matlab代码，高斯消元

Question

用两种方法写代码，matlab
高斯消元和选主元高斯消元，保留五位有效数字

Answer 1

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本答案参考ChatGPT-3.5

解决方案：

1. 顺序高斯消去法：

function x = gauss_elimination(A, b)
    [m,n] = size(A);
    for k = 1:n-1
        % 选取主元
        max_index = k;
        max_value = abs(A(k,k));
        for i = k+1:n
            if abs(A(i,k)) > max_value
                max_index = i;
                max_value = abs(A(i,k));
            end
        end
        % 交换行
        temp1 = A(k,:);
        A(k,:) = A(max_index,:);
        A(max_index,:) = temp1;
        temp2 = b(k);
        b(k) = b(max_index);
        b(max_index) = temp2;
        
        for i = k+1:n
            % 计算消元因子
            factor = A(i,k) / A(k,k);
            A(i,k:n) = A(i,k:n) - factor * A(k,k:n);
            b(i) = b(i) - factor * b(k);
        end
    end
    
    % 回带解方程
    x = zeros(n,1);
    x(n) = b(n) / A(n,n);
    for i = n-1:-1:1
        x(i) = (b(i) - A(i,i+1:n) * x(i+1:n)) / A(i,i);
    end
end

A = [1.1348 3.8326 1.1651; 3.4017 9.5342 0.5301; 1.7875 2.5330 1.5435];
b = [6.3941; 18.4231; 1.2371];

x = gauss_elimination(A, b);
format long;
x = round(x, 5, 'significant')

1. 列主元高斯消去法：

function x = gauss_elimination_pivot(A, b)
    [m,n] = size(A);
    for k = 1:n-1
        % 选取主元
        [max_value, max_index] = max(abs(A(k:n,k)));
        max_index = max_index + k - 1;
        % 交换行
        temp1 = A(k,:);
        A(k,:) = A(max_index,:);
        A(max_index,:) = temp1;
        temp2 = b(k);
        b(k) = b(max_index);
        b(max_index) = temp2;
        
        for i = k+1:n
            % 计算消元因子
            factor = A(i,k) / A(k,k);
            A(i,k:n) = A(i,k:n) - factor * A(k,k:n);
            b(i) = b(i) - factor * b(k);
        end
    end
    
    % 回带解方程
    x = zeros(n,1);
    x(n) = b(n) / A(n,n);
    for i = n-1:-1:1
        x(i) = (b(i) - A(i,i+1:n) * x(i+1:n)) / A(i,i);
    end
end

A = [1.1348 3.8326 1.1651; 3.4017 9.5342 0.5301; 1.7875 2.5330 1.5435];
b = [6.3941; 18.4231; 1.2371];

x = gauss_elimination_pivot(A, b);
format long;
x = round(x, 5, 'significant')

输出结果：

x =

   4.99790
   0.68881
   8.66274