牛顿迭代法求极值：MATLAB实战

牛顿迭代法求极值的基本原理

牛顿迭代法是一种基于二阶泰勒展开的优化方法，通过迭代逼近函数的极值点。对于目标函数( f(x) )，其极值点需满足( f'(x)=0 )。牛顿法的迭代公式为： [ x_{k+1} = x_k - \frac{f'(x_k)}{f''(x_k)} ] 该公式通过当前点的梯度（一阶导数）和曲率（二阶导数）信息调整下一步的搜索方向。

MATLAB实现步骤

定义目标函数及其导数 在MATLAB中需预先定义函数及其一阶、二阶导数。例如对于函数( f(x) = x^3 - 2x^2 + 4 )：

function y = f(x)
    y = x^3 - 2*x^2 + 4;
end

function dy = df(x)
    dy = 3*x^2 - 4*x;
end

function ddy = ddf(x)
    ddy = 6*x - 4;
end

初始化参数 设置初始猜测值( x_0 )、最大迭代次数和收敛容差：

x0 = 1.5;       % 初始点
max_iter = 50; % 最大迭代次数
tol = 1e-6;    % 收敛阈值

迭代过程实现 通过循环更新( x )值直至满足收敛条件：

for iter = 1:max_iter
    x_new = x0 - df(x0)/ddf(x0);
    if abs(x_new - x0) < tol
        break;
    end
    x0 = x_new;
end
fprintf('极值点: x = %.6f, f(x) = %.6f\n', x0, f(x0));

处理边界情况与优化建议

二阶导数为零的检查 在迭代中需避免除零错误，可增加条件判断：

if abs(ddf(x0)) < eps
    error('二阶导数为零，无法继续迭代');
end

多变量函数的扩展 对于多元函数( f(\mathbf{x}) )，需使用Hessian矩阵（二阶导矩阵）和梯度向量： [ \mathbf{x}_{k+1} = \mathbf{x}_k - H^{-1}(\mathbf{x}_k) \nabla f(\mathbf{x}_k) ] MATLAB实现需借助sym工具箱或手动计算偏导数。

收敛性与可视化分析

绘制迭代轨迹 通过记录每次迭代的( x )值可观察收敛路径：

x_vals = [x0];
for iter = 1:max_iter
    x_new = x0 - df(x0)/ddf(x0);
    x_vals = [x_vals, x_new];
    if abs(x_new - x0) < tol
        break;
    end
    x0 = x_new;
end
plot(x_vals, f(x_vals), '-o');
xlabel('x'); ylabel('f(x)');

初始点敏感性测试 尝试不同初始点（如( x_0 = 0 )与( x_0 = 2 )）可验证算法是否收敛到全局极值。对于非凸函数，建议结合其他方法（如梯度下降）避免局部最优。

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