# User:Brmcvet

#### A few things I have learned about Wikipedia and Wikiversity

• Some pages cannot be edited except by proposing a change to an editor with the power to enact a page edit.
• Five pillars guide the operations of Wikipedia[1]:
• Wikipedia is an encyclopedia
• Wikipedia is written from a neutral point of view
• Wikipedia is free content
• Editors should interact in a civil manner
• Wikipedia does not have firm rules
• Wikiversity is for learners and educators of all ages and academic levels.
• Wikipedia content includes more than just encyclopedia-type articles. It also makes use of outlines, timelines, and lists.
• Wikiversity has lesson plans on various topics and could prove to be a real tool to me in the future!

#### A Mathematical Formula

##### Cauchy's Residue Theorem

If Γ is a simple closed positively oriented contour and ${\displaystyle f}$ is analytic inside and on Γ except at the points ${\displaystyle z_{1},z_{2},...,z_{n}}$ inside Γ, then

${\displaystyle \int _{\Gamma \ }f(z)\,dz=2i\pi \ \sum _{j=1}^{n}Res(z_{j})}$ [2][3]

#### A Talk page Edit

Numerical Example? http://en.wikipedia.org/wiki/Talk:Secant_method#Numerical_Example.3F

Based on the comment to my edit of the secant method page, I added 2 more iterations to my example in the hope that the method will be clearer and the pattern will be more apparent. I decided to not actually edit the Wikipedia page at this time, however. I would like to see if I receive any feedback on the change to the example. Receiving none or positive responses, I will make the edit at a later time. In editing this page, I have learned that examples should be made as clear as possible with the addition of potentially more iterations that I initially thought were needed. In the future, I will aim for completeness and clarity.

#### A Simple Example

What is the commutative property of addition?

#### A Simple Quiz

1

 What is one of the seven colors of a rainbow?

2

 Name 1 type of seismic wave.

3

 are a type of seismic wave that cannot travel through water.

4

 Which wave type is faster--S-waves or P-waves?

5

 S-wave velocity is dependent upon which modulus?

#### Editing learning materials at the numerical analysis wikiversity page

I edited the page dealing with Newton's method for approximating roots. I began by editing for flow and corrected a couple of spelling mistakes. The page has a simple derivation of the method and mentions the rate of convergence, but I thought it was lacking in a description of why you should choose or avoid the method. Thus I created the advantages and disadvantages section on the page.

#### Final Project Proposition

For my final project, I plan to create a page for the secant method on Wikiversity's Numerical Analysis page. I hope to include several examples applying the method, a simple derivation, a code which can be used to implement the method, and several exercises to give readers the opportunity to practice using the method.

Sounds fine. Mjmohio (talk) 16:42, 7 November 2012 (UTC)

## Project Report for User:brmcvet

For Introduction to Numerical Analysis, Fall 2012.

### Introduction

My final project for Introduction to Numerical Analysis was to add content devoted to the secant method to Wikiversity's Numerical Analysis pages. The Secant method is an important topic to understand because it is well-known and widely used in approximating the roots of a function. The secant method page on Wikipedia was adequete in that it provided a good description of the method, several easy-to-understand techniques for deriving the method, and graphs that helped to clarify the method. However, I found it difficult to completely understand the secant method soley from this page. I found the page to be lacking in numerical examples and details of the convergence of the secant method, particularly that it converges on the order of the golden ratio.

As an attempt to improve what I found to be lacking on the page on Wikipedia and to add to the learning materials on Wikiversity, I created a secant method page consisting of a numerical example applying the method, several exercises to provide practice in applying the recurrence relation, a short quiz focused on comparing the secant method to the bisection method and Newton's method in terms of computational cost and efficiency, pseudo code showing how to implement the secant method, and a proof detailing the convergence of the secant method on the order of the golden ratio.

### Contribution

I created this page associated with the secant method to supplement the material that can be found on Wikipedia's secant method page. First, I included the algorithm associated with the secant method. Although this is available on Wikipedia, I chose to include it in order to emphasize it and make it as clear as possible (by putting it in a box) and make the examples I provided easier to follow.

Next, I provided a proof of the convergence of the secant method on the order of the golden ratio. I provided this proof because Wikipedia barely mentions convergence and I think it's better to show the whole story than to just mention it. Also, this allows us to better compare the secant method to Newton's method and the bisection method.

I provided a numerical example applying the secant method algorithm to find a root of ${\displaystyle f(x)=\sin x+xe^{x}}$ located between -3 and -4 accurate to 4 decimal places. In this example, I showed the evaluation of the given function at my initial points and then several iterations of the recurrence relation before organizing subsequent iterations in a table. I chose to do this example because I find it helpful to see an algorithm being used before using it myself. I hope it will help others similarly.

I next provided pseudo code which can be used to aid the reader in writing a program which will carry out the secant method. The code I provided hints at using a while loop with the recurrence relation inside which repeats until a certain tolerance is met. This code can be easily modified to run for a set number of iterations instead. I provided this code because I often find it difficult to figure out where to start when attempting to write a program for something. I think the pseudo code provided will give a good start to writing the full program.

I wanted the reader to have the opportunity to practice using the secant method algorithm, so I also provided two exercises. The first exercise asks the reader to find an approximation to ${\displaystyle {\sqrt {5}}}$ correct to four decimal places using the secant method on ${\displaystyle f(x)=x^{2}-5}$. I chose this example because it is fairly simple to do, yet gives the reader legitimate practice in applying the algorithm until a given tolerance is met. My solution shows two iterations of the method and then the results of the next two iterations organized in a table. The second exercise asks the reader to find a root of ${\displaystyle f(x)=x+e^{x}}$ by performing five iterations of the secant method beginning with x0 = -1 and x1 = 0. I chose this example because I wanted the reader to have practice applying the method a set number of iterations. In my solution, I provide details for each iteration. Similar exercises can be found at this site, which I used as a model for my exercises and example.

Finally, I provided a short quiz which asks the reader to compare the computational costs and efficiencies of the secant method to the bisection method and Newton's method for approximating roots of a given function. I felt this was an important topic to include because we must be able to decide which method is most appropriate to use when approximating roots, and computational cost and efficiency can be an important factor in that decision.

### Future Work

I think it may be beneficial to add a second proof detailing the convergence of the secant method. I think some details in the proof can be confusing, but if there was another proof to compare to, the details could be a bit clearer. It may also be beneficial to provide graphs for each of the examples and exercises showing that the secant lines do indeed converge. Pictures can be powerful learning tools, so could aid in the understanding of the method.

### Conclusions

In this project, I attempted to add to the materials already present for the secant method on Wikipedia. I provided examples, exercises, pseudo code, and a proof of the convergence of the secant method. I think these were valuable contributions because the reader now has concrete examples of applying the algorithm and details of the convergence of the secant method as opposed to the briefest mention of convergence that Wikipedia provides.