Note that the computation of uses only those elements of that have already been computed and only those elements of that have yet to be advanced to iteration. Rather, an element-based approach is used: This matrix expression is mainly of academic interest, and is not used to program the method. Where and, , and represent the diagonal, lower triangular, and upper triangular parts of the coefficient matrix. In matrix terms, the the Gauss-Seidel iteration can be expressed as We seek the solution to a set of linear equations: The method is similar to the Jacobi method and in the same way strict or irreducible diagonal dominance of the system is sufficient to ensure convergence, meaning the method will work. The method is named after the German mathematician Carl Friedrich Gauss and Philipp Ludwig von Seidel. The Gauss-Seidel method is a technique used to solve a linear system of equations.

This code was originally written for Jacobi, and I provided the modification for making it into Gauss-Seidel.Gauss-Seidel method - CFD-Wiki, the free CFD reference If you want, there is another way to calculate Jacobi and Gauss-Seidel, and you can take a look at that post here: Jacobi iteration doesn't end. Once we pass this point, then we use the variables calculated from the current iteration from the first variable up to the point where we are calculating, then use the rest of the variables for calculating the solution. If we are working on calculating the first variable, then we don't have any new information yet, and so we simply just update the solution using all of the previous information from the previous iteration. If idx = 1 %// Case where we are solving for the first variable Therefore: function z=gaussseidel(A,B) %// Change the function name This has to be done in the while loop, because that's where you're doing the iterations. For Gauss-Seidel, for each variable that you solve for, you must use the solutions of the previous variables calculated from the current iteration as part of the solution for the variable you are focusing on.Īs such, for your particular version of the code (though not optimal.), you simply need to add in a for loop where we solve for each variable one at a time, then keep feeding this information into the other variables. For Jacobi, you are simply using the previous iteration's solution to formulate the current solution.

, x_) as part of the solution for the current variable x_i. The difference between a Jacobi solver and a Gauss-Seidel solver is that when you're solving for the solution of a variable x_i at the current iteration, you need to use the information from the previous variables ( x_1, x_2.

Python gauss seidel