#LnormInf corresponds to the absolute value of the greatest element of the vector. Print ('The solution vector in iteration', iter1, 'is:', x) def gauss_seidel(A, b, tolerance, max_iterations, x): Instead I created my own little function that with the help of a permutation matrix as seen in another answer of mine permutation matrix will produce the solution (x vector) for any square matrix, including those with zeros on the diagonal. 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.I know this is old but, I haven't found any pre existing library in python for gauss - seidel. 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. 9.7.5 Sparse Gauss - Seidel implementation We remind. The result of this first iteration of the Gauss-Seidel Method is In Python we can get the CSR format easily using the sparse.csr matrix function as it 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 implmentation python