Exercise · ZW-JACOBI-AND-GAUSS-SEIDEL-METHODS-010 · ★★★ Advanced
Given \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\) and \(\mathbf{b}=(3,3)\), what is \(x^{(2)}\) with Jacobi from \((0,0)\)?
Choose one option.
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- C. \(3/4\)
Explanation
Jacobi gives \(x^{(1)}=y^{(1)}=3/2\) and then \(x^{(2)}=(3-y^{(1)})/2=3/4\), using only the previous iteration.
Review the concept: Jacobi and Gauss-Seidel methods
The Jacobi and Gauss-Seidel methods are iterative procedures that approximate the solution of a linear system \(A\mathbf{x}=\mathbf{b}\) by repeatedly updating its components.
\[x_i^{(k+1)}=\frac{1}{a_{ii}}\left(b_i-\sum_{j\neq i}a_{ij}x_j^{(k)}\right)\]
More practice problems: jacobi and gauss-seidel methods
- ZW-JACOBI-AND-GAUSS-SEIDEL-METHODS-001 Which method updates the components using values already computed in the same iteration?
- ZW-JACOBI-AND-GAUSS-SEIDEL-METHODS-002 Which property of the matrix guarantees convergence of both methods?
- ZW-JACOBI-AND-GAUSS-SEIDEL-METHODS-003 In the update formula, why is division by a_ii performed?
- ZW-JACOBI-AND-GAUSS-SEIDEL-METHODS-004 What is the usual initial vector in these methods when no estimate is known?
- ZW-JACOBI-AND-GAUSS-SEIDEL-METHODS-005 What type of system is solved with these methods?
- ZW-JACOBI-AND-GAUSS-SEIDEL-METHODS-006 For 4x+y=9, x+3y=7, what is x^(1) with Jacobi from (0,0)?
- ZW-JACOBI-AND-GAUSS-SEIDEL-METHODS-007 For 4x+y=9, x+3y=7, with Gauss-Seidel from (0,0), what is y^(1)?
- ZW-JACOBI-AND-GAUSS-SEIDEL-METHODS-008 If a_22=0 in a 2x2 system, what happens when applying the update formula to the second unknown?