Gauss$-$Seidel iteration improves on Jacobi by simply using a new value of xi as soon as it is ready.

example row $12$ out of $100,$ numbered $0$ to $99,$ we are setting i $=12$ inside of an $$i$$ loop

$[12]=1$

$[12][12]([12]$

$<12$

$$

$=0$

$[12][][]$

$<100$

$$

$=13$

$[12][][])$