Write a function that can solve a linear system, $Ax=b,$ using the Richardson iteration. Add this function to your

HW$2.$py file and call it Richardson it$().$

The function takes in a matrix $A,$ a right$-$hand side vector $b,$ a scaling parameter $,$ an allowed tolerance, and a maximum iteration number and returns the approximate solution x$.$ Recall from lecture that the Richardson iteration can be defined as follows:

$xk+1=xk+(b-$Axk$)(1)$

Where $xk+1$ is the next approximate solution to the linear system. You will need to start with an initial guess for $x,$ for now assume this is just the $O$ vector. Let the measure of error at the kth iteration be the $12$ norm of the vector Axk $-$ b$.$ Continue the Richardson iteration until either the error is below the input error tolerance or the maximum number of iterations are hit. The function should take the following inputs in this order:

numpy array $A$ : the matrix that defines the linear system

numpy array b: the right hand side vector of the linear system

float omega: the scaling parameter of the Richardson iteration $($by default this can be $1)$

float epsilon: the error tolerance

int max it: the maximum allowed number of iterations

And produce the following outputs in this order: $*$ numpy array $x$ : the approximate solution to the linear system

int $n$ it: the number of iterations taken to reach solution

float error : the calculated error in the last iteration

Hint: The Richardson iteration converges only if III so if you are

testing out your method on a matrix, be sul darr is property is satisfied. You can

check the norm of a matrix, A using numpy.linalg.norm$($A$,$ord$=2).$