Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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Can you help me with the code in MATLAB?
![### Numerical Experiment and Interpretation
#### Objectives:
Interpret the results of the following numerical experiment and draw some conclusions.
#### Experiment Steps:
**a. Polynomial Interpolation with Equally Spaced Nodes**
- Define \( p \) to be the polynomial of degree 20 that interpolates the function \( f(x) = (1 + 6x^2)^{-1} \).
- Use 21 equally spaced nodes in the interval \([-1, 1]\), including the endpoints as nodes.
- Create a table of \( f(x) \), \( p(x) \), and \( f(x) - p(x) \) at 41 equally spaced points in the interval.
**b. Polynomial Interpolation with Chebyshev Nodes**
- Repeat the experiment using the Chebyshev nodes given by:
\[
x_i = \cos\left[\frac{(i-1)\pi}{20}\right], \quad (1 \leq i \leq 21)
\]
**c. Cubic Interpolating Spline**
- With 21 equally spaced knots, repeat the experiment using a cubic interpolating spline.
#### Explanation of Graphs/Diagrams:
This section would typically feature graphs showing:
- **Function \( f(x) \)**: A smooth curve representing the function \( (1 + 6x^2)^{-1} \).
- **Interpolated Polynomial \( p(x) \)**: A plot of the polynomial curve fitted through the 21 nodes.
- **Error Graph \( f(x) - p(x) \)**: A plot showing the difference between the original function and the interpolated polynomial at the 41 evaluation points.
These graphs could help illustrate how well the polynomial or spline approximates the function and analyze the error distribution.](https://content.bartleby.com/qna-images/question/ad0d55fe-d83b-4711-86a1-cee8ecea510f/ba77516f-df93-478c-9b23-c8de75f01a8f/p1co5xm_processed.png)
Transcribed Image Text:### Numerical Experiment and Interpretation
#### Objectives:
Interpret the results of the following numerical experiment and draw some conclusions.
#### Experiment Steps:
**a. Polynomial Interpolation with Equally Spaced Nodes**
- Define \( p \) to be the polynomial of degree 20 that interpolates the function \( f(x) = (1 + 6x^2)^{-1} \).
- Use 21 equally spaced nodes in the interval \([-1, 1]\), including the endpoints as nodes.
- Create a table of \( f(x) \), \( p(x) \), and \( f(x) - p(x) \) at 41 equally spaced points in the interval.
**b. Polynomial Interpolation with Chebyshev Nodes**
- Repeat the experiment using the Chebyshev nodes given by:
\[
x_i = \cos\left[\frac{(i-1)\pi}{20}\right], \quad (1 \leq i \leq 21)
\]
**c. Cubic Interpolating Spline**
- With 21 equally spaced knots, repeat the experiment using a cubic interpolating spline.
#### Explanation of Graphs/Diagrams:
This section would typically feature graphs showing:
- **Function \( f(x) \)**: A smooth curve representing the function \( (1 + 6x^2)^{-1} \).
- **Interpolated Polynomial \( p(x) \)**: A plot of the polynomial curve fitted through the 21 nodes.
- **Error Graph \( f(x) - p(x) \)**: A plot showing the difference between the original function and the interpolated polynomial at the 41 evaluation points.
These graphs could help illustrate how well the polynomial or spline approximates the function and analyze the error distribution.
Expert Solution
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Step 1: a. Generate interpolation nodes, calculate function values, perform polynomial interpolation...
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VIEW Step 3: b. A function using Chebyshev nodes and calculates the differences between the original function:
VIEW Step 4: b. Matlab script:
VIEW Step 5: c. using a cubic interpolating spline with 21 equally spaced knots,
VIEW Step 6: Matlab script:
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