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MANE 3351

Lecture 14

Classroom Management

Agenda

  • Simpson's Rule
  • Lab 5 due before 9:30 AM
  • Lab 6 Assigned today
  • Homework 3 due 10/22/2025 before 11:59 PM

Calendar

Week Monday Lecture Wednesday Lecture
8 10/20: Simpson's Rule 10/22: Romberg Integration
9 10/27: Gaussian Quadrature 10/29: Numerical Differentiation (not on Test 2)
10 11/3: Linear Algebra 11/5: Test 2 (Root Finding and Numerical Integration)

Resources

Handouts

Lecture 14

Today's topic is Simpson's Rule.

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

Definition of Simpson's Rule

  • Chapra and Canale(2015)1 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
\[ \begin{aligned} I&\approx \left(b-a\right)\frac{f(x_0)+4f(x_1)+f(x_2)}{6}\\ I&\approx\mbox{ width}\times\mbox{ average height} \end{aligned} \]

Simpson's Rule Error Analysis

  • Chapra and Canale (2015)1 provides the following definition:
\[ \begin{aligned} I&=\frac{h}{3}\left[f(x_0)+4f(x_1)+f(x_2)\right]-\frac{1}{90}f^{(4)}(\xi)h^5\\ I&=\mbox{Simpson's 1/3 approximation}-\mbox{ Truncation error} \end{aligned} \]

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

Pseudo-code: Simpson's 1/3 Rule

  • CodeSansar2 provides the following pseudo-code Simpson's 1/3 pseudo-code

Python Code for Multiple Simpson's 1/3 Rule

import math
def f(z):
    return (math.exp(-0.5*z**2)/((2.0*math.pi)**0.5))

n=10
a=-5.0
b=0.0
h=(b-a)/n
sum=f(a)+f(b)
for i in range(1,n):
   # print(i)
    k=a+i*h
    if i%2==0:
        #print("even number")
        sum=sum+2.0*f(k)
    else:
        #print("odd number")
        sum=sum+4.0*f(k)
sum=sum*h/3.0
print("The area is {} after {} iterations".format(sum,n))

Simpson's 3/8 Rule

  • Simpson's 3/8 rule is Newton-Cotes with \(n=3\)
  • Chapra and Canale (2015)1 illustrate Simpson's rule in the figure shown below Simpson's3/8 Rule

Definition of Simpson's 3/8 Rule

  • Chapra and Canale(2015)1 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
\[ \begin{aligned} I&\approx \left(b-a\right)\frac{f(x_0)+3f(x_1)+3f(x_2)+f(x_3)}{8}\\ I&\approx\mbox{ width}\times\mbox{ average height} \end{aligned} \]

Truncation Error of Simpson's 3/8 Rule

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

Classroom Coding: One Interval of Simpson's 3/8 Rule

รท

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

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

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

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

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