User:Jl214707

5 things I learned about Wikiversity

 * 1) Wikiversity has learning projects, Whereas Wikibooks has books and Wikipedia has articles. But it is not developed yet like wikipedia.
 * 2) You can browse Wikiversity by many different ways such as by school, educational level, learning projects, resource types etc.
 * 3) While you are developing your knowledge, you can share yours with others. Also you can get some help from others.
 * 4) When you have finished editing, you should write a short edit summary. Or your save editing might not work. You'd better do talk or discuss before editing.
 * 5) you can creat your own sandbox and share your own knowledge.

Formulas in Numerical Analysis
Taylor Series


 * $$ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^ {(n+1)}$$

Newton's method formula


 * $$ x_{n+1}=x_{n}-\frac {f(x_n)}{f^\prime(x_n)}\,$$

Edit for the Talk Pages
http://en.wikipedia.org/wiki/Talk:Newton%27s_method#Convergence_rate_proof

I added the solution of homework question to show how to find the rate of convergence of Newton's method.

Bad Idea
I tried to show the general proof using real question. So it helps to understand the general proof better.

I thought it might be better my article goes after 4.1 Proof of quadratic convergence for Newton's iterative method.

However it turns out this example does not show much more than the general proof.

Thus I decided not to edit actual wikipedia page.

How to Use Quiz Templates
{Which method is the better approximation of the integral? - The composite Trapezoidal rule + Romberg's method - Simpson's rule - Midpoint rule
 * type=""}

{
 * type="{}"}

Solve $$\int_{-1}^{1}{x^2} \ln(x)dx $$ by integral by parts method.

$$ \displaystyle \ u = $$ { ln(x) } $$ \displaystyle \ dv = $$ { x^2 } $$ \displaystyle \ du = $$  { 1/x dx } $$ \displaystyle \ v = $$  { x^3/3 }

how to use Exercise Templates
Using a Lagrange interpolating polynomial method, find the interpolating polynomial which passes through the points $$ (2,4)$$, $$(3,9)$$, $$ (4,16)$$. Solution:

The interpolation polynomial in the Lagrange form is
 * $$L(x) = \sum_{j=0}^{2} y_j \ell_j(x)$$

Thus,we have
 * $$L(x) =4*\frac{(x-3)}{(2-3)} \frac{(x-4)}{(2-4)}+9*\frac{(x-2)}{(3-2)} \frac{(x-4)}{(3-4)}+ 16*\frac{(x-2)}{(4-2)} \frac{(x-3)}{(4-3)}$$
 * $$    =-2(x-3)(x-4)-9(x-2)(x-4)+8(x-2)(x-4)$$
 * $$    =x^2$$

Edit Wikiversity:Numerical Analysis
I worked on Vandermonde example[]

I fixed the matrix into augmented matrix form. I fixed the rightarrow sign into the right double arrow sign.

I added general form of vandermonde matrix before you plug in the data.

I rewrote the number in the power form at the beginning,

so people can understand better how to plug in the given points.

I also mentioned about the number of data(n) and expected degree of polynomial(n-1)

Project Plan
ODE in Vector Form


 * 1) I want to explain in general, how to convert higher order differential equation to a system of the first order differential equation.
 * 2) Then I want to show in general, how to convert the equation to the vector form.
 * 3) Then I want to show our homework#7(2) for the example using Euler method, backward Euler method, Midpoint method, and Adams-bashforth two step method using $$y_1$$ from the midpoint method.


 * Sounds good. Mjmohio (talk) 16:22, 7 November 2012 (UTC)

Project Report for User:jl214707
Topic:Numerical_Analysis/ODE_in_vector_form_Exercises

For Introduction to Numerical Analysis, Fall 2012.

Introduction
My final project is about ODE in Vector Form[]

The topic how to convert the higher order differential equation to the first order differential equation is important.

It will be convenient to solve if you convert the higher order differential equation to the first order differential equation.

The reason is that all of the standard methods for solving ordinary differential equations are intended for first order equations. Wikipedia simply explained that it can be written as a vector form. However,this is not enough for me to apply this knowledge to solve my homework question

which was a second order differential equation. It is difficult for me to come up with the idea how to approach this question.

Also Wikiversity only has one exercise about a first order differential equation but not about higher order differential equation.

Therefore, to facilitate learning of this topic, I added the explanation about how to convert higher order differential equation to a first order diffferential

equation and how to rewrite a system of first order differential equations in a vector form. Then for the exercise, I added second order differential equation.

Contribution
I created Topic:Numerical Analysis/ODE in vector form Exercises.

I linked wikipedia pages of reduction of order and math3600 lecture of converting a general higher order equation.

I believe that addition of the second order differential equation intitial problem will be beneficial for the students as well as the explaination about how to

apply methods such as the Euler's method, the Backward Euler's method,the Midpoint method, and the Two-step adams-Bashforth method.

Future Work
Although I added the example, it would be beneficial if people add more various examples.

Conclusions
In this project I explained how to convert higher order differential equation to a first oreder equation and showed how to solve by various methods.

I think this is a valuable contribution because it can be a good source for the students who wants to know how to approximate the higher order differential

equation. It was a exciting experience to find out the fact that I could be not only a reader but also an editor for Wikipedia and Wikiversity.

I was grateful while improving wikiversity. I hope I can keep contributing for the future.

Reference
http://en.wikipedia.org/wiki/Ordinary_differential_equation

http://www.math.ohiou.edu/courses/math3600/lecture29.pdf

http://www.ohio.edu/people/mohlenka/20131/4600-5600/hw7.pdf