1 sin (r) f(x) = x² = x cos(x)+ 0, with xo 2 4 (1) Does Newton's method converge quadratically to the root r Ti E (0, 1]? If not, explain why? (2) Find the multiplicity of the root r = r1 of f(x). (3) Write out the Modified Newton's Method such that we have quadratical convergence.
1 sin (r) f(x) = x² = x cos(x)+ 0, with xo 2 4 (1) Does Newton's method converge quadratically to the root r Ti E (0, 1]? If not, explain why? (2) Find the multiplicity of the root r = r1 of f(x). (3) Write out the Modified Newton's Method such that we have quadratical convergence.
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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Solvef(x) =x2−xcos(x) +14−sin2(x)4= 0,withx0=π2.(1) Does Newton’s method converge quadratically to the rootr=r1∈[0,1]? If not, explain why?(2) Find the multiplicity of the rootr=r1off(x).(3) Write out the Modified Newton’s Method such that we havequadratical convergence.
![### Educational Content for Advanced Calculus
#### Problem Statement
Consider the function:
\[ f(x) = x^2 - x \cos(x) + \frac{1}{4} - \frac{\sin^2(x)}{4} = 0 \]
We begin with an initial approximation:
\[ x_0 = \frac{\pi}{2} \]
#### Questions
1. **Convergence of Newton's Method**
Does Newton’s method converge quadratically to the root \( r = r_1 \in [0, 1] \)? If not, explain why.
2. **Multiplicity of the Root**
Find the multiplicity of the root \( r = r_1 \) of \( f(x) \).
3. **Modified Newton’s Method**
Write out the Modified Newton’s Method such that we have quadratic convergence.
#### Explanation
For educational purposes, we explore:
- **Quadratic Convergence**: Quadratic convergence occurs when the error in the numerical approximation decreases quadratically as iterations proceed.
- **Multiplicity of Roots**: This refers to the number of times a particular root is repeated. It affects the speed of convergence in methods like Newton’s.
- **Modified Newton’s Method**: When a root has a multiplicity greater than one, the standard Newton's method might not converge quadratically. Adjustments are needed in the iterative formula to ensure rapid convergence.
#### Conclusion
Understanding these concepts is crucial for students studying numerical methods and their applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96344119-693c-49c6-b06a-a46154ac473d%2F24324a38-0cdd-4c81-bcf0-1f5a0380c973%2Fqiihg5_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Content for Advanced Calculus
#### Problem Statement
Consider the function:
\[ f(x) = x^2 - x \cos(x) + \frac{1}{4} - \frac{\sin^2(x)}{4} = 0 \]
We begin with an initial approximation:
\[ x_0 = \frac{\pi}{2} \]
#### Questions
1. **Convergence of Newton's Method**
Does Newton’s method converge quadratically to the root \( r = r_1 \in [0, 1] \)? If not, explain why.
2. **Multiplicity of the Root**
Find the multiplicity of the root \( r = r_1 \) of \( f(x) \).
3. **Modified Newton’s Method**
Write out the Modified Newton’s Method such that we have quadratic convergence.
#### Explanation
For educational purposes, we explore:
- **Quadratic Convergence**: Quadratic convergence occurs when the error in the numerical approximation decreases quadratically as iterations proceed.
- **Multiplicity of Roots**: This refers to the number of times a particular root is repeated. It affects the speed of convergence in methods like Newton’s.
- **Modified Newton’s Method**: When a root has a multiplicity greater than one, the standard Newton's method might not converge quadratically. Adjustments are needed in the iterative formula to ensure rapid convergence.
#### Conclusion
Understanding these concepts is crucial for students studying numerical methods and their applications.
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