pleaseeeee solve this question as soon as possibleeee pleaseeee Chapter $7$: Solving Linear System of Equations

Due Date: $\frac{27}{05}\frac{?}{2024}$

Max. Mark: $20$

Objective:

The objective of this assignment is for the learners to demonstrate that they can

Apply different numerical methods to solve linear systems.

Carry out work$/$assignments responsibly.

Interpret the results of the numerical methods

CILOs:

This assignment measures CILOs $2,$ and $6.$

CILO $2$: Implement numerical methods to approximate the solution of nonlinear equations and systems of equations. $($This assignment focuses on solving linear systems by iterative methods$)$ CILO $6$: Write a computer program to implement the numerical algorithms to solve mathematical problems.

Instructions:

Name the assignment file as: ID$\_$First$\_$Name$\_$Assignment$\_4($ex$.20241234\_$Thuraya$\_$Assignment$\_4).$

Number the pages.

The cover page must include:

a$)$ Your name, Student ID$,$ and Assignment number.

b$)$ The declaration: Hereby, I declare that I did the assignment by myself $q,$

Solve the problem using Matlab or Octave.

Explain your programming steps using comments.

Print the program and the result. Also, take a screenshot of the programs.

Write the references and resources you used, such as books or websites.

Note that the Safe Assign is enabled to check for originality.

This is a summative assessment, which means it will be included in your final grade as part of the $12\%$ assignment grade.

Upload your assignment on Blackboard. You have $3$ attempts to submit the assignment.

TJ

Sem II$-2023/2024$

Page $1$

Question: $[20$ Marks$]$

Consider the following linear system

$4x_1+x_2-x_3+x_4=-2$

$x_1+4x_2-x_3-x_4=-1$

$-x_1-x_2+5x_3+x_4=0$

$x_1-x_2+x_3+3x_4=1$

a$)$ Solve the system using Matlab$/$Octave rref, backslash, and the inverse commands. Call the solution $x.$

$[4]$

b$)$ Use Matlab$/$Octave code to solve the system using the Jacobi method, starting with the initial value $x^{(0)}=(0,0,0,0).$ Call the solution $xJ.$ Print $5$ iterations.

c$)$ Use Matlab$/$Octave code to solve the system using the Gauss$-$Seidel method, starting with the initial value $($:$x^{(0)}=(0,0,0,0)\}.$ Call the solution $xGS.$ Print $5$ iterations.

d$)$ Use Matlab$/$Octave code to compute the $||x-xJ|{|}_2$ and $||x-xGS|{|}_2$ for the errors in the approximation.

e$)$ Are the two numerical methods converging to the solution? Which method is better to use? Justify your answer.

$3$Pl