求解非线性单调方程组的多元谱投影法

Multivariate Spectral Projection Method for Solving Non-linear Monotone Equations

结合拟牛顿型算法和共轭梯度型投影法,提出一种用于求解非线性单调方程组的多元谱投影算法.因为拟牛顿型算法需要储存F的Jacobian矩阵,且需要通过求解线性方程组得到搜索方向,所以求解大型非线性单调方程组通常具有一定的局限性.因此,谱梯度法被提出,用一个数量矩阵近似F的Jacobian矩阵,以达到减少储存量,提高计算效率的目的.为了更好的近似F的Jacobian矩阵,考虑用一个对角矩阵近似F的Jacobian矩阵.文中从两个不同的角度,提出两种用对角矩阵近似F的Jacobian矩阵的方法,对应的产生两种多元谱梯度算法.

Combining quasi-Newtonian algorithm and conjugate gradient projection method,a multivariate spectral projection algorithm for solving nonlinear monotone equations is proposed.Because the quasi-Newtonian algorithm requires the stored Jacobian matrix,and needs to obtain the search direction by solving the linear equations,the solution usually has certain limitations.Therefore,the spectral gradient method is proposed to use a Jacobian matrix approximated by a quantity matrix to reduce the storage and improve the computational efficiency.In this paper,from two different angles,two methods of approximating Jacobian matrix with diagonal matrix are proposed to generate two kinds of spectral gradient algorithms.