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  • 线性方程组的分解法——列主元消去法

      1.代码

    %%列主元消去法
    function ECPE = Elimination_of_column_pivot_entries(M,b)
    global n;
    [n,n] = size(M);
    B  =[M,b];
    R_A = rank(M);R_B = rank(B);
    if R_A ~= R_B
        disp('方程无解');
    elseif (R_A == R_B)&&(R_A == n)
        disp('此方程有唯一解');
        for k = 1:n-1
            B = Column_pivot_transformation(B,k);
            B = Elimination_method(B,k);
        end
        X = Upper_trig_iterative_solution(B);
    else
        disp('方程有无穷多组解');
    end
    disp('解向量为:');
    ECPE = X;
    %%列主元变换
    %%指定p列,找出列最大值,以此值为主元进行行变换
        function CPT = Column_pivot_transformation(M,p)
            [m,n] = size(M);
            s = max(M(p:m,p));
            [x,y] = find(M(p:m,p) == s);
            H = x+p-1;
            Ch1 = M(H,:);
            Ch2 = M(p,:);
            M(H,:) = Ch2;
            M(p,:) = Ch1;
            CPT = M;
        end
    %%p列消元函数
        function EM = Elimination_method(M,p)
            [m,n] = size(M);Div = zeros(1,m);
            for i = p+1:m
                Div(i) = M(i,p)/M(p,p);
            end
            for j = p:n
                for i = p:m
                    M(i,j) = M(i,j)-M(p,j)*Div(i);
                end
            end
            EM = M;
        end
    %%上三角迭代法
        function UTIS = Upper_trig_iterative_solution(M)
            [m,n] = size(M);
            A = M(:,1:n-1);ba = M(:,n);
            x = zeros(1,m);
            x(m) =ba(m)/A(m,m);
            for i = m-1:-1:1
                sum = 0;
                for j = i+1:1:m
                    sum = sum+A(i,j)*x(j);
                end
                x(i) = (ba(i)-sum)/A(i,i);
            end
            UTIS = x';
        end
    end
    

      2.例子

    clear all
    clc
    M = rand(9)
    b = reshape(rand(3),9,1)
     
    S = Elimination_of_column_pivot_entries(M,b)
    
    M
    

      结果

    M =
      列 1 至 7
        0.2089    0.3502    0.8699    0.6473    0.4046    0.1389    0.7413
        0.7093    0.6620    0.2648    0.5439    0.4484    0.6963    0.5201
        0.2362    0.4162    0.3181    0.7210    0.3658    0.0938    0.3477
        0.1194    0.8419    0.1192    0.5225    0.7635    0.5254    0.1500
        0.6073    0.8329    0.9398    0.9937    0.6279    0.5303    0.5861
        0.4501    0.2564    0.6456    0.2187    0.7720    0.8611    0.2621
        0.4587    0.6135    0.4795    0.1058    0.9329    0.4849    0.0445
        0.6619    0.5822    0.6393    0.1097    0.9727    0.3935    0.7549
        0.7703    0.5407    0.5447    0.0636    0.1920    0.6714    0.2428
      列 8 至 9
        0.4424    0.3309
        0.6878    0.4243
        0.3592    0.2703
        0.7363    0.1971
        0.3947    0.8217
        0.6834    0.4299
        0.7040    0.8878
        0.4423    0.3912
        0.0196    0.7691
    b =
        0.3968
        0.8085
        0.7551
        0.3774
        0.2160
        0.7904
        0.9493
        0.3276
        0.6713
    此方程有唯一解
    解向量为:
    S =
       19.6917
        7.6005
       17.0314
       -1.6699
       -5.7675
      -12.1059
      -19.9661
       10.5792
      -18.0751
    ans =
       19.6917
        7.6005
       17.0314
       -1.6699
       -5.7675
      -12.1059
      -19.9661
       10.5792
      -18.0751
    

      

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  • 原文地址:https://www.cnblogs.com/guliangt/p/12119393.html
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