PLEASE USE MATLAB AND DONT COPY SOMEONE ELSES ANSWER!!!Problem $3$

Consider the linear system $A_n=,$ where $A_n$ is an $nn$ matrix with $2\text{'}$s on the main diagonal, $-1\text{'}$s

directly above and below the main diagonal and $0\text{'}$s everywhere else. For example,

$A_5=([2,-1,0,0,0],[-1,2,-1,0,0],[0,-1,2,-1,0],[0,0,-1,2,-1],[0,0,0,-1,2]).$

This is a discretized version of Poisson's equation

$\frac{d^2(x)}{d{x}^2}=(x),$

which appears very often in physical applications. We will discuss discretizations and differential equations,

including the origin of the matrix $A_n,$ later in the class.

$(1)$ Setting up the matrix:

$($a$)$ Construct the matrix $A_{114}$ in MATLAB$/$python$.($You should be able to do this in only a few

lines of code with the help of the diag or np$.$diag function. In particular, you should figure

out what the commands diag$(v),$diag$(v,1)$ and diag$(v,-1)$ do when $v$ is a vector or $1D$

array.$)$ Save a copy of this matrix in a variable named A$9($remember$,$ in python you need to use

A$9=$ A $*copy().$

$($b$)$ Now construct the right hand side vector $.$ This should be a $1141$ vector such that the $j$ th

entry of $$ is

${}_j=2(1-cos(\frac{53}{115}))sin(\frac{53j}{115})$

Save a copy of this vector in a variable named A$10($remember$,$ in python you need to use

$A10=*copy().$

$(2)$ Jacobi method:

$($a$)$ The Jacobi method for this problem can be written as ${}_k=M{}_{k-1}+c,$ where $M$ is a $114114$

matrix that we discussed in lecture. $($Note that ${}_k$ in this equation means the $k^{th}$ guess for the

vector $$ and it is an entire vector. It does not mean the $k$ th entry of $).$ Use the Jacobi method

to solve for $.$ Your initialization should be a $1141$ vector of all ones, and you should use a

tolerance of ${10}^{-5}.$ That is$,$ you should stop when max using the abs $()$ and

max$()$ functions. Save a copy of your final iteration in a $1141$ column vector named A$11.$ Save

the total number of iterations required $($including the initial iteration$)$ in a variable named A$12.$

$($b$)$ The true solution to the system of equations $A=$ is the $1141$ vector $$ whose $j^{th}$ entry is

defined according to the formula

${}_j=sin(\frac{53j}{115}).$

To test the efficacy of the Jacobi method on this problem, find the maximum error in absolute

value between your final iteration and the true solution using the abs$()$ and max$()$ functions.

Save your result in a variable named A$13.$

$(3)$ Gauss$-$Seidel method:

$($a$)$ The Gauss$-$Seidel method for this problem can be written as ${}_k=M{}_{k-1}+c,$ where $M$ is a

$114114$ matrix that we discussed in lecture. Use the Gauss$-$Seidel method to solve for $.$ Your

initial iteration should be a $1141$ vector of all ones, and you should use a tolerance of ${10}^{-5}.$

Save a copy of your final iteration in a $1141$ column vector named A$14.$ Save the total number

of iterations required $($including the initial iteration$)$ in a variable named A$15.$

$($b$)$ To test the efficacy of the Gauss$-$Seidel method on this problem, find the maximum error in

absolute value between your final iteration and the true solution from the previous part. Save

your result in a variable named A$16.$