In this situation, the direction that we 'sweep' is important - if we step though the solution vector in the opposite direction, the solution moves away from the chosen initial condition (zero everywhere in the interior) more quickly. The following table gives the results of 10 iterations with 5 nodes (3 interior and 2 boundary) as well as norm error. Note that (in this scheme), is from the previous iteration, while is from the current iteration: Then, stepping through the solution vector from to, we can compute the next iterate from the two surrounding values. However, for any given for, we can write Where is the (discrete) solution available at uniformly spaced nodes. The standard second-order finite difference discretization is Consider the simple heat equation problem In some cases, we need not even explicitly represent the matrix we are solving. It is necessary, however, that the diagonal terms in the matrix are greater (in magnitude) than the other terms. The Gauss-Seidel method sometimes converges even if this condition is not satisfied. Strict row diagonal dominance means that for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms: The method will always converge if the matrix A is strictly or irreducibly diagonally dominant. The iteration is generally continued until the changes made by an iteration are below some tolerance. Thus, unlike the Jacobi method, we do not have to do any vector copying should we wish to use only one storage vector. While this might seem like a rather minor concern, for large systems it is unlikely that every iteration can be stored. This means that no additional storage is required, and the computation can be done in place ( replaces ). 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. Gauss-Seidel method - CFD-Wiki, the free CFD reference

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