Modify the following cubic spline MATLAB code into quadratic spline. Do NOT use AI for this task. Clearly explain your reasoning with necessary equations. Also provide an example plot at the end of the code. Thank you!

function yy $=$ natspline$($x$,$ y$,$ xx$)$

$\%$ natspline$($x$,$y$,$xx$)$:

$\%$ uses a natural cubic spline interpolation to find yy$,$ the values

$\%$ of the underlying function y at the points in the vector xx$.$

$\%$ The vector x specifies the points at which the data y is given.

n $=$ length$($x$)$;

m $=$ length$($xx$)$;

aa $=$ zeros$($n$)$;

bb $=$ zeros$($n$,1)$;

$\%$ Boundary conditions for natural spline

aa$(1,1)=1$;

aa$($n$,$ n$)=1$;

bb$(1)=0$;

bb$($n$)=0$;

$\%$ Build the system of equations

for i $=2$:n$-1$

aa$($i$,$ i$-1)=$ h$($x$,$ i $-1)$;

aa$($i$,$ i$)=2*($h$($x$,$ i $-1)+$ h$($x$,$ i$))$;

aa$($i$,$ i$+1)=$ h$($x$,$ i$)$;

bb$($i$)=3*($fd$($i $+1,$ i$,$ x$,$ y$)-$ fd$($i$,$ i $-1,$ x$,$ y$))$;

end

$\%$ Solve for c

c $=$ aa $\backslash$ bb;

$\%$ Calculate spline coefficients a$,$ b$,$ d

a $=$ y$(1$:n$-1)\text{'}$;

b $=$ zeros$($n$-1,1)$;

d $=$ zeros$($n$-1,1)$;

for i $=1$:n$-1$

b$($i$)=$ fd$($i $+1,$ i$,$ x$,$ y$)-$ h$($x$,$ i$)/3*(2*$ c$($i$)+$ c$($i $+1))$;

d$($i$)=($c$($i $+1)-$ c$($i$))/3/$ h$($x$,$ i$)$;

end

$\%$ Evaluate the spline at the points in xx

yy $=$ zeros$($m$,1)$;

for i $=1$:m

yy$($i$)=$ SplineInterp$($x$,$ n$,$ a$,$ b$,$ c$,$ d$,$ xx$($i$))$;

end

end

$\%$ Helper function to calculate intervals

function hh $=$ h$($x$,$ i$)$

hh $=$ x$($i $+1)-$ x$($i$)$;

end

$\%$ Helper function to calculate finite differences

function fdd $=$ fd$($i$,$ j$,$ x$,$ y$)$

fdd $=($y$($i$)-$ y$($j$))/($x$($i$)-$ x$($j$))$;

end

$\%$ Helper function for spline interpolation

function yyy $=$ SplineInterp$($x$,$ n$,$ a$,$ b$,$ c$,$ d$,$ xi$)$

for ii $=1$:n$-1$

if xi $>=$ x$($ii$)-0\mathrm{.}000001$ && xi $<=$ x$($ii $+1)+0\mathrm{.}000001$

yyy $=$ a$($ii$)+$ b$($ii$)*($xi $-$ x$($ii$))+$ c$($ii$)*($xi $-$ x$($ii$))^2+$ d$($ii$)*($xi $-$ x$($ii$))^3$;

break;

end

end

end