matlab实现牛顿迭代法求解二元函数最优点并绘制动态图像

深藏阁楼爱情的钟 2022-12-28 04:22 112阅读 0赞

1. 定义目标函数，及其雅可比矩阵和海塞矩阵：

function [f_val, f_grad_val, f_hess_val] = V_func(x_val, y_val)
        syms x y;
        
        f = (x-2)^4+((x-2)^2)*(y^2)+(y+1)^2;
        
        f_grad = jacobian(f, [x; y]);
        f_hess = jacobian(f_grad, [x, y]);
        
        f_val = subs(f, {x, y}, {x_val, y_val});
        f_grad_val = subs(f_grad, {x, y}, {x_val, y_val});
        f_hess_val = subs(f_hess, {x, y}, {x_val, y_val});
    end

2. 迭代求解并绘制图像

clear
    clc
    
    init_point = [12, 15]';
    iter_max = 30;
    [f_init, f_grad_init, f_hess_init] = V_func(init_point(1), init_point(2));
    
    % 由于是每次迭代需要更新两个变量，因此需要2*x大小的数组存储
    x_steps_val = zeros(2, iter_max);
    f_steps_val = zeros(1, iter_max);
    x_steps_val(:, 1) = init_point;
    f_steps_val(:, 1) = f_init;
    
    err_val = inf;
    err_max = 1e-6;
    iter = 1;
    residual = zeros(1, iter_max);
    
    while err_val > err_max 
        if iter == iter_max
            disp('已经达到最大迭代次数');
            break;
        end
        x = x_steps_val(:, iter);
        [f_val_prev, f_grad_val, f_hess_val] = V_func(x(1), x(2));
        x_steps_val(:, iter + 1) = x_steps_val(:, iter) - inv(f_hess_val)*(f_grad_val');
        x_2 = x_steps_val(:, iter + 1);
        [f_val_cur, f_grad_val, f_hess_val] = V_func(x_2(1), x_2(2));
        residual(:, iter) = norm(f_val_prev - f_val_cur);
        err_val = residual(:, iter);
    
        if err_val < err_max
            disp('找到满足精度要求的解');
            disp(x_2);
            disp(['迭代次数为',num2str(iter)]);
            break;
        end
        iter = iter + 1;
    end
    
    % 定义三维图像显示范围
    x = -15:0.5:15;
    y = -15:0.5:15;
    
    % 绘制三维函数图像
    [xx, yy] = meshgrid(x, y);
    [f, f_grad, f_hess] = V_func(xx, yy);
    figure;
    mesh(xx, yy, double(f));
    alpha(0.3)
    hold on;
    
    % 绘制迭代点图像
    for i = 1 : iter
        span = 1 : 1 : i-1;
        scatter3(x_steps_val(1,i), x_steps_val(2,i), f_steps_val(i), 'MarkerEdgeColor','k','MarkerFaceColor',[0 .75 .75])
        plot3(x_steps_val(1,span), x_steps_val(2,span), f_steps_val(span), 'r-', 'LineWidth', 3)
        pause(1);
        M(i) = getframe;
    end