Back substitution help examining code?

I am examining this code for performing back substitution on naive gaussian elimination, and I can't seem to figure out where the x(j) is defined at? Or what exactly it's suppose to be prior to finding the solution x. I know that x is our solutions / solution vector to the system of equations when performing naive gaussian elimination. But I am not sure what exactly the value of a(i,j) * x(j) is at this step; is our solution vector x just suppose to be all zeros?? I am just not following how the solution vector is used in this step. Thanks for the help.

for i = n : -1 : 1

for j = i+1 : n

b(i) = b(i) - a(i,j)*x(j);

end

x(i) = b(i)/a(i,i);

end

Answer 1

[Sai Sri Pathuri]

Consider the system of equations as Ax = B where A is the coefficient matrix and B is the constant matrix.

In your code, the matrices a,b correspond to the matrices A, B after Gaussian elimination. For i = n, the value of j is n+1. Hence, the code inside second for loop is not executed. When j = n (the code inside second for loop run for the first time), the value of x(j) is defined as

x(j) = x(n) = b(n)/a(n,n); 

And the remaining indices (1:n) of x are set to zero. The values of x for index < n is obtained by using the previous index value. For example, x(n) is used in the computation of x(n-1).