%----------------------------------------------------------%
%- @description:    computes the LU-Decompositon of a 
%-                  matrix  without using pivot method
%- @input:  A   -   regular square matrix
%- @output: L   -   a lower triangle matrix
%-          R   -   an upper triangle matrix
%----------------------------------------------------------%

function [L,R] = lr(A)

R = A;
n = size(R,1);

for k = 1:n
    L(k,k) = 1;
    for i = k+1:n
        %compute l_ik
        L(i,k) = R(i,k) / R(k,k);      
        for j = k+1:n
            %compute a_jk^(k+1)
            R(i,j) = R(i,j) - L(i,k) * R(k,j);            
        end        
        R(i,k) = 0;
    end    
end