# User:Poojap

## Quiz template

1

${\displaystyle \sum _{i=0}^{3}(5+{\sqrt {4^{i}}})}$

 45 35 30 25

2

 What is the full form of ODE?

3

Solution for ODE ${\displaystyle y^{\prime }-2ty=t}$ is ${\displaystyle Ce^{t^{2}}-1/2}$

 TRUE. FALSE.

What is 2+2?

## Project Report for User:PoojaP

For Introduction to Numerical Analysis, Fall 2010.

### Introduction

My final project is about Romberg integration method. Romberg integration method is one of the useful methods used for approximating integrals of the form
${\displaystyle \int _{a}^{b}f(x)\,dx}$
where f(x) is called the integrand
a = lower limit of integration
b = upper limit of integration
This method uses two phases of calculation. In the first phase composite trapezoidal rule is used for a series a approximations. While in the second phase the approximations are improved using Richardson Extrapolation.

### Initial Experience

Embarking on the exciting final project,helped me learn essential things about Numerical Analysis.Before I could begin with this project I did some smaller projects as part of homework assignments. This was a good learning curve for me.

Initially I had started with my proposal for changes on Fixed point iteration method. Did lot of reading and tried to make some changes into the current topic present within the Wikipedia page. Unfortunately this wasn't accepted and helped me learn that it needs even bigger effort to actually make changes into pages that are under public surveillance.

I found the existing page on fixed point iteration very brief so I added an excerpt on Methodological error.This basically described how to check if a function is converging or not.The excerpt was not satisfying because the WP tone used to write the extract was not fitting under the rules. The content was also described to be a bit confusing.

"Methodological Error"
The link to the source of this claim has been collected from [1]
The error of the fixed point iteration method can be shown as:
${\displaystyle E_{n+1}=|g'(r)|E_{n}}$
This equation shows us that fixed point iteration is a first order scheme provided ${\displaystyle g'(r)!=0}$ This equation shows us if the scheme converges or not.
The error reduces if ${\displaystyle |g'(r)|<1}$, the scheme converges,
The error increases if ${\displaystyle |g'(r)|>1}$, the scheme diverges.

When this proposal was posted to the discussion page someone there deleted the content and added the reason as described earlier.

I realized that I need more time for this topic to be prepared and added to the current Wikipedia page.The topic is also missing out on the code snippet for evaluating fixed point. This proposal has been updated in the discussion page.

"Sample Code"

function x = fixedpoint(f,x0,tol,n)
% f is the function to be iterated
% x0 is the initial estimate
% tol is the tolerance
% n is the maximum number of iterations
x(1) = x0;
for i=2:n
x(i) = feval(f,x(i-1));
error = abs(x(i) - x(i-1));
relative_error = error/abs(x(i));
if(error<tol)|(relative_error<tol),break;
end
if i == n
disp('maximum number of iterations exceeded')
end
end


After few hours this has also been deleted from the topic.

### Main Project

Main Project includes editing the Numerical Analysis content on Wikiversity. I have chosen Romberg Integration method as the topic for editing the webpage content.There are few pages added for Romberg's Integration method. First the method is described and the formulas are listed out. There is also an inclusion of sample code. Also there is an example and an exercise for students to practice and understand the method. Quiz is also included related to this method.

#### Conclusion

I have tried to cover as much content as possible for Romberg's method.There is Wikipedia reference added along with the method explanation.The formulas are consolidated so that it would be easier for students to solve problems referring to them.There are few problems solved in detail with steps clearly mentioned. This would help students get a clear idea about the method.Also there is sample matlab code included. Quiz on Romberg's method is also part of the topic which could be taken once the students are familiar with the method.

#### Resouces

1. Numerical Analysis by Burden
2. Numerical methods using Matlab by Mathews