3. Consider the linear system: a. Write out the equations for the Jacobi iterative method for solving this system (don't actually do any iterations). XAeI = (3 -y)/ (14 e) Jour =(4-6)/(He) b. Write out the equations for the Gaus-Seidel iterative method for solving this system. Xary - (3 ya)/(He) e. True or False: Ife > 0, the Jacobi iterative method (3a) will converge for any starting vector (To, p0). Give a reason for your answer. tre, A in diegural demint d. Find the condition number of the above matrix (using the L norm). If you were to solve the above linear system using Gaus sian elimination with partial pivoting, would you expect serious roundoff errors, if e is very small? Hìnt: The inverse of At= I Ze+c* lAl = ?+¢ %3D - Ite cond (A) you, Jeriar rombf
3. Consider the linear system: a. Write out the equations for the Jacobi iterative method for solving this system (don't actually do any iterations). XAeI = (3 -y)/ (14 e) Jour =(4-6)/(He) b. Write out the equations for the Gaus-Seidel iterative method for solving this system. Xary - (3 ya)/(He) e. True or False: Ife > 0, the Jacobi iterative method (3a) will converge for any starting vector (To, p0). Give a reason for your answer. tre, A in diegural demint d. Find the condition number of the above matrix (using the L norm). If you were to solve the above linear system using Gaus sian elimination with partial pivoting, would you expect serious roundoff errors, if e is very small? Hìnt: The inverse of At= I Ze+c* lAl = ?+¢ %3D - Ite cond (A) you, Jeriar rombf
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Answer is given BUT need full detailed steps and process since I don't understand the concept.
![### Consider the Linear System
Given the system:
\[
\begin{bmatrix}
1 + \varepsilon & 1 \\
1 & 1 + \varepsilon \\
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
\end{bmatrix}
=
\begin{bmatrix}
3 \\
4 \\
\end{bmatrix}
\]
#### a. Jacobi Iterative Method
Write out the equations for the Jacobi iterative method:
\[ x_{n+1} = \frac{(3 - y_n)}{(1 + \varepsilon)} \]
\[ y_{n+1} = \frac{(4 - x_n)}{(1 + \varepsilon)} \]
*(Don’t actually do any iterations.)*
#### b. Gauss-Seidel Iterative Method
Write out the equations for the Gauss-Seidel iterative method:
\[ x_{n+1} = \frac{(3 - y_n)}{(1 + \varepsilon)} \]
\[ y_{n+1} = \frac{(4 - x_{n+1})}{(1 + \varepsilon)} \]
#### c. Convergence Discussion
**True or False:** If \(\varepsilon > 0\), the Jacobi iterative method (3a) will converge for any starting vector \((x_0, y_0)\).
- **Answer:** True, as the matrix is diagonally dominant.
#### d. Condition Number and Roundoff Errors
Find the condition number of the above matrix (using the \(L_{\infty}\) norm).
Matrix \(A\):
\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
- \(\| A \| = 2 + \varepsilon\)
Inverse of A:
\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
Calculate:
\[ A^{-1} = \frac{1}{2\varepsilon + \varepsilon^2} \begin{bmatrix} 1 + \varepsilon & -1 \\ -1 & 1 + \varepsilon \end{bmatrix} \]
Condition number:
\[ \| A^{-1} \| = \frac{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2F600bcdbe-9d52-4179-a37a-8a33911f3bc7%2Fhndbkra_processed.png&w=3840&q=75)
Transcribed Image Text:### Consider the Linear System
Given the system:
\[
\begin{bmatrix}
1 + \varepsilon & 1 \\
1 & 1 + \varepsilon \\
\end{bmatrix}
\begin{bmatrix}
x \\
y \\
\end{bmatrix}
=
\begin{bmatrix}
3 \\
4 \\
\end{bmatrix}
\]
#### a. Jacobi Iterative Method
Write out the equations for the Jacobi iterative method:
\[ x_{n+1} = \frac{(3 - y_n)}{(1 + \varepsilon)} \]
\[ y_{n+1} = \frac{(4 - x_n)}{(1 + \varepsilon)} \]
*(Don’t actually do any iterations.)*
#### b. Gauss-Seidel Iterative Method
Write out the equations for the Gauss-Seidel iterative method:
\[ x_{n+1} = \frac{(3 - y_n)}{(1 + \varepsilon)} \]
\[ y_{n+1} = \frac{(4 - x_{n+1})}{(1 + \varepsilon)} \]
#### c. Convergence Discussion
**True or False:** If \(\varepsilon > 0\), the Jacobi iterative method (3a) will converge for any starting vector \((x_0, y_0)\).
- **Answer:** True, as the matrix is diagonally dominant.
#### d. Condition Number and Roundoff Errors
Find the condition number of the above matrix (using the \(L_{\infty}\) norm).
Matrix \(A\):
\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
- \(\| A \| = 2 + \varepsilon\)
Inverse of A:
\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
Calculate:
\[ A^{-1} = \frac{1}{2\varepsilon + \varepsilon^2} \begin{bmatrix} 1 + \varepsilon & -1 \\ -1 & 1 + \varepsilon \end{bmatrix} \]
Condition number:
\[ \| A^{-1} \| = \frac{
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