线性方程组的分解法——列主元消去法

Posted 2020-11-24 guliangt

　　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