. Janssen, KTH, 2014) Consider the linear system 0.550x + 0.423y = 0.127 0.484x +0.372y = 0.112 1.7 Suppose we are given two possible solutions, u = [_1,91] and v - 1.01 -0.99 a. Decide based on the residuals b – Au and b – Av which of the two possible solutions is the 'better' solution. b. Calculate the exact solution x. c. Compute the errors to the exact solution. That is, compute the infinity norms of u-x and v-x. Do the results change your answer to 7a?
. Janssen, KTH, 2014) Consider the linear system 0.550x + 0.423y = 0.127 0.484x +0.372y = 0.112 1.7 Suppose we are given two possible solutions, u = [_1,91] and v - 1.01 -0.99 a. Decide based on the residuals b – Au and b – Av which of the two possible solutions is the 'better' solution. b. Calculate the exact solution x. c. Compute the errors to the exact solution. That is, compute the infinity norms of u-x and v-x. Do the results change your answer to 7a?
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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Question
![**Consider the Linear System (Source: B. Janssen, KTH, 2014)**
Given the linear system of equations:
\[
0.550x + 0.423y = 0.127
\]
\[
0.484x + 0.372y = 0.112
\]
We are provided with two possible solutions:
\[
\mathbf{u} = \begin{bmatrix} 1.7 \\ -1.91 \end{bmatrix}
\]
and
\[
\mathbf{v} = \begin{bmatrix} 1.01 \\ -0.99 \end{bmatrix}
\]
**Tasks:**
a. **Determine the "Better" Solution:**
- Use the residuals \(\mathbf{b} - A\mathbf{u}\) and \(\mathbf{b} - A\mathbf{v}\) to decide which of the two solutions is the "better" one.
b. **Calculate the Exact Solution \(\mathbf{x}\):**
- Solve the given system to determine the exact values of \(x\) and \(y\).
c. **Compute Errors to the Exact Solution:**
- Calculate the infinity norms of \(\mathbf{u} - \mathbf{x}\) and \(\mathbf{v} - \mathbf{x}\).
- Assess if these results alter the conclusion from part (a).
**Explanation of the Linear Equations and Solutions**
1. **Equations:** The system consists of two equations with two unknowns \(x\) and \(y\). Each equation is a linear combination of \(x\) and \(y\) set equal to a constant.
2. **Residuals:** The residuals are calculated to measure the accuracy of the solutions \(\mathbf{u}\) and \(\mathbf{v}\). Mathematically, the residuals for solution \(\mathbf{u}\) are found by computing \(\mathbf{b} - A\mathbf{u}\), where \(\mathbf{b}\) is the vector of constants on the right-hand side of the equations.
3. **Exact Solution:** The exact solution \(\mathbf{x}\) is obtained by solving the system of equations using methods such as substitution, elimination, or matrix operations.
4. **Infinity Norms:** The infinity norm of a vector is the maximum absolute value of its components. Calcul](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe6e77107-ef1d-4e39-badf-4decafa47f1b%2Feeaf060f-2d73-4581-9fc7-796e39648e3c%2Fj9ab5b_processed.png&w=3840&q=75)
Transcribed Image Text:**Consider the Linear System (Source: B. Janssen, KTH, 2014)**
Given the linear system of equations:
\[
0.550x + 0.423y = 0.127
\]
\[
0.484x + 0.372y = 0.112
\]
We are provided with two possible solutions:
\[
\mathbf{u} = \begin{bmatrix} 1.7 \\ -1.91 \end{bmatrix}
\]
and
\[
\mathbf{v} = \begin{bmatrix} 1.01 \\ -0.99 \end{bmatrix}
\]
**Tasks:**
a. **Determine the "Better" Solution:**
- Use the residuals \(\mathbf{b} - A\mathbf{u}\) and \(\mathbf{b} - A\mathbf{v}\) to decide which of the two solutions is the "better" one.
b. **Calculate the Exact Solution \(\mathbf{x}\):**
- Solve the given system to determine the exact values of \(x\) and \(y\).
c. **Compute Errors to the Exact Solution:**
- Calculate the infinity norms of \(\mathbf{u} - \mathbf{x}\) and \(\mathbf{v} - \mathbf{x}\).
- Assess if these results alter the conclusion from part (a).
**Explanation of the Linear Equations and Solutions**
1. **Equations:** The system consists of two equations with two unknowns \(x\) and \(y\). Each equation is a linear combination of \(x\) and \(y\) set equal to a constant.
2. **Residuals:** The residuals are calculated to measure the accuracy of the solutions \(\mathbf{u}\) and \(\mathbf{v}\). Mathematically, the residuals for solution \(\mathbf{u}\) are found by computing \(\mathbf{b} - A\mathbf{u}\), where \(\mathbf{b}\) is the vector of constants on the right-hand side of the equations.
3. **Exact Solution:** The exact solution \(\mathbf{x}\) is obtained by solving the system of equations using methods such as substitution, elimination, or matrix operations.
4. **Infinity Norms:** The infinity norm of a vector is the maximum absolute value of its components. Calcul
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