Linear equations decomposition method - out PCA elimination method

　　1. Code

%% main-element elimination
function ECPE = Elimination_of_column_pivot_entries(M,b)
No overall;
[n,n] = size(M);
B  =[M,b];
R_A = rank(M);R_B = rank(B);
if R_A ~= R_B
    DISP ( 'equation has no solution');
elseif (R_A == R_B)&&(R_A == n)
    DISP ( 'This equation has a unique solution');
    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 ( 'equation has infinite number of solutions');
end
DISP ( 'solution vector is:');
ECPE = X;
%% out PCA transform
%% column designated p, find the maximum value of the column, this value is used for the main line change element
    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 columns elimination function
    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
%% upper triangular iterative method
    function UTIS = Upper_trig_iterative_solution(M)
        [m,n] = size(M);
        = M (:, 1: n-1); b = 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. Examples

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

M\b

　　result

M =
  Column 1-7
    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
  Column 8-9
    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
This equation has a unique solution
Solution vector is:
S =
   19.6917
    7.6005
   17.0314
   -1.6699
   -5.7675
  -12.1059
  -19.9661
   10.5792
  -18.0751
years =
   19.6917
    7.6005
   17.0314
   -1.6699
   -5.7675
  -12.1059
  -19.9661
   10.5792
  -18.0751