MANE 3351
Lecture 11, September 27
Classroom Management
Agenda
- Update on Raspberry Pi VM
- 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
- No lab today!
Resources
Handouts
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 |
Assignments
- Homework 3 (assigned 9/22/2021, due 9/30/2021)
Looking Ahead
| Lecture | Date | Content |
|---|---|---|
| 12 | September 29 | Numerical Integration, Newton-Cotes |
| 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 |
Lecture 11, September 27
Today's topic is the secant method. The secant method does not utilize derivative information as Newton's method does. The secant method is also similar to the false position method.
Geometric Inspiration
Cheney and Kincaid (2004)2 demonstrate the geometric inspiration for secant method
Secant Method
- The formula is simply
\[
x_{n+1}=x_n - g(x_n)\frac{x_n-x_{n-1}}{g(x_n)-g(x_{n-1})}
\]
Example Problem Used for Newton's Method
- Consider a new function, \(f(x)=e^x+2^{-x}+2\cos(x)-6=0\)
Pseudo-code
Brin (2020)1provides the following pseudo-code
Convergence
- Brin (2020)1 study the performance of Secant Method
- The secant method converges with order \(\frac{1+\sqrt{5}}{2}=1.62\)
- Not quite as fast as Newton's Method (quadratic)
Similarity to False Position
- Secant method: \(x_{n+1}=x_n - g(x_n)\frac{x_n-x_{n-1}}{g(x_n)-g(x_{n-1})}\)
- False Position method: \(x_r=x_u-\frac{f(x_u)(x_l-x_u)}{f(x_l)-f(x_u)}\)
- Secant method is not guaranteed to bracket result when starting
Speed of Convergence
Chapra and Canale (2015)3 compared the performance of various root finding methods
