 Gauss-Seidel iteration for a two-variable positive-definite system

Consider the original linear systemDirect substitution verifies the solution. The matrix is a symmetric positive-definite matrix: completing the square in its quadratic form gives $4x^2+2xy+3y^2=4(x+y/4{)}^2+11y^2/4>0$ for every nonzero vector. It is also a strictly diagonally dominant matrix.

Starting from $(x^{(0)},y^{(0)})=(0,0)$, update the first row and then the second:The first sweep gives $x^{(1)}=1/4$ and $y^{(1)}=(2-1/4)/3=7/12$. The next gives $x^{(2)}=(1-7/12)/4=5/48$ and $y^{(2)}=(2-5/48)/3=91/144$. The second row uses $1/4$ during the first sweep, even though the initial first coordinate was zero.

With a stale first coordinate, the Jacobi method would instead give $(1/4,2/3)$ after the first sweep. Both methods have the same fixed point, but changing when coordinates become available changes the entire sequence of vectors.