Word Count : 526

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We are given the system of equations: \[ \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ -2 \end{pmatrix} \] which corresponds to: \[ x + 2y = 4 \] \[ 2x + y = -2 \] We will set up the Gauss-Jacobi and Gauss-Seidel iteration schemes and analyze their convergence. --- ### Gauss-Jacobi Iteration The Jacobi method is based on solving for each variable in terms of the other variables from the diagonal elements. From the first equation: \[ x = 4 - 2y \] From the second equation: \[ y = -2 - 2x _________ _________ ____ _________ ______ ___ _________ _________ ______.
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