线性方程组的分解法——LU分解法

Posted guliangt

tags:

篇首语:本文由小常识网(cha138.com)小编为大家整理,主要介绍了线性方程组的分解法——LU分解法相关的知识,希望对你有一定的参考价值。

  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
>> 

  

以上是关于线性方程组的分解法——LU分解法的主要内容,如果未能解决你的问题,请参考以下文章