In polynomial$\_$interpolation. py you will find a function with definition

def compute$\_$piecewise$\_$errors$($a$,$b$,$n$\_$m$,$ M$,$ n$\_$p$,$ P$,$f$)$

The function should return as output:

u$\_$error$\_$matrix, nu$\_$error$\_$matrix $-$ two numpy.ndarrays, each of shape

$($n$\_$m$,$n$\_$p$).$

u$\_$fig, nu$\_$fig $-$ two matplotlib.figure.Figures.

Complete the function to compute $($approximations to$)$ the error

$ma{x}_{\text{x}\text{i}\text{n}[a,b]}|S_{p_k}^{m_l}(x)-f(x)|$ for a range of polynomial degrees $P=\{p_k{\}}_k=0^{n_p-1}$ and

numbers of subintervals $M=\{m_l{\}}_l=0^{n_m-1}.$ Row $l$ of u$\_$error$\_$matrix should

contain the errors when a uniform set of interpolating nodes is used in each

subinterval, while row $l$ of nu$\_$error$\_$matrix should contain the errors when the

nonuniform set of interpolating nodes is used in each subinterval.

Compute the approximation to the error by evaluating the error $|S_{p_k}^{m_l}(x)-f(x)|$

for $2500$ equally spaced points over $a,b$ and take the maximum of those.

The function should create two plots, one for the uniform errors and one for

the nonuniform errors. Each plot should be of the errors against $\{m_l{\}}_l=0^{n_m-1},$ have

a separate line for each $\{p_k{\}}_k=0^{n_p-1},$ and use plt$.loglog.$ The figures should be

returned in u$\_$fig and nu$\_$fig.

Add a description at the top of your compute$\_$piecewise$\_$errors function

as a "docstring" that, when help$($p$\_$int.compute$\_$piecewise$\_$errors$)$ is run,

will briefly comment on and explain the results when $P=\{1,2,3,4,5,6\}$ and

$M=\{1,2,4,8,16,32,64,128,256,512,1024\},$ with

$($a$)f(x)=sin(x)+\frac{1}{10}cos(8x),[a,b]=[-1,1].$

$($b$)f(x)=\sqrt[\phantom{2}]{|x-\frac{1}{3}|},[a,b]=[-1,1].$

You should also comment briefly in the "docstring" on the relative effectiveness

of increasing $p$ and $m$ at reducing the error.