import numpy as np$$

from copy import copy

copy your vector norm function here

$$

def VNorm$($v$,$p$)$:

$$ nrm $=0$

$$ if p $==$ np$.$inf:

$$ nrm $=$ np$.$max$($np$.$abs$($v$))$ # Infinity norm $($max absolute value in v$)$

$$ else:

$$ nrm $=$ np$.$sum$($np$.$abs$($v$)**$p$)**(1/$p$)$ # p$-$norm calculation

$$

$$ return$($nrm$)$

$$

$1)$ define a function

$$ name : Jacobi

$$ inputs : A $(2$d numpy array$)$ size nxn$,$ positive definite

$$ x$0(1$d numpy array$)$ starting value

$$ b$(1$d numpy array$)$ right$-$hand$-$side vector

$$ output : x $(1$d numpy array$)$ approximate solution to Ax$=$b

$$k $($integer$)$ number of iterations performed

$$ note$$ : do not use the matrix formulation of the algorithm

$$ note$$ : The function should work for any size rectangular array

$$ note$$ : use a tolerance of $1$e$-15$ as a stopping criterion

$$

$$

def Jacobi$($A$,$x$0,$b$)$:

$$ # add code here

$$ return$($x$,$k$)$

$$

$$

$2)$ define a function

$$ name : Jacobi$\_$matrix

$$ inputs : A $(2$d numpy array$)$ size nxn$,$ positive definite

$$ x$0(1$d numpy array$)$ starting value

$$ b$(1$d numpy array$)$ right$-$hand$-$side vector

$$ output : x $(1$d numpy array$)$ approximate solution to Ax$=$b

$$ k $($integer$)$ number of iterations performed

$$ note$$ : use the matrix formulation of the algorithm

$$ note$$ : The function should work for any size rectangular array

$$ note$$ : use a tolerance of $1$e$-15$ as a stopping criterion

$$

$$

def Jacobi$\_$matrix$($A$,$x$0,$b$)$:

$$ # add code here

$$ return$($x$,$k$)$

$3)$ define a function

$$ name : Gauss$\_$Seidel

$$ inputs : A $(2$d numpy array$)$ size nxn$,$ positive definite

$$ x$0(1$d numpy array$)$ starting value

$$ b$(1$d numpy array$)$ right$-$hand$-$side vector

$$ output : x $(1$d numpy array$)$ approximate solution to Ax$=$b

$$ k $($integer$)$ number of iterations performed

$$ note$$ : do not use the matrix formulation of the algorithm

$$ note$$ : The function should work for any size rectangular array

$$ note$$ : use a tolerance of $1$e$-15$ as a stopping criterion

$$

$$

def Gauss$\_$Seidel$($A$,$x$0,$b$)$:

$$ # add code here

$$ return$($x$,$k$)$

$$

$4)$ define a function

$$ name : Gauss$\_$Seidel$\_$matrix

$$ inputs : A $(2$d numpy array$)$ size nxn$,$ positive definite

$$ x$0(1$d numpy array$)$ starting value

$$ b$(1$d numpy array$)$ right$-$hand$-$side vector

$$ output : x $(1$d numpy array$)$ approximate solution to Ax$=$b

$$ k $($integer$)$ number of iterations performed

$$ note$$ : use the matrix formulation of the algorithm

$$ note$$ : The function should work for any size rectangular array

$$ note$$ : use a tolerance of $1$e$-15$ as a stopping criterion

$$

$$

def Gauss$\_$Seidel$\_$matrix$($A$,$x$0,$b$)$:

$$ # add code here

$$ return$($x$,$k$)$