Word Count : 526

---

We are given a linear system: $$ \begin{pmatrix}1 & 2\\ 2 & 1\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}4\\ -2\end{pmatrix} $$ That is, $$ \begin{aligned} x + 2y &= 4 \quad \text{(1)} \\ 2x + y &= -2 \quad \text{(2)} \end{aligned} $$ --- ### Step 1: Rewrite equations for iterative methods We rewrite each equation solving for the leading variable: From (1): $$ x = 4 - 2y \quad \text{(A)} $$ From (2): $$ y = -2 - 2x \quad \text{(B)} $$ --- ### Step 2: Jacobi Iteration Scheme In Jacobi, each new value is calculated using only values from the previous iteration. Using equations (A) and (B), the iterative formulas become: $$ \begin{aligned} x^{(k+1)} &= 4 - 2y^{(k)} \\ y^{(k+1)} &= -2 - 2x^{(k)} \end{aligned} $$ Or in matrix form: $$ \begin{pmatrix} x^{(k+1)} ______ _____ ____ _____ _________ _______ ____ _____ _____ _________ ____.
_____ _________ ____ _______ _____ _________.
_____ ________ ________ ____ _________ _________ _______ ______ ____.
_______ _______ _______ ___ ____ _______ __________ ________ _______ ______ __________.
_________ ___ ____ _____ ____ _________ ____ __________ ______.
___ ________ ______ ___ ________ _________ ________ ___ __________ ______.
____ ________ _______ _____ _____ ____ _____ _________ ________ _______ _______.
__________ _________ _______ ___ ____ _______ __________ ______ _________ ______ ______ __________.
_____ _______ _______ _________ ________ _________ ________.
___ ___ _______ _________ ___ _______ _________ ________ ________ ______ ___.
______ _________ _______ _____ ___ _____ _________ _____.
________ _______ _______ ______ __________ ______ ______.
___ ____ _____ _____ _____ ________ _________.
__________ ____ ________ ____ ____ ______ ______ ___ ___ _______ ___ _______.
_________ _______ _____ _________ _______ ___ __________ ___ _____ ___ _____ _______.
____ ______ _______ ____ ____ _______ ______.
__________ _________ ____ _____ ________ ___ ___ _________ ________.
_______ _______ _______ ________ __________.
______ _______ _____ ___ _____ ______ _____ ______ _________ __________ __________ ____.
________ _______ __________ ___ ______ ___ _________ _____ _________.
____ ____ _____ ______ __________ __________ _____ ___ ____ ____ ______ ____.
_________ ________ ______ ______ __________ __________ _______ ________ __________ __________ ________.
__________ _____ _________ ____ ___ ___ _________ ______ ________.
__________ _________ _________ ________ ____ ___ ___ ______ _________ _______ _________.
________ __________ ________ ______ ________.
______ _________ ______ ________ _____ _____.
___ _______ _________ __________ _________ ______.
_____ ___ _______ _________ _____ ___ ______ ________ ________ ____.
_________ _________ ____ _________ _______ __________ __________ ________ __________ ___ ____ __________.
______ __________ ___ _______ ________.
______ _____ ____ ___ ___ ______ __________ _____ __________ _____ ______ ____.
______ ___ _______ ________ ________ _________ _________ ________ _______ ________.
_______ _________ ____ ______ ___ _____ ______.
________ ___ __________ ______ _______ __________ ___ _________ _____ ______ ___ _________.
______ _____ ___ __________ _________ ______ _____ ___.
_________ ___ ____ ______ ___ ____ _______ _________ _________ _______ __________ ___.
__________ __________ _______ _______ ________.
_______ _______ ____ _______ ________ _________ _________ ___ _____ _____.
_________ ________ _________ ____ ____ ______ ____ ___.
______ _________ _________ ___ _______ __________ ______ _____.
_______ ___ __________ ___ ______ _____.
_____ ______ ___ ________ ______ ______ _______ _______ _________ _______ ____.
__________ _____ _______ ______ ______ _______ _____ _________ _____ _________.
________ _________ ______ _________.
Get Full Answer on WhatsApp