Modify the following code to obtain a Python function for performing Newton's method for a given function $f.$

Input Arguments

Your function should take the following as input:

Functions for evaluating $f(x)$ and its derivative $f^{\text{'}}(x).$

Initial solution $x_0.$

Maximum number of iterations and stopping tolerance.

Output Arguments

Your function should return the following as output:

The finite sequence of solutions $\{x_0,x_1,$dots,$x_k\},$ where $x_k$ is the solution when the algorithm is terminated.

The sequence of absolute differences in consecutive iterates $\{|x_1-x_0|,|x_2-x_1|,$dots,$|x_k-x_{k-1}|\}.$

The number of iterations performed.

The function below gives the general form of Newton's Method but is missing code for executing several steps, e$.$g$.,$ calculating $x_k$ from $x_{k-1},$ updating the residuals $|x_k-x_{k-1}|.$ You will need to modify the given code to include these steps.

$[]$: def Newton$(f,df,x0,$ nmax, tol$)$:

$""$n$"$

Performs Newton$-$Raphson Method to calculate a root of function $f.$

$f$ : function for evaluating $f$

df: function for evaluating $f^{\text{'}}$

$x0$ : initial iterate.

nmax: maximum number of iterates to perform.

tol: stopping tolerance

return sequence of approximate roots and number of iterations performed.

$""$n

# Import numpy.

import numpy as np

# Initialization.

iter $=0$ # Number of iterations is $0.$

$fx=$ tol $+1$ # Initialize function value to be greater than tol.

errs $=$ np$.$ones $(nmax+1)$ # Initial vector of errors.

# Update iterate until a root is found or max # of iterations are performed. while $($iter $$ nmax and errs$[$iter$]>$ tol$)$:

# Calculate function value and derivative value.

iter