function sol=gauss(a,b)
%  File name:gauss.m      
%  Author:Yan Anxin
%  ID number:081810
%  Date:finished on 2008.10.29
%  Description:  
%       Here defines a function,applying Gaussian main-element elimination to solve
%       linear equations by the standard form Ax=b.
%       In the syntax sol=gauss(a,b),
%       a----input,matrix of coefficient
%       b----input,right vector 
%       sol----output,returns the solution of linear equation     

siz=size(a);   %gets the dimension of the coefficient matrix 
n=siz(1);
a=[a b];       %forms augmented matrix


for i=1:n
    maxyuan=0;          %the largest element of the column vector
    maxyuanabs=0;       %absolute value of maxyuan
    for k=i:n
        if(abs(a(k,i))>maxyuanabs)    %to find out the main-element and its column
            maxyuanabs=abs(a(k,i));
            maxyuan=a(k,i);
            maxn=k;        
        end 
    end
    
    %exits the loop if there is no nonzero column element,or exchanges rows  
    if(maxyuan==0) ,continue,end   
    
    change=a;                          %exchanges rows
     for k=i:(n+1)
         change(i,k)=a(maxn,k);    
         change(maxn,k)=a(i,k);
         a=change;        
     end
     
     for p=(i+1):n                     %eliminate elements,forming triangular matrix
         L=a(p,i)/a(i,i);
         for t=i:(n+1)             
             a(p,t)=a(p,t)-a(i,t)*L;
         end 
     end
    
end

x(n)=a(n,n+1)/a(n,n);                  %back substitutes to solve the equations set 
for i=1:(n-1)
    s=0;
    for j=((n-i)+1):n
        s=s+a(n-i,j)*x(j);
    end
    x(n-i)=(a(n-i,n+1)-s)/a(n-i,n-i);
end

sol=x';