def uniform$\_$poly$\_$interpolation$($a$,$b$,$p$,$n$,$x$,$f$,$produce$\_$fig$)$:

### Example of creating a figure object

fig $=$ plt$.$figure$()$ # This line is required

#plt$.$plot$([0,1],[2,3])$ # Delete this line and replace

#Remove the following line when you have completed the code

#interpolant $=$ None

xhat $=$ np$.$linspace$($a$,$ b$,$ p $+1)$

# Call lagrange$\_$poly with tol $=1\mathrm{.}0$e$-10$

lagrange$\_$matrix, error$\_$flag $=$ lagrange$\_$poly$($p$,$ xhat, n$,$ x$,1\mathrm{.}0$e$-10)$

if error$\_$flag $==1$:

print$("$Error: Nodal points are not distinct."$)$

return None, None

# Calculate interpolant

interpolant $=$ np$.$dot$($f$($xhat$),$ lagrange$\_$matrix$)$

if produce$\_$fig:

# Plot the function f and the interpolant

fig, ax $=$ plt$.$subplots$()$

ax$.$plot$($x$,$ f$($x$),$ label$=\text{'}$f$($x$)\text{'})$

ax$.$plot$($x$,$ interpolant, label$=$f'Interpolant $($p$=\{$p$\})\text{'})$

ax$.$legend$()$

plt$.$show$()$

else:

fig $=$ None

return interpolant, fig

#$\%\%$

def nonuniform$\_$poly$\_$interpolation$($a$,$b$,$p$,$n$,$x$,$f$,$produce$\_$fig$)$:

#Remove the following two lines when you have completed the code

#interpolant $=$ None

#fig $=$ None

xhat $=$ np$.$cos$((2*$ np$.$arange$($p $+1)+1)/(2*($p $+1))*$ np$.$pi$)$

# Map nodal points to the interval $[$a$,$ b$]$

xhat $=0\mathrm{.}5*($b $-$ a$)*$ xhat $+0\mathrm{.}5*($a $+$ b$)$

# Call lagrange$\_$poly with tol $=1\mathrm{.}0$e$-10$

lagrange$\_$matrix, error$\_$flag $=$ lagrange$\_$poly$($p$,$ xhat, n$,$ x$,1\mathrm{.}0$e$-10)$

if error$\_$flag $==1$:

print$("$Error: Nodal points are not distinct."$)$

return None, None

# Calculate interpolant

interpolant $=$ np$.$dot$($f$($xhat$),$ lagrange$\_$matrix$)$

if produce$\_$fig:

# Plot the function f and the interpolant

fig, ax $=$ plt$.$subplots$()$

ax$.$plot$($x$,$ f$($x$),$ label$=\text{'}$f$($x$)\text{'})$

ax$.$plot$($x$,$ interpolant, label$=$f'Interpolant $($p$=\{$p$\})\text{'})$

ax$.$legend$()$

plt$.$show$()$

else:

fig $=$ NonePiecewise Polynomial Interpolation

Recall, given a function $f$:$[a,b]R,$ we can construct its piecewise polynomial

interpolant of order $p$ by splitting $a,b$ up into uniform subintervals and applying the

Lagrange interpolant of order $p$ on each subinterval. Hence, using $m$ subintervals

$\{[tilde(x{)}_{i-1},$tilde$(x{)}_i]{\}}_i=1^m,$ the piecewise interpolant $S_p^m(x)$ satisfies:

$S_p^m(x){|}_{[tilde(x{)}_{i-1},\text{t}\text{i}\text{l}\text{d}\text{e}(x{)}_i]}=p_p^i(x),i=1,$dots,$m$

where $p_p^i(x)$ is the polynomial interpolant of $f(x)$ on tilde$(x{)}_{i-1},$tilde$(x{)}_i.$

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

def $piecewis{e}_i$nterpolation$(a,b,p,m,n,x,f,produc{e}_{f}ig)$

The function should return as output:

p$\_$u$\_$interpolant, p$\_$nu$\_$interpolant $-$ two numpy.ndarrays, each of

shape $,$

return interpolant, fig

Need the python code