 非线性方程组 分层多元谱梯度算法 数值分析 

分 类 号：O241.7[理学—计算数学]