These questions in Numerical Analysis.

Could you please make sure the code for Matlab is working? Thanks. $(1)($MATLAB: Lagrange interpolation$)$

We consider the function $f(x)=\frac{1}{1+x^2}$ on the interval $-5,5.$

$($a$)$ Implement a MATLAB function that compute the interpolating Lagrange polynomial

${}_{n}f$ of degree $n=2,4,8$ on equally spaced nodes $($you can use the function polyfit and

polyval$).$ Plot the graph of $f$ and the interpolating polynomials on the same figure $($use

$100$ sample points to evaluate $f$ and the Lagrange polynomial$).$ Comment your result.

$($b$)$ Explain briefly why can we use the MATLAB command polyfit to find Lagrange

interpolating polynomial.

$(2)($MATLAB $+$ natural cubic spline$)$

$($a$)$ Implement a MATLAB function called natural cubic spline that has the form:

$[$ xout, yout $]=$n atural$\_$cubic$\_$spline $($x$,$y$,$ nsample $)$

where the vectors $x$ and $y$ represents the nodes $x_i$ and data $y_i,$ the integer nsample is the

number of sampling points in each interval $x_i,x_{i+1}($use the command linspace to

generate them$).$ The function returns a vector xout with the coordinate of the sampling

point and a vector yout with the value of the natural cubic spline at the sampling points.

$($b$)$ Create a MATLAB script that use the above function with $x_i=-5+i,y_i=\frac{1}{1+x_i^2}$

for $i=0,$dots,$5$ and $n=11.$ Plot the output in a graph. Any comment compares to the

results you got in question $1.$a$?$