Lec4.3. Determine mmnimum nuneber of iterations so that ds J7 is estimated withn the accuracy o* Eclo25 applying Bisect ion Methed for a=1.5 be3 ,
Lec4.3. Determine mmnimum nuneber of iterations so that ds J7 is estimated withn the accuracy o* Eclo25 applying Bisect ion Methed for a=1.5 be3 ,
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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![### Lecture 4.3: Determining Minimum Number of Iterations
**Objective:**
Determine the minimum number of iterations required so that the value of \( \alpha = \sqrt{7} \) is estimated within an accuracy of \( \varepsilon = 10^{-25} \) using the **Bisection Method**.
**Given:**
- Goal: to estimate \( \alpha = \sqrt{7} \)
- Desired accuracy: \( \varepsilon = 10^{-25} \)
- Initial interval for the Bisection Method:
- \( a = 1.5 \)
- \( b = 3 \)
In this session, we will apply the Bisection Method to approximate the square root of 7 to the specified accuracy and determine the minimum number of iterations needed.
### Explanation of the Bisection Method:
The Bisection Method is an iterative numerical technique used to find the roots of a continuous function. It involves repeatedly dividing the interval and selecting a subinterval in which the function changes sign, thereby narrowing down the root.
### Procedure:
1. **Initialization**:
- Select initial points \( a \) and \( b \) such that \( f(a) \) and \( f(b) \) have opposite signs.
2. **Iteration**:
- Calculate the midpoint \( c = \frac{a + b}{2} \).
- Determine the function value at \( c \).
3. **Check for Convergence**:
- If \( f(c) \) is close enough to zero (within the tolerance \( \varepsilon \)), or the interval \([a, b]\) is sufficiently small, stop the iteration.
- If N iterations are required to achieve the desired accuracy, derive N using the formula for the Bisection method:
\[
N \geq \frac{\log_2\left(\frac{b - a}{\varepsilon}\right)}{\log_2(2)}
\]
### Practical Example to Illustrate:
Given the interval \([1.5, 3]\):
- Iteratively apply the Bisection Method until the interval's width is less than \( \varepsilon = 10^{-25} \).
By the end of this lesson, you should be comfortable with determining the number of iterations required to achieve a specific accuracy using the Bisection Method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc58b2eee-5b79-4c9d-923b-d4e913d44dbe%2F4dc40177-ddd1-4fc9-b39e-13ca20518dff%2Fyh22nf8_processed.png&w=3840&q=75)
Transcribed Image Text:### Lecture 4.3: Determining Minimum Number of Iterations
**Objective:**
Determine the minimum number of iterations required so that the value of \( \alpha = \sqrt{7} \) is estimated within an accuracy of \( \varepsilon = 10^{-25} \) using the **Bisection Method**.
**Given:**
- Goal: to estimate \( \alpha = \sqrt{7} \)
- Desired accuracy: \( \varepsilon = 10^{-25} \)
- Initial interval for the Bisection Method:
- \( a = 1.5 \)
- \( b = 3 \)
In this session, we will apply the Bisection Method to approximate the square root of 7 to the specified accuracy and determine the minimum number of iterations needed.
### Explanation of the Bisection Method:
The Bisection Method is an iterative numerical technique used to find the roots of a continuous function. It involves repeatedly dividing the interval and selecting a subinterval in which the function changes sign, thereby narrowing down the root.
### Procedure:
1. **Initialization**:
- Select initial points \( a \) and \( b \) such that \( f(a) \) and \( f(b) \) have opposite signs.
2. **Iteration**:
- Calculate the midpoint \( c = \frac{a + b}{2} \).
- Determine the function value at \( c \).
3. **Check for Convergence**:
- If \( f(c) \) is close enough to zero (within the tolerance \( \varepsilon \)), or the interval \([a, b]\) is sufficiently small, stop the iteration.
- If N iterations are required to achieve the desired accuracy, derive N using the formula for the Bisection method:
\[
N \geq \frac{\log_2\left(\frac{b - a}{\varepsilon}\right)}{\log_2(2)}
\]
### Practical Example to Illustrate:
Given the interval \([1.5, 3]\):
- Iteratively apply the Bisection Method until the interval's width is less than \( \varepsilon = 10^{-25} \).
By the end of this lesson, you should be comfortable with determining the number of iterations required to achieve a specific accuracy using the Bisection Method.
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