# Name: Sujit Kadam

## Project Report: Fixed point iteration

### Introduction

My final project is about Fixed point iteration. I got interested to Fixed point iteration when I learnt about Babylonian method. It’s a very old process but still used today and is powerful at the same time. I was amazed to know its application in Newton Raphson’s method and Secant method. I wanted to learn more about this iteration process. Hence, I opted for this process. This project is intended to get some more knowledge about Fixed point iteration and its application. The project explains about the different concepts and applications of Fixed point iteration using examples, exercises and quizzes.

### Initial Experience

After reading wiki page of Fixed Point Iteration, I felt something’s missing. I started to think and search various explanations on Fixed point iteration. I read few books online. I found that many things can be added to improve Fixed point iteration on Wikipedia. After some initial research on topic I decided to write my project proposal that described about the things I will be adding. In fact, initially I was a little bit confused about Instructor's expectations. I thought that I need to add whatever I have learned and some extra information on Fixed point iteration. Accordingly, I made my project proposal.

### I proposed to add points mentioned below

1. Introductory definition and explanation about the iteration: Introduction, definition, formula and explanation about Fixed point iteration.
2. Supporting theorems and derivations: Banach fixed point theorem, Babylonian method and any other important derivation.
3. Rate of convergence: Proof and explain convergence speed of the iteration method.
4. Examples to explain fixed point iteration: Some exercises on fixed point iteration based on different scenarios.
5. Applications and usefulness of the iteration: Explain application of the iteration in Newton’s method and Secant method.
6. Disadvantages or failure of this iteration: Explore and explain any cases of failure of this iteration method.
7. Exercises and quizzes on fixed point iteration: Few exercises based on fixed point iteration.
8. Important links and external study help materials: If I find any important interesting link related to the topic then I will post it in the project.

### Instructor's input

My instructor pointed out that it would be much better if I don’t duplicate material and add something to the topic that is not present in Wikipedia or any online resources. The explanation of Fixed point iteration, derivation and theorems are already explained in Wikipedia pages are pretty good enough to understand. Instructor also pointed out that I should concentrate more on examples, exercises and quizzes. The examples should clear some basic and new concepts. Exercises should help readers to apply the concepts. Quizzes should test readers’ concept on Fixed point iteration. Instructor's comment were really helpful for me to know which points exactly I need to concentrate. It really add some good new information to the content.

### Main Project

#### Motivation

After thinking about the inputs from the instructor, I started some in-depth research on Fixed point iteration material. I checked out some books and online material and found that Wikipedia is missing on some parts in Fixed point iteration section. I thought to add algorithm of Fixed point iteration, a new concept related to derivative of function and example that can be used in application of Fixed point iteration i.e. Newton Raphson’s method, difficult exercises and quizzes. In our Numerical analysis course we studied many methods and iterations to find root. But I think Fixed point iteration is really a powerful yet simple tool because of which it was used new methods. It's accuracy is good, convergence rate is linear to quadratic (in some cases) and chances of failure is less compared to other methods. It offers really a good deal.

#### Proposed changes

1. Introductory definition and explanation about the iteration: Introduction, definition, formula and explanation about Fixed point iteration.
2. Supporting theorems and derivations: Banach fixed point theorem, Babylonian method and any other important derivation.
3. Rate of convergence: Proof and explain convergence speed of the iteration method.
4. Examples to explain fixed point iteration: Some exercises on fixed point iteration based on different scenarios.
5. Applications and usefulness of the iteration: Explain application of the iteration in Newton’s method and Secant method.
6. Disadvantages or failure of this iteration: Explore and explain any cases of failure of this iteration method.
7. Exercises and quizzes on fixed point iteration: Few exercises based on fixed point iteration.
8. Important links and external study help materials: If I find any important interesting link related to the topic then I will post it in the project.

#### Actual changes

1. Added Algorithm for Fixed point iteration that was not included in Wikipedia.
2. Added a simple example that finds the square root of 2 correct till 3 decimal places.
3. Added an example that proves a new concept not mentioned in Wikipedia: If |g'(x)| > 1 then sequence will not converge to desired value. If |g'(x)| < 1 then sequence will converge to desired value. This concept can be used in Newton Raphson method.
4. Mentioned about the applications of Fixed point iteration: Newton Raphson’s method and Secant method.
5. Added exercise that calculates the square root of 0.5 correct till 3 decimal places. The reason to choose this problem was to show that if a square root of a number near to zero is chosen then by using Babylonian method it first diverges and then converges to the desired value. The exercise displays the iteration steps.
6. Added quizzes based on the concepts of Fixed point iteration.

#### Conclusion

Concluding, my project work I found that there are still many concepts to be learned in Fixed point iteration. I have tried to cover some concepts but there are still some concepts yet to be touched and learned. Its concepts are used in Newton Raphson and Secant method. Fixed point iteration is really an effective way of finding square root.