三项谱共轭梯度法matlab代码

很抱歉，根据提供的引用内容，没有找到关于三项谱共轭梯度法的 Matlab 代码。但是，我可以为您提供一些关于共轭梯度法的 Matlab 代码和介绍。

共轭梯度法是一种用于求解大型线性方程组的迭代方法。它的优点是收敛速度快，尤其适用于对称正定矩阵。以下是一个简单的 Matlab 代码示例，用于解决对称正定矩阵的线性方程组：

% 定义矩阵A和向量b
A = [4, -1, 0; -1, 4, -1; 0, -1, 4];
b = [1; 2; 3];

% 初始化向量x和r
x = zeros(size(b));
r = b - A*x;

% 初始化共轭向量p和残差向量r
p = r;
rsold = r'*r;

% 迭代求解
for i = 1:length(b)
    Ap = A*p;
    alpha = rsold / (p'*Ap);
    x = x + alpha*p;
    r = r - alpha*Ap;
    rsnew = r'*r;
    if sqrt(rsnew) < 1e-10
        break;
    end
    p = r + (rsnew/rsold)*p;
    rsold = rsnew;
end

% 输出结果
disp(x);

相关问题

共轭梯度法matlab代码

下面是共轭梯度法的MATLAB代码：

function [x, fval] = conjugate_gradient(A, b, x0, tol)
% 求解线性方程组 Ax = b，使用共轭梯度法
% A: 系数矩阵
% b: 右侧向量
% x0: 初始解向量
% tol: 计算精度
% x: 解向量
% fval: 目标函数值

n = size(A, 1);  % 系数矩阵维度
x = x0;  % 初始解向量
r = b - A * x;  % 初始残差向量
d = r;  % 初始方向向量
fval = 0.5 * r' * r;  % 初始目标函数值

for k = 1:n
    Ad = A * d;
    alpha = (r' * r) / (d' * Ad);
    x = x + alpha * d;
    r_new = r - alpha * Ad;
    beta = (r_new' * r_new) / (r' * r);
    d = r_new + beta * d;
    r = r_new;
    fval = 0.5 * r' * r;
    if fval < tol
        break
    end
end

end

其中，输入参数 A、b、x0 和 tol 分别为线性方程组的系数矩阵、右侧向量、初始解向量和计算精度。输出参数 x 和 fval 分别为解向量和目标函数值。

最优化共轭梯度法matlab代码

以下是两种MATLAB编写的最优化共轭梯度法的代码：

1. 使用黄金分割法精确一维搜索的最优化共轭梯度法代码：

function [x, fval] = conjugate_gradient_golden(x0, epsilon)
    % 初始化
    x = x0;
    fval = objective_function(x);
    grad = gradient(x);
    d = -grad;
    
    % 迭代
    while norm(grad) > epsilon
        alpha = golden_section_search(x, d);
        x = x + alpha * d;
        grad_new = gradient(x);
        beta = norm(grad_new)^2 / norm(grad)^2;
        d = -grad_new + beta * d;
        grad = grad_new;
        fval = objective_function(x);
    end
end

function fval = objective_function(x)
    % 目标函数
    fval = x(1)^2 + 2 * x(2)^2;
end

function grad = gradient(x)
    % 数值微分法计算梯度
    h = 1e-6;
    grad = zeros(size(x));
    for i = 1:length(x)
        x_plus_h = x;
        x_plus_h(i) = x_plus_h(i) + h;
        grad(i) = (objective_function(x_plus_h) - objective_function(x)) / h;
    end
end

function alpha = golden_section_search(x, d)
    % 黄金分割法精确一维搜索
    a = 0;
    b = 1;
    rho = 0.618;
    epsilon = 1e-6;
    
    while abs(b - a) > epsilon
        x1 = a + (1 - rho) * (b - a);
        x2 = a + rho * (b - a);
        f1 = objective_function(x + x1 * d);
        f2 = objective_function(x + x2 * d);
        
        if f1 < f2
            b = x2;
        else
            a = x1;
        end
    end
    
    alpha = (a + b) / 2;
end

2. 使用Wolfe-Powell非精确一维搜索的最优化共轭梯度法代码：

function [x, fval] = conjugate_gradient_wolfe_powell(x0, epsilon)
    % 初始化
    x = x0;
    fval = objective_function(x);
    grad = gradient(x);
    d = -grad;
    
    % 迭代
    while norm(grad) > epsilon
        alpha = wolfe_powell_search(x, d);
        x = x + alpha * d;
        grad_new = gradient(x);
        beta = norm(grad_new)^2 / norm(grad)^2;
        d = -grad_new + beta * d;
        grad = grad_new;
        fval = objective_function(x);
    end
end

function fval = objective_function(x)
    % 目标函数
    fval = x(1)^2 + 2 * x(2)^2;
end

function grad = gradient(x)
    % 数值微分法计算梯度
    h = 1e-6;
    grad = zeros(size(x));
    for i = 1:length(x)
        x_plus_h = x;
        x_plus_h(i) = x_plus_h(i) + h;
        grad(i) = (objective_function(x_plus_h) - objective_function(x)) / h;
    end
end

function alpha = wolfe_powell_search(x, d)
    % Wolfe-Powell非精确一维搜索
    alpha = 1;
    c1 = 0.1;
    c2 = 0.9;
    rho = 0.618;
    epsilon = 1e-6;
    
    while true
        f0 = objective_function(x);
        g0 = gradient(x);
        f1 = objective_function(x + alpha * d);
        
        if f1 > f0 + c1 * alpha * g0' * d || (f1 >= objective_function(x + alpha * d) && alpha < epsilon)
            alpha = zoom(x, d, alpha);
            break;
        end
        
        g1 = gradient(x + alpha * d);
        
        if abs(g1' * d) <= -c2 * g0' * d
            break;
        end
        
        if g1' * d >= 0
            alpha = zoom(x, d, alpha);
            break;
        end
        
        alpha = rho * alpha;
    end
end

function alpha = zoom(x, d, alpha_lo)
    % Wolfe-Powell非精确一维搜索中的zoom函数
    alpha_hi = alpha_lo * 2;
    c1 = 0.1;
    c2 = 0.9;
    epsilon = 1e-6;
    
    while true
        alpha = (alpha_lo + alpha_hi) / 2;
        f0 = objective_function(x);
        g0 = gradient(x);
        f1 = objective_function(x + alpha * d);
        
        if f1 > f0 + c1 * alpha * g0' * d || f1 >= objective_function(x + alpha * d)
            alpha_hi = alpha;
        else
            g1 = gradient(x + alpha * d);
            
            if abs(g1' * d) <= -c2 * g0' * d
                break;
            end
            
            if g1' * d * (alpha_hi - alpha_lo) >= 0
                alpha_hi = alpha_lo;
            end
            
            alpha_lo = alpha;
        end
        
        if abs(alpha_hi - alpha_lo) < epsilon
            break;
        end
    end
end