$(5)$ You will write a Python code that generates and evaluates Newton's Divided Differences

polynomials. This code should have two functions, with names and inputs as shown

below. The first function calculates the coefficients and the second define and evaluates

the polynomial. Upload your code to a file named HW$4\_$Question$5.$py

def divided$\_$differences$($node$\_$x: np$.$ndarray, node$\_$y: np$.$ndarray$)$

np$.$ndarray:

$t$

Calculates the coefficients of a Newton's Divided Difference

polynomial using

divided differences via the recurrence relation

:param node$\_$x: $x$ values for the nodes

:param node$\_$y: $y$ values for the nodes

:return: array of coefficients

def newtons polynomial $(x$ data: np$.$ndarray, node $x$ : np$.$ndarray, coeffs:

np$.$ndarray$)$ np$.$ndarray:

$"$n$"$

Defines and evaluates the Newton's Divided Difference Polynomial

:param $x$ data: $x$ values of the data on which we will evaluate the

polynomial

:param node$\_$x: $x$ values for the nodes

:param coeffs: array of coefficients

return: polynomial evaluated on $x$ data

$"$n$"$

$($a$)$ Generate a set of $x$ and $y$ values using the code below. These will be the nodes. Use

the code you wrote to generate a Newton's Divided Difference polynomial that fits to

these nodes, and evaluate that polynomial for $1000$ values of $x$ between $0$ and $3.$

Plot this polynomial evaluation and also plot the node points $($using a scatter plot$).$

Upload this plot to a file named $"$HW$4\_$Question$5$a$\_$plot.png$".$

import numpy $asnp$

$x=np*linspace(0,3**np*,10)$

$y=np*sin(x)$

$($b$)$ Evaluate this polynomial for the value of $x=3\mathrm{.}2.$ What is the result and what is the

absolute error in this result?