# User:Pa648412

1. what is wikiversity?

Wikiversity is different from wikipedia.Sometimes people may confused wikiversity with wikipedia. We can have account both in wikipedia and wikiversity. Wikiversity is a content free website which anyone can participate in, and multilingual encyclopedia collaborative program.

2. The Goal of wikiversity:

The goal of wikiversity is to build a complete, accurate and neutral encyclopedia.

3. How to create a wikiversity account:

To create an account, you can explore Wikiversity at the following website: http://en.wikiversity.org/. Then click on “create account” in the upper right and make one. If you already have an account, you can login in by click on “login in”.

4. How to use wikiversity

You can search whatever you want by input the key words you are interested in. And you can also find the latest new on the home page. Also you can click the link to get more details.

5. How to get Help

If you have want some help such as you want to know how to edit the formula. You can just input the keywords:formula in the "search" box, then click the icon or just click "enter" on your keyboard and you can always find your answer.

## Some complicated math formula

${\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\cdot \sum _{m=1}^{n}C_{j}^{(n)}f(x_{j})}$

## Edit the Talk page

My proposal was a bad idea.

There are some reasons that my proposal is a bad idea.First, I can't find the precise resource of the theorem,I just remeber it and find it on my notes. So,in this way, it is not useful and I don't give a proof of it. Second, the therem should go on the fixed point theorem rather than the newton's method theorem as Dr. Mohlenka pointed out.

I learned a lot from this process.For instance,I have learned the knowlege about the order of newton's method and how to edit the talk page on wikipedia, with which I am not familiar.I also find it is important to give the precise resouce of the proposal I give which is rather necessary and important for others to have a refernce.Finally, next time when I do some wikipedia edit work or some other kind of work, I will give the precise resouce where I find it and what have helped me in the process and put the edit on the proper page.

## Exercise/Example

Here is an Example with a solution if the reader wants to check the answers.

What is 2*7?

## Quiz

1

Which of the following statement is right?

 every function has a inverse function a function must be an odd function or even function a one to one function has a inverse function None of those

2

Type text here or a no-break space code

 The solution of ${\displaystyle \ x-3=0}$ is .

3

 Here, we have Ax = y ${\displaystyle {\begin{bmatrix}7&4\\2&1\end{bmatrix}}}$ X ${\displaystyle {\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}$ = ${\displaystyle {\begin{bmatrix}1\\2\end{bmatrix}}}$ Use backward substitution we have: ${\displaystyle \displaystyle \ x_{1}=}$ ${\displaystyle \displaystyle \ x_{2}=}$

## Edit Wikiversity page by using the Good Problem Logic and Flow skills

Here is the link of which page I edit.Lagrange_exercise. On this page, I rewrite the solution of the exercise to make it easy to understand. I use logic connectives such as the words since, therefore, thus and so on to make the logic more resonable. Also, I give more explaination when using a fomula rather than just give the calculation without any explanations. In this way, the solution looks much more clearly.

This is the original solution given on the page:

${\displaystyle p(x)=2{\frac {(x-3)}{(1-3)}}{\frac {(x-5)}{(1-5)}}{\frac {(x-7)}{(1-7)}}+4{\frac {(x-1)}{(3-1)}}{\frac {(x-5)}{(3-5)}}{\frac {(x-7)}{(3-7)}}+6{\frac {(x-1)}{(5-1)}}{\frac {(x-3)}{(5-3)}}{\frac {(x-7)}{(5-7)}}+8{\frac {(x-1)}{(7-1)}}{\frac {(x-3)}{(7-3)}}{\frac {(x-5)}{(7-5)}}}$

${\displaystyle p(x)={\frac {5}{12}}x^{3}-{\frac {65}{12}}x^{2}+{\frac {247}{12}}x-{\frac {163}{12}}.}$

This is the solution after I edit it:

By using the Lagrange method, we need to find the lagrange basis polynominals first.Since we know

{\displaystyle {\begin{aligned}x_{0}&=1&&&&&f(x_{0})&=2\\x_{1}&=3&&&&&f(x_{1})&=4\\x_{2}&=5&&&&&f(x_{2})&=6\\x_{3}&=7&&&&&f(x_{3})&=8.\end{aligned}}}

So we can get the basis polynominals as following:

${\displaystyle \ell _{0}(x)={x-x_{1} \over x_{0}-x_{1}}\cdot {x-x_{2} \over x_{0}-x_{2}}\cdot {x-x_{3} \over x_{0}-x_{3}}=-{1 \over 48}(x-3)(x-5)(x-7)}$
${\displaystyle \ell _{1}(x)={x-x_{0} \over x_{1}-x_{0}}\cdot {x-x_{2} \over x_{1}-x_{2}}\cdot {x-x_{3} \over x_{1}-x_{3}}={1 \over 16}(x-1)(x-5)(x-7)}$
${\displaystyle \ell _{2}(x)={x-x_{0} \over x_{2}-x_{0}}\cdot {x-x_{1} \over x_{2}-x_{1}}\cdot {x-x_{3} \over x_{2}-x_{3}}=-{1 \over 16}(x-1)(x-3)(x-7)}$
${\displaystyle \ell _{3}(x)={x-x_{1} \over x_{0}-x_{1}}\cdot {x-x_{2} \over x_{0}-x_{2}}\cdot {x-x_{3} \over x_{0}-x_{3}}={1 \over 48}(x-1)(x-3)(x-5).}$

Thus the interpolating polynomial then is:

{\displaystyle {\begin{aligned}L(x)&=f(x_{0})\ell _{0}(x)+f(x_{1})\ell _{1}(x)+f(x_{2})\ell _{2}(x)+f(x_{3})\ell _{3}(x)\\[10pt]&=2\cdot {1 \over 16}(x-1)(x-5)(x-7)+4\cdot {-1 \over 16}(x-1)(x-3)(x-7)+6\cdot {-1 \over 16}(x-1)(x-3)(x-7)+8\cdot {1 \over 48}(x-1)(x-3)(x-5)\\[10pt]&={\frac {5}{12}}x^{3}-{\frac {65}{12}}x^{2}+{\frac {247}{12}}x-{\frac {163}{12}}.\end{aligned}}}

Therefore, we get the Lagrange form interpolating polynomial:

${\displaystyle L(x)={\frac {5}{12}}x^{3}-{\frac {65}{12}}x^{2}+{\frac {247}{12}}x-{\frac {163}{12}}.}$

## Final Project

I plan to add some examples, exercises and quiz about the Gaussian quadrature to make it more easy to understand. In the example I will give the solution as detailed as possible from the method to the answer and write it logically. Also, I will make a quiz comprehensively to help the reader test themselves.

There is already material on Topic:Numerical Analysis/Gaussian Quadrature. You will need to choose something else. Mjmohio (talk) 16:54, 7 November 2012 (UTC)

## Project Report for User:Pa648412

For Introduction to Numerical Analysis, Fall 2012.

### Introduction

My final project is about Stability of multistep method. The topic region of absolute stability is important because it allows one to find the appropriate step size h under which the multistep method is absolute stable. It is difficult to understand using only Wikipedia because there is no clear definition of the region of absolutely. And there is no methods about how to find the region of absolute stability. And no examples and exercises are provided to help understand this topic.

To facilitate learning of this topic I give introduction of the concepts of stable method , absolute stability and region of absolute stability. I also give examples illustrating if a given multistep is stable or not. And examples state how to find the region of absolute stability of a given multistep method. exercises to develop skill at judging if a stable method and finding the region of absolute stability. quiz about basic concepts on stable multistep method, root condition and region of stability to reinforce the basic concepts.

### Contribution

I created stability of multistep method stability of Multistep methods which contains basic concepts about what root condition is , what a stable multistep method is, what characteristic polynominal of the given multistep method is and how to find the region of absolute stability of the multistep method by using the characteristic polynominal.

Also, some examples have been given to illustrate the topic.I chose these particular examples because they are easy to understand and they are typical examples through whih the readers can quickly master the topic.

### Future Work

I decided that although add more content on absolute stability and how to find the absolute stability region of multi-step method would be good it was too much for this project, so I just made an outline that others can fill in.

In the future, it would be beneficial if someone can add more examples and give more specific details of how to find the region of stability of the muitisetp method. And it would be good to compare RK4 method and the multistep method on how to find the region of absolute stability. Also, it will be good to add the graphs of the region of stability.Then the readers will easily understand the material and have a good sense of what the region is.

### Conclusions

In this project I fist talked about what the stable multistep method is through the concept of root condition. Second, I introduce the concept of absolute stability and give the method to find the region of absolute stability and give an example to illustrate this method.At last, I give some exercises and quizzes about the concepts of stable method and finding the region of stability of a given multistep method.

I think this is a valuable contribution because this can help reader easily understand what the region of absolute stability is and how to find the region quickly.