工程数学实验三

function [k, x, val] = dampnm(fun, gfun, Hess, x0, epsilon)

% 输入:% fun - 被优化的函数;% gfun - 目标函数的梯度;% Hess - 目标函数的Hessian矩阵;% x0 - 初始点;% epsilon - 收敛阈值;% 输出:% k - 迭代次数;% x - 极值点;% val - 极值点的函数值;

k = 1; % 初始化迭代计数器

gk = feval(gfun, x0); % 计算初始梯度和Hessian矩阵

Gk = feval(Hess, x0);

while norm(gk) > epsilon

    dk = -Gk\gk;               % 求解线性方程组以找到搜索方向

    x0 = x0 + dk;               % 更新点

    gk = feval(gfun, x0);    % 计算新点的梯度和Hessian矩阵

    Gk = feval(Hess, x0);

     

     k = k + 1;    % 迭代计数器递增

     if k > 10000   % 如果迭代次数太多，退出循环

        warning('The method did not converge after 10000 iterations.');

        break;

    end

end

x = x0; % 输出最终迭代结果

val = feval(fun, x0);

end

脚本：

% 使用符号变量定义目标函数、梯度和Hessian矩阵

syms x1 x2 x3 x4;

funf_sym = (x1 + 10*x2)^2 + 5*(x3 - x4)^2 + (x2 - 2*x3)^4 + 10*(x1 - x4)^4;

gradf_sym = gradient(funf_sym, [x1, x2, x3, x4]);

hessf_sym = hessian(funf_sym, [x1, x2, x3, x4]);

 

% 将符号表达式转换为函数句柄，明确设置为接受列向量

funf = matlabFunction(funf_sym, 'Vars', {[x1; x2; x3; x4]});

gradf = matlabFunction(gradf_sym, 'Vars', {[x1; x2; x3; x4]});

hessf = matlabFunction(hessf_sym, 'Vars', {[x1; x2; x3; x4]});

 

% 初始化初始点和精度

x0 = [2; 2; 2; 2]; % 初始点

epsilon = 1e-6;    % 收敛阈值

 

% 调用牛顿法

[k, x_opt, f_opt] = dampnm(funf, gradf, hessf, x0, epsilon);

 

% 输出结果

fprintf('迭代次数: %d\n极值点 x: [%f, %f, %f, %f]\n极值点的函数值: %f\n', ...

        k, x_opt(1), x_opt(2), x_opt(3), x_opt(4), f_opt);