User:XueGong

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

Introduction
My final project is about Cubic Spline Interpolation. This is a interpolating method that gives smoother results than interpolating polynomials in Lagrange form or Newton form and it can avoid Runge's phenomenon. This method is said to have smaller error than the other methods mentioned above also. So I think it would be good to explain more details of this method and illustrate how to use it more on the wikiversity page.

It is difficult to understand using only Wikipedia because the wikipedia page about Spline interpolation does not explain this method very clearly. It has no formal definition and the algorithm given there is actually the Cubic Hermite spline method. This is a good method to find the cubic spline, but I found another method that is more intuitive and can be illustrated using simple examples.

Contribution
I created http://en.wikiversity.org/wiki/Cubic_Spline_Interpolation which contains detailed formal definition for the cubic spline and its three types of boundary conditions. I also explained one method for finding such cubic splines given two different types of boundary conditions. After this, I wrote one example showing the procedure of finding the cubic spline according to the type I boundary condition and one exercise illustrating how to find the spline according to the type II boundary condition.

I chose these two particular examples because they can show clearly how this method works. I chose the data set given to be 4 points so that there will be 3 cubic polynomials to interpolate. It's not too much work but also can illustrate the idea of cubic spline perfectly. Meanwhile, I used the same data points for the second example so that there will be some replicated work that we can skip. By choosing the same data points, I am also able to show the graphs of those two results in one coordinate plane in order to compare them.

Future Work
It would be good if I can show the expression for error term in cubic spline interpolation. We can compare the error with what we know for polynomial interpolation error to show that cubic spline is truly a better method considering the error is smaller. One proof I know to show the error expression needs to use the Cubic Hermite spline expression and its uniqueness.I think it would be too much for this project. So I didn't put them on the wikiversity page. I hope that in the methods section, the Cubic Hermite spline can be introduced in the future, then one can also discuss the error expression later.

Conclusions
In this project I contributed to the cubic spline interpolation page in wikiversity. I think this is a valuable contribution because cubic spline interpolation is a very important interpolating method when requiring certain smoothness and accuracy of the interpolating function. I hope this can help people have a better understanding of the method.

Things I Learned about Wikipedia/Wikiversity

 * 1) Anyone can edit my page anonymously.
 * 2) Editing this page here is simple. I can upload images and sound files very conveniently. Also, I can edit math formulas same as how I do it in Latex.
 * 3) I can see the pages which people recently edited by going to "Recent changes". I selected one page there and found that the person is also writing about five  Wikiversity is a wiki-based learning community. Here you can use online courses and also create courses yourself.things he learned.
 * 4) The English language Wikiversity website opened on August 15, 2006. There are some other languages Wikiversity website. But no Chinese website is available for people in China to share and learn Chinese educational content.
 * 5) I can use online courses on Wikiversity and also create courses by myself.

A Complicated Math Formula
Weibull distribution has the density function


 * $$f(x;\lambda,k)=\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}}, x \geq 0$$.

Wikipedia Talk Page
http://en.wikipedia.org/wiki/Talk:Fixed-point_iteration#When_will_the_fixed-point_iteration_converge.3F

Homework 5
I edited the Wikipedia page on fixed-point iteration. I added the proof of a theorem which I proposed in the talk page and combined it with the property section in the article page. Also, I did some minor correction about the proof and reordered the paragraphs to make it flow better with the materials which are already on the page. My contributions are listed at http://en.wikipedia.org/wiki/Special:Contributions/XueGong.

Example
Differentiate $$y=x^2 \sin x$$. Solution: Using the product rule, we have

$$\frac{dy}{dx}=x^2\frac{d}{dx}(\sin x)+\sin x\frac{d}{dx}(x^2)$$

$$=x^2\cos x+2x\sin x$$.

Quiz
{$$\frac{d}{dx} \sin(-x)=$$ - $$-\cos(x)$$. + $$-\cos(-x)$$. - $$\cos(-x)$$. - $$\cos(x)$$.
 * type=""}

{ Evaluate $$f(x)=\frac{x^3}{\sqrt{x-1}}$$ at $$x=2$$ { 8 }
 * type="{}"}

Homework 7
I edited the wikiversity page on the exercises about LU decomposition at http://en.wikiversity.org/wiki/Topic:Numerical_analysis/LU_decomposition_exercises.

I corrected several following minor mistakes first:


 * 1) using "X" as the cross symbol for multiplication.
 * 2) period signs
 * 1) period signs

There was a mistake in the second equation in the solution of Exercise 2 that the order of matrices $$L\times U$$ was wrong. I corrected it so that we can derive the third equation.

Class Project: Invertible Matrix
I am going to write some examples on wikiversity page about how to find inverse of a matrix. Also, I will write a quiz to test some properties on inverible matrix and non-invertible matrix. At last, I will discuss some numerical methods to find the inverse of a matrix.


 * Do not duplicate material already on Invertible matrix,  Invertible_matrix,  Gaussian_elimination,  Neumann series, etc. You may be able to do a project on invertible matrices, but before starting you should develop the proposal more to make sure it is viable. Mjmohio (talk) 16:50, 7 November 2012 (UTC)