最速下降法与牛顿法用matlab写出

最速下降法的matlab代码示例：

function [x, fval, exitflag] = GradientDescent(funct, x0, tol, max_iter)
% funct - 目标函数
% x0 - 初始点
% tol - 容忍误差
% max_iter - 迭代次数最大值
% x - 最优解
% fval - 目标函数在最优解处的取值
% exitflag - 迭代结束标志

% 初始化
x = x0;
fval = funct(x);
delta_f = inf;
iter = 0;

while delta_f > tol && iter < max_iter
% 计算梯度
grad = Gradient(funct, x);
% 计算步长
alpha = StepSize(funct, x, grad);
% 迭代
x_new = x - alpha * grad;
fval_new = funct(x_new);
% 更新状态
delta_f = abs(fval - fval_new);
x = x_new;
fval = fval_new;
iter = iter + 1;
end

% 判断迭代结束标志
if delta_f <= tol
exitflag = 0;
else
exitflag = -1;
end

end

function grad = Gradient(funct, x)
% 计算目标函数在x处的梯度

eps = 1e-6;
n = length(x);
grad = zeros(n, 1);

for i = 1:n
dx = zeros(n, 1);
dx(i) = eps;
grad(i) = (funct(x + dx) - funct(x - dx)) / (2 * eps);
end

end

function alpha = StepSize(funct, x, grad)
% 计算最速下降法的步长

g = -grad;
alpha = fminbnd(@(a) funct(x + a * g), 0, 1);

end

牛顿法的matlab代码示例：

function [x, fval, exitflag] = NewtonMethod(funct, x0, tol, max_iter)
% funct - 目标函数
% x0 - 初始点
% tol - 容忍误差
% max_iter - 迭代次数最大值
% x - 最优解
% fval - 目标函数在最优解处的取值
% exitflag - 迭代结束标志

% 初始化
x = x0;
fval = funct(x);
delta_f = inf;
iter = 0;

while delta_f > tol && iter < max_iter
% 计算梯度和海森矩阵
[grad, hess] = GradientHessian(funct, x);
% 计算步长
p = - hess \ grad;
alpha = StepSize(funct, x, p);
% 迭代
x_new = x + alpha * p;
fval_new = funct(x_new);
% 更新状态
delta_f = abs(fval - fval_new);
x = x_new;
fval = fval_new;
iter = iter + 1;
end

% 判断迭代结束标志
if delta_f <= tol
exitflag = 0;
else
exitflag = -1;
end

end

function [grad, hess] = GradientHessian(funct, x)
% 计算目标函数在x处的梯度和海森矩阵

eps = 1e-6;
n = length(x);
grad = zeros(n, 1);
hess = zeros(n, n);

% 计算梯度
for i = 1:n
dx = zeros(n, 1);
dx(i) = eps;
grad(i) = (funct(x + dx) - funct(x - dx)) / (2 * eps);
end

% 计算海森矩阵
for i = 1:n
for j = 1:n
dx = zeros(n, 1);
dx(i) = eps;
dx(j) = eps;
hess(i, j) = (funct(x + dx) - funct(x - dx) - 2 * grad(i) * dx(j)) / (2 * eps^2);
end
end

end

function alpha = StepSize(funct, x, p)
% 计算牛顿法的步长

g = Gradient(funct, x);
h = Hessian(funct, x);
alpha = - g' * p / (p' * h * p);

end

function hess = Hessian(funct, x)
% 计算目标函数在x处的海森矩阵

eps = 1e-6;
n = length(x);
hess = zeros(n, n);

for i = 1:n
for j = 1:n
dx = zeros(n, 1);
dx(i) = eps;
dx(j) = eps;
hess(i, j) = (funct(x + dx) - 2 * funct(x) + funct(x - dx)) / eps^2;
end
end

end

注意：以上代码仅为示例，具体实现方式可能会因函数形式、精度要求等因素而有所不同。如果要使用最速下降法和牛顿法求解实际问题，请谨慎选择应用场景、调整超参数，并进行充分的测试和验证。