MANE 3351

Lecture 13, October 4

Classroom Management

Agenda

• Lab today in ENGR 2.268 at 3:30 pm
• GitHub Accounts
  • Sign up for free student GitHub account at https://education.github.com/students
  • Register your student GitHub account with MANE 3351 GitHub classroom email
• Lecture

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Resources

Handouts

• Lecture 13 Slides
• Lecture 13 Marked Slides

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Calendar

Lecture	Date	Content
1	August 23	Welcome to Class, Syllabus
2	August 25	Error Analysis
3	August 30	Introduction to Jupyter Notebook
4	September 1	Markdown
5	September 6	Labor Day Holiday
6	September 8	Taylor Polynomials, Homework 1
7	September 13	Roots of Equations, Bisection Method
8	September 15	Bisection Error Analysis
9	September 20	False Position Algorithm
10	September 22	Newton Raphson Algorithm
11	September 27	Secant Method
12	September 29	Numerical Integration

Assignments

• Homework 3 (assigned 9/22/2021, due 9/30/2021)

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Looking Ahead

Lecture	Date	Content
13	October 4	Simpson's Rule
14	October 6	Romberg Integration
15	October 11	Gaussian Quadrature (end of material for Test 1)
16	October 13	Vectors
17	October 18	Vector Operations
18	October 20	Test 1

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Lecture 13, October 4

Today's topic is Simpson's Rule.

• Simpson's (1/3) rule is Newton-Cotes with \(n=2\)
• Chapra and Canale (2015)³ illustrate Simpson's rule in the figure shown below

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Definition of Simpson's Rule

• Chapra and Canale(2015)³ provide the following definition of Simpson's rule

\[ I=\frac{h}{3}\left[f(x_0)+4f(x_1)+f(x_2)\right] \]

where \(h=(b-a)/2\)

• Note that Simpson's rule is of the form

\[ \begin{aligned} I&\approx \left(b-a\right)\frac{f(x_0)+4f(x_1)+f(x_2)}{6}\\ I&\approx\mbox{ width}\times\mbox{ average height} \end{aligned} \]

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Simpson's Rule Error Analysis

• Chapra and Canale (2015)³ provides the following definition:

\[ \begin{aligned} I&=\frac{h}{3}\left[f(x_0)+4f(x_1)+f(x_2)\right]-\frac{1}{90}f^{(4)}(\xi)h^5\\ I&=\mbox{Simpson's 1/3 approximation}-\mbox{ Truncation error} \end{aligned} \]

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Multiple Applications of Simpson's Rule

• The number of segments must be even
• The formula is

\[ I\approx\left(b-a\right)\frac{f(x_0)+4\sum_{i=1,3,5}^{n-1}f(x_i)+2\sum_{j=2,4,6}^{n-2}f(x_j)+f(x_n)}{3n} \]

where \(h=\frac{b-a}{n}\)

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Pseudo-code: Simpson's 1/3 Rule

• CodeSansar⁴ provides the following pseudo-code

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Jupyter Notebook Demonstration

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Simpson's 3/8 Rule

• Simpson's 3/8 rule is Newton-Cotes with \(n=3\)
• Chapra and Canale (2015)³ illustrate Simpson's rule in the figure shown below

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Definition of Simpson's 3/8 Rule

• Chapra and Canale(2015)³ provide the following definition of Simpson's rule

\[ I=\frac{3h}{8}\left[f(x_0)+3f(x_1)+3f(x_2)+f(x_3)\right] \]

where \(h=(b-a)/3\)

• Note that Simpson's 3/8 rule is of the form

\[ \begin{aligned} I&\approx \left(b-a\right)\frac{f(x_0)+3f(x_1)+3f(x_2)+f(x_3)}{8}\\ I&\approx\mbox{ width}\times\mbox{ average height} \end{aligned} \]

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Truncation Error of Simpson's 3/8 Rule

\[ \begin{aligned} E_t&=-\frac{3}{80}h^5f^{(4)}(\xi)\\ &=-\frac{\left(b-a\right)^5}{6480}f^{(4)}(\xi) \end{aligned} \]

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Pseudo-code: Simpson's 3/8 Rule

CodeSansar⁵ provides the following pseudo-code for Simpson's 3/8 Rule

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1. Cheny, W., and Kincaid, D., (2004), Numerical Mathematics and Computer, 5th edition ↩

2. Kiusalaas, J. (2013), Numerical Methods in Engineering with Python 3 ↩

3. Chapra, S., and Canale, R., (2015), Numerical Methods for Engineers, 7th edition ↩↩↩↩↩

4. https://www.codesansar.com/numerical-methods/integration-simpson-1-3-method-algorithm.htm ↩

5. https://www.codesansar.com/numerical-methods/integration-simpson-3-8-method-pseudocode.htm ↩