Vv148408

Five things I learned about Wikipedia...

1. I learned how to actually edit a page in Wikipedia
2. I learned that there is a surprising amount of math related information on Wikipedia
3. I learned that German is second to English in number of actual articles written in the language (I would've guessed Spanish)
4. I learned that in addition to Wikipedia and Wikiversity, there are also other project like Wikiquote and Wikispecies
5. I learned that Wikipedia has portals, something I had never used before.

$f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }[a_{n}\cos(nx)+b_{n}\sin(nx)]$

I chose to edit slightly the loss of significance page under the section "Instability of the Quadratic Equation." I thought it could be better explained. The statement that was present before about how subtraction causes loss of significance I felt was kind of misleading, and might lead people to believe that loss of significance occurs when one nearly computes -b - b or calculates b^2 - 4ac rather than when one adds -b to a value very close to bMy edits are in boldface.

Notice that the solution of greater magnitude is accurate to ten digits, but the first nonzero digit of the solution of lesser magnitude is wrong.

This occurs because the value for c is very small, which leads the product 4ac to be small as well. When this product is added to the b squared term and the square root is taken, the result is a number that is very similar to b. Significance is lost when -b is added to the similar number, but not when the similar number is subtracted from -b, which is why the root of greater magnitude (the numerator of which is very close to -b minus b, or very close to -2b) is accurate while the other is not. See below for a better method to calculate the root of lesser magnitude when c is small.

Final Project Report

For my final project in Math 444, I chose to edit the Wikipedia page about the Euler method. I thought that the addition of examples and more details in the error section would be beneficial to readers.

Prior to the final project, my homework assignments required me to do various tasks, including proposing what I believed to be a necessary change to a Wikipedia page of my choice, with the intent of eventually making that change. I chose to propose an alteration to the Loss of Significance page. There is an example on the page of an instance for which the quadratic formula is unstable and computations of the roots using the traditional algorithm result in loss of significance, and I thought that it would be beneficial to try to specify which values of the coefficients $a$, $b$, and $c$ would result in loss of significance when using the traditional algorithm. I ultimately decided not to make my proposed edit, because it seemed like such a minor and very specific edit. I figured that with all of the material on Wikipedia, I could find better things to change if I had to, things that were more important and urgent than adding a short paragraph to the Loss of significance page.

When it was time to choose my topic for the final project, I had a bit of a hard time deciding on a topic, since we had gone over so many over the course of the quarter. I finally decided to make the Euler method the focus of my project because it is a topic I felt comfortable with, and I genuinely felt that I could make useful contributions to this particular page.

My proposed additions dealt with the addition an example and additional details in the section about error. I wanted the example that I would place on the webpage to include step by step explanations to support the Informal geometric description portion of the page. I also was toying with the idea of including a chart with the computations as a part of the example. This is a method of organizing computations that I believe to be quite helpful, since the Euler method is repetitive and a bit tedious, and therefore one could easily make a small error in their computations that would cause the final answer to be incorrect. At the time of the proposal, I also was not sure about the specifics of what I would add to the error section, although I did want to include something about error bounds. My professor ended up suggesting that I include a proof about error accumulation that I had presented in class or a similar one.

The actual changes I made were in line with my proposed changes for the most part. I added an examples section with a simple example, and also the simple chart I wanted to include. My other changes were a slight modification of the proof in the error section, the addition of error computations, and the addition of an error bound section.

The example I chose to add was the differential equation $y'=y$ with the initial point $y(0)=1$. While this is a very simple problem, I chose this particular one because the graphics on the page used this equation, and also because this differential equation can easily be solved, so the error can be computed. I performed the calculations with step size $h=1$, as was the case in one of the graphs on the right side of the page. The goal of the example was to approximate $y(3)$. Along with the calculations themselves, I explained what each part of the Euler method was doing. For instance, instead of simply showing the calculation $h*f(t,y)$ I explained that since $f(t,y)$ represents the slope of the solution curve at the point being input, when one multiplies this by $h$ which is the change in $t$, a change in $y$ is obtained. Below the computations I inserted the chart to encourage those who may be doing these computations by hand to organize their work.

To the error section I made a few changes. First, I slightly changed the proof for determining the order of the method. Instead of leaving the proof as is, I used the fact that $y'=f(t,y)$ to calculate $y''$ and then I substituted both into the Taylor expansion to show that when the equations were subtracted, the $y''$ truly would not cancel, meaning that the local error would be of order $h^{2}$. Also, the orders of higher order methods can be determined in a similar manner, so this simple problem could be used by a reader as a simple example to base the proof for the order of a higher order method off of. In addition, to support the statement about how decreasing the step size $h$ will decrease the error, I calculated the error for $y_{3}$ of the example problem with two different step sizes.

Last, I added the error bound section. I ultimately decided not to include the proof that I presented in class in my edits. I think that this proof, or any other one similar one I would have found online, would have been too lengthy to include in its entirety, but too complicated to summarize while still expecting readers to follow along with the steps. I think such a lengthy proof would be better suited for Wikiversity, rather than Wikipedia. In any case, I found a pdf file showing the proof online, and while I didn't post the entire thing, I included the expression for the error bound. I also included a link to the pdf in the references section, so that anyone who would like to read more about how the error bound was derived can do so. I then used the error bound to further support statements on the page that suggest using alternative higher order methods.

Overall, I am pleased with my edits, though I have gone back and tweaked a few things since my presentation, and I hope that my changes are helpful to readers.