一种保持对称性、稀疏性的拟Newton法

考虑非线性方程组 (1)F(x)=0其中,F(x)=(f_1(x),…,f_n(x))~T,x∈R~n.当F(x)为梯度算子时,F(x)的Jacobian是对称的.这类问题在实际计算工作中大量存在,比如近些年来研究很多的非线性泛函极小化问题就是如此,因此,人们自然地想到要把对于解非线性方程组很有效的Broyden方法发展到对称情形.1970年Powell提出PSB修正:     

A COLUMN-SECANT UPDATE TECHNIQUE FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS

In this paper,we present a column-secant modification of the SCC method,which is called the CSSCC method.The CSSCC method uses function values more efficiently than the SCC method,and it is shown that the CSSCC method has better local q-convergence and r-convergence rates than the SCC method.The numerical results show that the CSSCC method is competitive with some well known methods for some standard test problems.     

关于换列拟Newton法

由于拟Newton法具有超线性收敛性,而其每步的计算量仅为Newton法的O(1/n),因此,被认为是解多变量非线性方程组的有效方法。近十余年来,人们对这类方法已经提出各种计算方案,并且把它们应用于解非线性方程组和优化问题。此外,由于大量常见的非线性方程组的稀疏性特征,近年来人们对于解稀疏非线性方程组的拟Newton法的研究日益增加。