User:Whlitt

What am I doing?
I will be working in Numerical Analysis.

I will also start contributing to Economics.

About Wikipedia & Wikiversity
The Wikiverse, as it were, is an extensive collection of information. I reckoned this much was clear, but it is actually larger than expected.

Though I have used, and still use Wikipedia often, my first insight is that there exists Wikimedia Foundation: inclusive of not just Wikipedia, but other sites like Wikiversity, Wikibooks, Wikiquote, and more.

Another important discovery was Wikipedia's Five Pillars. These are:
 * 1) Wikipedia is an encyclopedia.
 * 2) Wikipedia is written from a neutral point of view.
 * 3) Wikipedia is free content that anyone can edit, use, modify, and distribute.
 * 4) Editors should interact with each other in a respectful and civil manner.
 * 5) Wikipedia does not have firm rules.

Wikipedia also encourages users to be bold. This can be carried over into related wikis, where being bold is about collaboration, fixing problems, and development.

Wikipedia has incredible amounts of information. However, there are many things that Wikipedia does not have; and many things that Wikipedia is not. Wikipedia is not a blog, a dictionary, or a soap box. One interesting note, is that Wikipedia is not a textbook.

Lucky for those in search of a textbook-like, Wikipedia-type tool, Wikiversity is available! Wikiversity is a community for those interested in learning, from the young to the old, and covering multiple topic areas like Economics, German, Differential Equations,Linear Algebra and Topic:Numerical analysis.

Some particularly helpful pages for working within Wikipedia and Wikiversity include:


 * a "Cheatsheet" for some basic Wikipedia syntax and usage tools. (If you follow the link, this refers to the other Cheetsheet to which Wikipedia says you may be referring just below the page title.)


 * a guide on creating sections and table of contents within pages,


 * a guide to referencing within pages, and among many other pages,


 * a page on using mathematical formulas. See below for a formula putting this guide to use.

Euler's Identity
For my first formula, I will pick one that is both simple and complex, at the same time. This is Euler's identity. Euler's identity is short; there are really only a few numbers in this formula. However, there is much more going on with this formula. Wikipedia's page on the identity includes a section that discusses this formula's beauty. Without further ado:


 * $$ e^{\pi i} + 1 = 0 $$

Bisection Method page
I suggested the use of an example for the Wikipedia page on Bisection method.
 * Update: After reviewing comments on the talk page then ensuring the math formulas' flow were consistent and built into the sentence structure, I added this example to the main Bisection Method page on Wikipedia.

Homework 7: Improve ODE Exercises Page

 * I created subheadings for the exercises to improve the flow and organization of the page. This also makes it easier to edit any particular problem.  Now all of the wikipedia "code" does not show up when trying to edit, making it easier to navigate to and edit exercises, as needed - which brings me to my next edit.
 * I improved the flow of the forth exercise (RK4), correcting some of the  formatting, and adding narrative to explain what is happening during the course of the problem, as well as including more sentence structure within the problem.
 * Similar edits could be made the previous three examples.
 * Another helpful addition may be as follows. The last three exercises use the same differential equation (DE).  It would be great if
 * 1) the first exercise used the same DE,
 * 2) the exact solution for the DE was solved as a separate exercise, and
 * 3) errors for each method were found and displayed.
 * Perhaps this could be a project for someone. My proposed project, however, is discussed in the next section.

Final Project: Iterative Refinement
I would like to discuss iterative refinement: what it is, how it can improve ill-conditioned matrices, how it works, and some Matlab code that can be used to implement the technique.


 * This topic gives a nice connection between roudoff error, condition number, and Gaussian elimination. I presume you found a good source, such as a text, to get more details than is on the Wikipedia page. Remember to avoid duplicating the Wikipedia page iterative refinement and to incorporate learning materials such as examples, exercises, or a quiz. Mjmohio (talk) 14:24, 9 November 2012 (UTC)

Link to the new Wikiversity page here.

Project Report for User:Whlitt
For Introduction to Numerical Analysis, Fall 2012.

Introduction
My final project is about iterative refinement and its connection with condition number. The Wikiversity page I created gives the reader a more in-depth look at iterative refinement and provides an opportunity for them to see it used in an example. It also explains how the iterative refinement process can help approximate the condition number of matrix. While a high condition number may often be remedied by changing the design of a matrix, having another tool at our disposal could serve useful when certain situations present themselves. Wikipedia, and even general online searches, had little to say about iterative refinement. It is also remarkably hard to find an example online, so this addition on Wikiversity should prove beneficial, especially to those readers that having the opportunity to see and do actual problems increases their understanding.

To facilitate the learning of this topic I gave a brief overview of what iterative refinement is and provided what I think is a clearer description of the process than what Wikipedia presents. Before I go on to examples, I provide a little more theory surrounding the approximation of the condition number and show that it will use vectors found during the iterative refinement process. I gave examples to demonstrate the processes and the connection between iterative refinement and approximating the condition number, and finally I gave a quiz with questions relevant to the previous examples, as well as questions that some may notice are similar to homework questions from our class. By doing this, readers will hopefully be able to see connections in their work and be encouraged to explore new connections.

Contribution
I created Iterative Refinement which contains theory surrounding the topic, examples, and a quiz.

After briefly explaining some theory behind the topics, I provide an example of both iterative refinement and approximating the condition number to demonstrate how they are completed. In the iterative refinement exercise, I go through the first iteration step-by-step to ensure a thorough understanding. In the following iterations, I simply display the results, leaving the calculations up to zealous readers. The example I gave is a good introductory example because we are using two-digit rounding, simplifying the calculations so that the readers can focus on the process.

For condition number, I use the same linear system that I used for iterative refinement so that the reader can see and understand the connection between the two topics. Again, this will simplify understanding of the actual process instead of getting lost in new systems and calculations. Providing this example offers readers an easy way to approximate the condition number if the norm-calculation isn't necessary.

Following the examples, I added a quiz which draws from examples provided above and questions somewhat similar to homework problems. I think the questions I asked all provide value, and upon completion will deepen the reader's understanding of what iterative refinement is, when and how to use it, and how to use the process to calculate an approximation for the condition number. Question 1 is theory-based and ensure that the reader knows when iterative refinement is appropriate to use; Question 2 makes the reader apply Question 1 and determine which of the matrices have high condition numbers - and therefore require iterative refinement; Question 3 shows the reader that eventually, iterative refinement will converge to an approximate solution; and Question 4 makes the reader do a more complex iterative refinement calculation, and determine the approximate and norm-based condition numbers, this question drawing from all parts of what they have learned.

Future Work
Iterative refinement is covered little on Wikipedia and online, and could benefit from several more additions to either the Wikipedia or Wikiversity pages. Showing a more complex example could further demonstrate iterative refinement's usefulness. Having MatLab or even just pseudo-code for the process would also be a nice addition. Given this, more applicable examples could be provided in which the computer program would be necessary to find the solution.

Finally, more theory around the iterative refinement process and approximating the condition number may be useful, including more definitions, descriptions, and perhaps a proof of important theory. Though these ideas fell beyond the scope of my project, future Wikipedia and Wikiversity users, and students of Numerical Analysis may benefit from these proposed additions.

Conclusions
My additions to Wikiversity covering iterative refinement and approximating the condition number have added helpful descriptions, and especially examples that are hard to find elsewhere. Iterative refinement is a powerful process that can greatly improve, or even lead to exact solutions to a linear system that is ill-conditioned. The examples I provided are simple enough to develop an understanding of the process, and readers can build upon this foundation during the quiz and more-complex examples on their own. While there is much that can still be done, I hope these additions prove valuable to future readers and students and serve a platform for future work on the topic.

Quiz
{Numerical Analysis is primarily concerned with (check all that apply) - Elementary arithmatic + Algorithms + Accuracy - What numbers look like - What numbers are made up of + Failure + Efficiency - Logic
 * type="[]"}

{In root-finding, Bisection method is generally faster than Newton's method. - TRUE. + FALSE.
 * type=""}

{ The solutions to the system of equations $$ x + 2y = 1 $$ and $$ 4x + 6y = 0 $$, given by matrix $$\left[\begin{array}{c c c}1 & 2 & 1\\4 & 6 & 0 \end{array} \right]$$
 * type="{}"}

are x = { -3 }. and y = { 2 }.