First the system is rearranged to the form: The Gauss-Seidel method with a stopping criterion of will be used. The same stopping criterion as in the Jacobi method can be used for the Gauss-Seidel method. If we start with nonzero diagonal components for, then can be used to solve the system using forward substitution: The left hand side can be decomposed as follows:Įffectively, we have separated into two additive matrices: where is an upper triangular matrix with zero diagonal components, while is a lower triangular matrix with its diagonal components equal to those of the diagonal components of. First notice that a linear system of size can be written as: This is different from the Jacobi method where all the components in an iteration are calculated based on the previous iteration. In the Gauss-Seidel method, the system is solved using forward substitution so that each component uses the most recent value obtained for the previous component. The Gauss-Seidel method offers a slight modification to the Jacobi method which can cause it to converge faster. Open Educational Resources Iterative Methods: Derivatives Using Interpolation Functions.
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