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  • 线性方程组的分解法——LU分解法

      1.代码

    %%LU分解法
    function LUDM = LU_Decomposition_method(A,b)
    global n;global B;global U;global L;global M;
    [n,n] = size(A);
    B = [A,b];
    R_A = rank(A);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');
        M = LU_decomposition(A);
        L = M(:,:,1);U = M(:,:,2);
        matrix1 = [L b];
        Y = Lower_trig_iterative_solution(matrix1);
        matrix2 = [U Y];
        X = Upper_trig_iterative_solution(matrix2);
        disp('LU分解中L=');
        L
        disp('LU分解中U=');
        U
    else
        disp('方程有无穷多组解');
    end
    disp('解向量为:');
    LUDM = X;
    
    %%矩阵的LU分解
        function LUD = LU_decomposition(A)
            [n,n] = size(A);
            M = Elementary_transformation_of_the_lower_triangle(A);
            L = M(:,:,n);U=A;
            for i = 1:1:n-1
                U = M(:,:,i)*U;
            end
            LUD(:,:,1) = L;
            LUD(:,:,2) = U;
        end
    %%下三角初等变换
        function ETLT = Elementary_transformation_of_the_lower_triangle(A)
            [n,n] = size(A);
            L = zeros(n,1,n);
            for i = 1:1:n
                for j = 1:1:n
                    for k = 1:1:n
                        if j == k
                            L(j,k,i) = 1;
                        end
                    end
                end
            end
            for i = 1:1:n-1
                for j = 1:1:n
                    for k = 1:1:n
                        if j > k
                            if i == k
                                L(j,k,i) = -A(j,k)/A(k,k);
                            end
                            L(i+1:n,i,n) = -L(i+1:n,i,i);
                        end
                    end
                end
                A = L(:,:,i)*A;
            end
            ETLT = L;
        end
    %%下三角迭代法
        function LTIS = Lower_trig_iterative_solution(M)
            [m,n] = size(M);
            B  =M(:,1:n-1);ba = M(:,n);
            y = zeros(1,m);
            y(1) = ba(1);
            for i = 2:1:m
                sum = 0;
                for j = 1:1:i-1
                    sum = sum+B(i,j)*y(j);
                end
                y(i) = ba(i)-sum;
            end
            LTIS = y';
        end
    %%上三角迭代法
        function UTIS = Upper_trig_iterative_solution(M)
            [m,n] = size(M);
            B = M(:,1:n-1);ba = M(:,n);
            x = zeros(1,m);
            x(m) =ba(m)/B(m,m);
            for i = m-1:-1:1
                sum = 0;
                for j = i+1:1:m
                    sum = sum+B(i,j)*x(j);
                end
                x(i) = (ba(i)-sum)/B(i,i);
            end
            UTIS = x';
        end
    end
    

      2.例子

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

      结果

    M =
      列 1 至 7
        0.5944    0.4709    0.4076    0.4235    0.5181    0.0680    0.6022
        0.0225    0.6959    0.8200    0.0908    0.9436    0.2548    0.3868
        0.4253    0.6999    0.7184    0.2665    0.6377    0.2240    0.9160
        0.3127    0.6385    0.9686    0.1537    0.9577    0.6678    0.0012
        0.1615    0.0336    0.5313    0.2810    0.2407    0.8444    0.4624
        0.1788    0.0688    0.3251    0.4401    0.6761    0.3445    0.4243
        0.4229    0.3196    0.1056    0.5271    0.2891    0.7805    0.4609
        0.0942    0.5309    0.6110    0.4574    0.6718    0.6753    0.7702
        0.5985    0.6544    0.7788    0.8754    0.6951    0.0067    0.3225
      列 8 至 9
        0.7847    0.1917
        0.4714    0.7384
        0.0358    0.2428
        0.1759    0.9174
        0.7218    0.2691
        0.4735    0.7655
        0.1527    0.1887
        0.3411    0.2875
        0.6074    0.0911
    b =
        0.5762
        0.6834
        0.5466
        0.4257
        0.6444
        0.6476
        0.6790
        0.6358
        0.9452
    此方程有唯一解
    LU分解中L=
    L =
      列 1 至 7
        1.0000         0         0         0         0         0         0
        0.0379    1.0000         0         0         0         0         0
        0.7155    0.5352    1.0000         0         0         0         0
        0.5261    0.5762  -74.4491    1.0000         0         0         0
        0.2717   -0.1391 -136.4397    1.7669    1.0000         0         0
        0.3008   -0.1074  -74.0359    0.9200    0.6765    1.0000         0
        0.7115   -0.0228   42.5434   -0.5996    0.3838 -141.0829    1.0000
        0.1585    0.6728   -1.3001   -0.0414    0.8852  -70.1396    0.4925
        1.0070    0.2658  -39.5864    0.4476    1.3552   49.3425   -0.3788
      列 8 至 9
             0         0
             0         0
             0         0
             0         0
             0         0
             0         0
             0         0
        1.0000         0
        5.1107    1.0000
    LU分解中U=
    U =
      列 1 至 7
        0.5944    0.4709    0.4076    0.4235    0.5181    0.0680    0.6022
             0    0.6781    0.8045    0.0748    0.9240    0.2522    0.3640
             0         0   -0.0039   -0.0765   -0.2275    0.0404    0.2903
             0         0         0   -5.8101  -16.7848    3.4944   21.0900
       -0.0000         0         0         0   -1.1550    0.1988    2.6992
        0.0000         0         0         0         0   -0.0074    0.5483
        0.0000   -0.0000         0         0         0         0   76.6535
        0.0000    0.0000         0   -0.0000         0         0         0
       -0.0000   -0.0000         0    0.0000         0         0         0
      列 8 至 9
        0.7847    0.1917
        0.4416    0.7312
       -0.7621   -0.2857
      -57.2283  -20.8735
       -2.2924   -1.7782
       -1.9343    0.0429
     -274.3037    6.4447
       -1.9999   -0.0598
             0    0.7768
    解向量为:
    S =
       -0.9496
        2.2130
        0.5483
        1.9595
       -3.8859
       -0.4632
        0.4453
        0.3978
        2.6573
    ans =
       -0.9496
        2.2130
        0.5483
        1.9595
       -3.8859
       -0.4632
        0.4453
        0.3978
        2.6573
    >> 
    

      

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  • 原文地址:https://www.cnblogs.com/guliangt/p/12119385.html
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