Sage codes for blog.ttheng.com
Dynamical Systems: Course Works
These are some interactive Sage code blocks for one of the articles in my blog site, titled Dynamical Systems: Course Works, which serve as a demonstration of code implementations that are part of the workings of some exercises of a course about dynamical systems.
Due to the technical limitations of the Docusaurus framework that makes embedding interactive Sage code blocks there fairly involved, I have decided to put the code blocks here instead, as this site is built using raw HTML and CSS, which is more straightforward to work with.
Chapter 1, Question 1
Note: The following code blocks are linked, so previous code blocks need to be executed to ensure that the following ones run properly.
Chapter 1, Question 2
Note: Please run the first code block (that defines compose_iter_2) beforehand so that this code block can execute properly.
The context of this code block is to implement Heron's method of computing square roots, by starting from a pair of positive integers \((1,z)\), then iteratively replacing the first and second coordinate by their arithmetic and harmonic means respectively. After a sufficient number of iterations, one can then see that the pair of numbers tends to \((\sqrt{z},\sqrt{z})\), yielding better approximations of the square root.
Chapter 1, Question 3
The following code snippet demonstrates a specific case of the periodicity of the last digits (before the decimal point) of \(x_n\), where \(x_n=n^2p/q\) and \((p,q)=(8,3)\) (which are coprime).Here's another 'email to me' button!
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