User:Nm160111

Project
For my final project, I intend to add a page of examples of and exercises related to numerical differentiation. Specifically, I intend to show examples of derivations of methods of several orders in one and two variables, as well as example C++ code for some low-order methods. I haven't yet decided how best to create the exercises.
 * Sounds good. Mjmohio (talk) 18:34, 7 November 2012 (UTC)

Project Report for User:Nm160111
For Introduction to Numerical Analysis, Fall 2012.

Introduction
My final project is about derivations of numerical differentiation methods, particularly finite-differences methods. It is difficult to understand using only Wikipedia because the Wikipedia article only provides examples of methods, but does not provide information on how they are obtained. Additionally, it does not address partial derivatives of multivariate functions. The examples I've provided solve both of those deficiencies simultaneously.

To facilitate learning of this topic, I have provided examples illustrating derivation of the methods, checking the order of the the errors of methods, and applying the multivariate Taylor's theorem to similar multivariate problems, and sample code showing how one of the simpler methods might be implemented.

Contribution
I created Topic:Numerical Analysis/Differentiation/Examples which contains examples of deriving 2 methods and checking the order of two more. Of these 4, one is for a mixed derivative.

I chose these particular examples because they are simple enough that the basic idea may be shown, but not so simple that they fail to show how to solve such problem. In particular, the places where functions are evaluated in the more complex methods are chosen to be optimal for that number of points.

Future Work
I decided that although examples of fourth-order mixed derivatives would be good it was too much for this project, so I just made an outline that others can fill in. More specifically, I listed the steps to take and gave an example of the multivariate Taylor's theorem.

It would be beneficial if someone created a quiz or set of exercises to complement the project.

Conclusions
In this project I showed derivations of finite-difference methods and applied the multivariate Taylor's theorem to find finite-difference approximations of mixed derivatives.

I think this is a valuable contribution because it seems to be the first article on either Wikipedia or Wikiversity to show derivations of finite-difference methods, although some articles may link to derivations. It also shows multivariate methods, which Wikipedia and Wikiversity do not mention.

Wikiversity Improvements
Changed article: The original article had several mathematical mistakes, as well as a pointless (and misleading) chart. I rewrote it to use correct formulas, and added a brief explanation of how Wikipedia's formula for $$R_{k,0}$$ is derived.

Example, Quiz
Given that $$f(1) = 1$$ and $$f(-1) = 0$$, estimate $$f'(0)$$. Solution: $$f'(0) \approx \frac{1}{2}$$

{Which of the following bound the error of the example above, assuming that f is 6 times continuously differentiable on [-1, 1]? - $$\left|\frac{f^{(2)}(\xi)}{6}\right|$$, for some $$\xi \in \left[0, 1\right]$$ + $$\left|\frac{f^{(3)}(\xi)}{6}\right|$$, for some $$\xi \in \left[-1, 1\right]$$ - $$\left|\frac{f^{(3)}(\xi)}{3}\right|$$, for some $$\xi \in \left[-1, 0\right]$$ - 0
 * type="[]"}

{What is the probability of picking the right answer to this question with a random guess? - 25% - 50% + 25% - 0%
 * type=""}

{ $$\sqrt{1} = $$ { 1|-1 _3 }
 * type="{}"}

List of Poor Quality Topics

 * Round-off Error: (Was) Poorly written, makes incorrect statements based on assumed decimal representation. Should probably be re-written by an undergrad.
 * Guard Digit: Needs rewrite.