I have a Gauss-Seidel linear system solver that has always been able to solve many kinds of linear systems. However, it struggles when solving this very simple system: $$ 22.7986\,x_1 + 4981.2903 \,x_2 = 25.48488 \\ 10047.4775\,x_1 -4981.2903\,x_2 = 6.05489 $$
My initial guess is $(0.01,0.01)$ but I have tried changing it to other values and it's still is not converging.
Basically the new guess that the algorithm computes in each iteration consistently becomes larger, and so is the error. I was checking the Burden book to see if there are special cases in which Gauss-Seidel does not converge but there is little on that.
I know that there exists a solution to the above system because I have solved it using Excel's functionality to invert matrices and then use matrix multiplication. Particularly, the solution to the system is $(0.003131967,0.005101786)$.
Any help would be welcome.