求解非线性方程组之一般牛顿法

一、理论基础

       
       
       
       

二、 算法设计

       
       
       
       

三、算法伪代码

四、老师课件

五、matlab程序实现

例子：

function ordinary_newton
% date : 2020.10.15
% 一般牛顿法求解非线性方程组
% author ： mw_1422102031
clear;
% clc;
x0 = [0.1, 0.1, -0.1]';
eps1 = 1e-8 ;
eps2 = 1e-8 ;
KMAX = 20;

x = x0;
k = 0;
fprintf('******************newton iterate history**********************\n\n')                    %\n ： 表示换行
for k = 0:KMAX
    b = F(x);
    norm_b = norm(b,Inf);
    if norm_b<eps1
        fprintf('%d: norm_Fx = %10.8f,         norm_dx = %10.8f\n',k,norm_b,norm_dx);
        break
    end
    A = Jac(x);
    dx = -A\b;
    x = x+dx;
    norm_dx = norm(dx,Inf);
    if norm_dx < eps2
        fprintf('%d: norm_Fx = %10.8f,         norm_dx = %10.8f\n',k,norm_b,norm_dx);
        break
    end
    fprintf('%d: norm_Fx = %10.8f,         norm_dx = %10.8f\n',k,norm_b,norm_dx);
end
x

end
% ~~~~~~~~~~~~~~~~~~~~~subroutines(子程序)~~~~~~~~~~~~~~~~~~~~~~~
function y = F(x)
y = zeros(size(x));
x1 = x(1);
x2 = x(2);
x3 = x(3);

y(1) = 3*x1 - cos( x2*x3 ) -1/2;
y(2) = x1^2 - 81*(x2+0.1)^2 + sin(x3) +1.06;
y(3) = exp( - x1*x2 ) + 20*x3 + ( 10*pi-3 )/3;
end

function M = Jac(x)
M = zeros(3,3);
x1 = x(1);
x2 = x(2);
x3 = x(3);
M = [3 x3*sin(x2*x3) x2*sin(x2*x3);...
        2*x1 -162*(x2+0.1) cos(x3);...
        -x2*exp(-x1*x2) -x1*exp(-x1*x2) 20];
end