This function solves a linear system Ax=b using the Gaussian elimination method with pivoting. The algorithm is outlined below:
1) Initialize a permutation vector r = [1, 2,...,n] where r(i) corresponds to row i in A.
2) For k = 1,...,n-1 find the largest (in absolute value) element among a(r(k),k),a(r(k+1),k),...,a(r(n),k).
3) Assume r(j,k) is the largest element. Switch r(j) and r(k).
4) For i=1,...,k-1,k+1,...,n calculates:
zeta = a(r(i),k) / a(r(k),k)
5) For j=k,...,n calculate:
a(r(i),j)=a(r(i),j)-a(r(k),j)*zeta
b(r(i)) = b(r(i))-b(r(k))*zeta
6) Steps 1 through 6 has effectively diagonalized A.
7) Each element in the solution vector is:
x(r(i)) = b(i)/a(i,i);

Reference:
(1) http://mathworld.wolfram.com/GaussianElimination.html