# User:Nraymoss

Simple Exercise: What is 10^2?

#### Gaussian Quadrature Example

I realized that there was insufficient information example on Gaussian Quadrature after the derived and solve sample thus i took the pain to edit this wikiversity page by adding a solved example to the information already on there and below is what i factored in.

Find the constants Co, C1, and X so that the quadrature formula

$\int _{0}^{1}f(x)\,dx=Cof(0)+C1f(x_{1}).$ This has the highest possible degree of precision.

Solution.

Since there are three unknowns, Co, C1 and X1, we will expect the formula to be exact for

$f(x)=1,x,and\,\ x^{2}$ Thus

$f(x)=1,\ \int _{0}^{1}f(x)\,dx=1=C_{0}+C_{1}\ \qquad \ Equation1,\qquad f(x)=x,\ \int _{0}^{1}f(x)\,dx={\frac {1}{2}}=C_{1}x_{1}\ \qquad \ Equation2.\qquad f(x)=x^{2},\ \int _{0}^{1}f(x)\,dx={\frac {1}{3}}=C_{1}x_{1}^{2}.$ Equation 2 and 3 will yield.

${\frac {c_{1}x_{1}}{c_{1}x_{1}^{2}}}=x_{1}={\frac {2}{3}}.\qquad c_{1}={\frac {3}{4}}.\qquad c_{0}={\frac {1}{4}}.$ Hence

$\int _{0}^{1}f(x)\,dx={\frac {1}{4}}f(0)+{\frac {3}{4}}f({\frac {2}{3}})$ Now,

$f(x)=x^{3}.\qquad \int _{0}^{1}x^{3}\,dx={\frac {1}{4}}$ And

${\frac {1}{4}}(0)^{3}+{\frac {3}{4}}({\frac {2}{3}})^{3}={\frac {2}{9}}.$ Thus the degree of the precision is 2

### WHAT I PLAN TO DO.

I had wanted to draw inferences from several definitions of the subject from some Mathematical schools of thought and buttress this with why the need for this Method and the detailed step by step approach in the computation of this method. In furtherance, i would state and prove the Fixed point theorem and give the fixed point iteration algorithm, and would be adding up some more examples and quizzes to enlightened my readers.

There is already material at Fixed Point Iteration. (For some reason the link from Topic: Numerical Analysis was missing.) Between that and w: Fixed-point iteration I think this subject is covered and you need a new one. Mjmohio (talk) 14:06, 16 November 2012 (UTC)

## Simple Quiz

1

f(x)=mx+c is a

 constant number slope variable

2

 The determinant of $\left[{\begin{array}{c c}8&1\\4&1\end{array}}\right]$ is .

## Final Project Report

### Introduction

My final project was based on the topic matrix Norm. This topic was strategically chosen to highlight some equally important proofs that could not be found on Wikipedia . Matrix norm is very important topic because even though it is not well known as compared to others like the Secant Method and Newton's method yet they provide vector spaces and their linear operators with measures of size, length and distance. The matrix norm on Wikipedia was adequate enough based on the fact that they provided step by step approach to their methods and gave examples to most of their definitions and properties, however there were very few of them that i could not adequately understand because there wasn't much information, examples and proofs for better understanding and comprehension for people like me. It is against this background that the proposal of this project was accepted, that is to redefine the Induced norm same as on Wikipedia and provide the proofs of the Sub-multiplicative Matrix Norm, the maximum absolute row sum of a matrix and show some examples of Norm Equivalence for better understanding.

### Contribution

I was able to create this page which contains some detailed definitions , theorems and examples to supplement the material we already have on Wikipedia. Even though some of these theorems are already available on Wikipedia, certain important proofs of these that will help various readers to understand and comprehend them better were not present.

To commerce with, i decided to revisit the various definitions of Matrix Norms, Induced Matrix Norms and Equivalence of Norms which are already defined on Wikipedia so i can interconnect them with their appropriate theorems and examples. I followed the definition of an Induced Norm by giving a proof that shows that all Induced norms are Sub-multiplicative and buttress the definition by giving a step by step explanation of an example that ends up to explain how the definition of Induced Norms can be used in solving some induced norm problems.

In addition, i picked the maximum absolute row sum of the Matrix $\{\left\|A\right\|_{\infty }=\max \limits _{1\leq i\leq m}\sum _{j=1}^{n}|a_{ij}|\}$ which falls under then definition of induced norms and wrote a prove of that theorem together with an example to enable the readers to understand it better.

Last but not least, i thought it wise to include Norm and its Equivalence and then gave some examples to demonstrate how they are arrived at.

### Future Work

It would have been better if not the best to have shown some proves on the Norm Equivalence and how they behave the way they do but that would have been too much for this project so thus i think it would be beneficially for future interested students to build on following outline below;

i. Define Norm Equivalence

ii. Prove the definition of Norm Equivalence

iii. Demonstrate some more examples of Norm Equivalence

iv. Give some quizzes to facilitate better understanding.

### Conclusions

In a nutshell, this project is attempting to provide additional material to facilitate the understanding of the Induced Norm definition, the sub- multiplicative of the Matrix, Equivalence of Norms and their examples and i think this is a valuable contribution because it provide proofs and examples to theorems that could not be found on Wikipedia.