Aitken's Method and Superconvergence
Initial Publication Date: October 10, 2024
Summary
This activity aims to explore Newton's method's efficiency for various cases. We also provide some ideas on improving or accelerating the convergence through numerical experiments.
CollapseLearning Goals
Students understand the order of convergence of a numerical sequence.
Explore when Newton's method fails to achieve the order of convergence 2.
Explore how to fix the slow convergence issues (both geometrically and analytically) and accelerate the convergence by adopting Aitken's method.
Explore when Newton's method fails to achieve the order of convergence 2.
Explore how to fix the slow convergence issues (both geometrically and analytically) and accelerate the convergence by adopting Aitken's method.
Context for Use
This material was used for undergraduate Numerical Analysis class.
It was presented after we introduced Newton's method and the order of convergence.
We assumed a computer lab setting with Matlab is available for each computer.
The majority of the class was either math or engineering majors.
It was presented after we introduced Newton's method and the order of convergence.
We assumed a computer lab setting with Matlab is available for each computer.
The majority of the class was either math or engineering majors.
Description and Teaching Materials
Newton's method and the order of convergence are standard material in almost all Numerical Analysis classes. We just added a few exploring computer lab activities for when it may fail to work as expected and how to improve the situation.
Almost all standard textbooks cover the topic material.
Aitken's Method (Acrobat (PDF) 28kB Oct10 24)
Teaching Notes and Tips
This lab (GroupWork) is designed to work as a group (2 or 3 people per group). As each individual has a different level of coding skill, the group work goes much more smoothly. Also, I value exploring and discovering discussion through cooperative learning.
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Assessment
After exploring and experimenting with Matlab, each group presented what they discovered or had explored from the computation experiments. This computer lab is intended for active cooperative learning.
References and Resources
Many Numerical Analysis textbooks covered Newton's method, order of convergence, and Aitken's method. One may also refer to:
https://en.wikipedia.org/wiki/Newton%27s_method
https://en.wikipedia.org/wiki/Aitken%27s_delta-squared_process