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I found that the relaxation methods for solving linear system as an iterative sequence are derived from the Gauss-Seidel method and not from the Jacobi method. I understand that the Gauss-Seidel method might look better behaved and in many cases have a smaller spectral radius than that of the Jacobi method. However, once we introduce a relaxation parameter $\omega$, things might change. Or, if they do not improve can anyone show why doing a Jacobi-based relaxation would be a bad idea? That is \begin{equation} \bf{x}_{k+1} = (\rm 1 - \omega) \bf{x}_k + \omega \, Jacobi\_Iteration( \bf{x}_k). \end{equation} does not converge faster in any situation that \begin{equation} \bf{x}_{k+1} = (\rm 1 - \omega) \bf{x}_k + \omega \, Gauss\_Seidel\_Iteration( \bf{x}_k) ? \end{equation}

Thanks.

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I think, the term relaxation has confused you. The weighted Jacobi exists, see e.g. https://en.m.wikipedia.org/wiki/Jacobi_method#Weighted_Jacobi_method but the term relaxation w.r.t. Gauss-Seidel means that the method minimizes a certain energy functional and can be viewed as a variational iterative method. Moreover, look here: https://ocw.mit.edu/courses/mathematics/18-086-mathematical-methods-for-engineers-ii-spring-2006/readings/am62.pdf where Gilbert Strang makes a strong point about using weighted Jacobi with the weight 2/3

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