非线性方程组的Newton流线法

NEWTON FLOW METHOD FOR NONLINEAR SYSTEMS OF EQUATIONS

为求解非线性方程组F(x)=0,研究了Newton流方程x_t(t)=V(x)=-(DF(x))^(-1)F(x),x(0)=-x^0,及数值Newton流x^(j+1)=x^j+hV(x^j),h∈(0,1].导出了减幅指标g_j(h)=‖F(x^(j+1)‖/‖F(x^j)‖=1-h+h^2d_j(h)<1和m重根x~*附近的表示g_j(h)=(1-h/m)~m+h^2O(‖x^j-x~*‖).最后基于4个可计算量g_j,d_j,K_j,q_j,提出了新的Newton流线法,如果投入大量的随机初始点,能找到所有实根、重根和复根.

To solve nonlinear systems of equations F（x） =0, Newton＇s flow equation xt（t） = V （ x ） =- （ D F （ x ） ）^-1 F （ x ） , x （0 ） = x^0 and its numerical flow x^j＋1 = x^j ＋ h V （ x^j） for h ∈ （0, 1] are studied. The damped index gj（h） =‖F（x^j＋1）‖/‖F（x^j）‖ = ‖ - h ＋ h^2dj（h）｜ 〈 1 and refine expression gj （h） = （1 - h/m）^m ＋ h2O（‖x^j - x^＊‖） near the m-ple root x^＊ are derived. Finally based on fourth computable quantities gj, dj, Kj, qj, a new Newton flow algorithm is proposed, which can find all real, multiple and complex roots, if put into a large number of stochastic initial points.