This is generally expected, since the Gauss-Seidel Method uses new values as we find them, rather than waiting until the subsequent iteration, as is done with the Jacobi Method.ĭavid M.In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems. Notice that this sequence of iterations converges to the true solution (1, -2, 1) much more quickly than we found in Example 1 using the Jacobi Method. We iterate this process to generate a sequence of increasingly better approximations x (0), x (1), x (2), … and find results similar to those that we found for Example 1.Īs in Example 1, we stop iterating after x ( k) − x ( k-1), e ( k), and || e ( k)|| are all 0 to three decimal places. The result of this first iteration of the Gauss-Seidel Method is We then solve for x 2 (1) in the second equation, using the new value of x 1 (1) = 0.750, and find thatįinally, we solve for x 3 (1) in the third equation, using the new values of x 1 (1) = 0.750 and x 2 (1) = 1.750, and find that We first solve for x 1 (1) in the first equation and find that Let us be clear about how we solve this system. To compare our results from the two methods, we again choose x (0) = (0, 0, 0).

Let's apply the Gauss-Seidel Method to the system from Example 1:Īt each step, given the current values x 1 ( k), x 2 ( k), x 3 ( k), we solve for x 1 ( k+1), x 2 ( k+1), x 3 ( k+1) in This technique is called the Gauss-Seidel Method - even though, as noted by Gil Strang in his Introduction to Applied Mathematics, Gauss didn’t know about it and Seidel didn’t recommend it.

So x 1 ( k+1) is found as in Jacobi's Method, but in finding x 2 ( k+1), instead of using the old value of x 1 ( k) and the old values x 3 ( k), …, x n ( k), we now use the new value x 1 ( k+1) and the old values x 3 ( k), …, x n ( k), and similarly for finding x 3 ( k+1), …, x n ( k+1). Where the true solution is x = ( x 1, x 2, …, x n), if x 1 ( k+1) is a better approximation to the true value of x 1 than x 1 ( k) is, then it would make sense that once we have found the new value x 1 ( k+1) to use it (rather than the old value x 1 ( k)) in finding x 2 ( k+1), …, x n ( k+1). Let us take Jacobi’s Method one step further.

Gauss seidel with relaxation python