秩一拟牛顿法的matlab代码

下面是一个简单的秩一拟牛顿法的 Matlab 代码示例：

function [x, fval, exitflag, output] = quasi_newton(fun, x0, options)
% fun: 目标函数
% x0: 初始点
% options: 选项结构体，可用于设置迭代终止条件等

% 默认选项
default_options = struct('MaxIter', 1000, 'TolFun', 1e-6, 'TolX', 1e-6);

% 合并选项
if nargin == 3
    options = merge_options(default_options, options);
else
    options = default_options;
end

% 初始化
x = x0;
fval = fun(x);
n = length(x);
H = eye(n);
exitflag = 1;

% 迭代
for iter = 1:options.MaxIter
    % 梯度
    grad = grad_fun(fun, x);

    % 终止条件
    if norm(grad) < options.TolFun
        break;
    end

    % 搜索方向
    d = -H * grad;

    % 线搜索
    alpha = backtracking(fun, x, d, grad);

    % 更新
    s = alpha * d;
    x_new = x + s;
    y = grad_fun(fun, x_new) - grad;
    rho = 1 / (y' * s);
    H = (eye(n) - rho * s * y') * H * (eye(n) - rho * y * s') + rho * s * s';

    % 终止条件
    if norm(s) < options.TolX
        break;
    end

    % 更新
    x = x_new;
    fval = fun(x);
end

% 输出
output.iterations = iter;
output.funcCount = iter + 1;
output.algorithm = 'quasi_newton';
output.message = '';

% 终止条件
if iter == options.MaxIter
    exitflag = 0;
    output.message = 'Maximum number of iterations exceeded.';
elseif norm(grad) > options.TolFun
    exitflag = 0;
    output.message = 'Function value did not converge.';
elseif norm(s) > options.TolX
    exitflag = 0;
    output.message = 'Step size did not converge.';
end
end

function grad = grad_fun(fun, x)
% 计算梯度
h = 1e-6;
grad = zeros(size(x));
for i = 1:length(x)
    x1 = x;
    x1(i) = x1(i) + h;
    f1 = fun(x1);
    x1(i) = x1(i) - 2 * h;
    f2 = fun(x1);
    grad(i) = (f1 - f2) / (2 * h);
end
end

function alpha = backtracking(fun, x, d, grad)
% 线搜索
alpha = 1;
rho = 0.5;
c = 1e-4;
fval = fun(x);
while true
    x_new = x + alpha * d;
    fval_new = fun(x_new);
    if fval_new <= fval + c * alpha * grad' * d
        break;
    end
    alpha = rho * alpha;
end
end

function options = merge_options(default_options, options)
% 合并选项
if isfield(options, 'MaxIter')
    default_options.MaxIter = options.MaxIter;
end
if isfield(options, 'TolFun')
    default_options.TolFun = options.TolFun;
end
if isfield(options, 'TolX')
    default_options.TolX = options.TolX;
end
options = default_options;
end

注意：这是一个简单的示例，可能不能处理所有情况。在实际应用中，需要根据具体问题进行适当修改。