$1.($a$)$ Write a MATLAB function $($without loops$)$ to perform cubic spline interpolation using the $$not a knot$$ end conditions. The first line of your code should be

function s $=$ myspline$($y$,$r$)$

where y is the column vector of data values, and the positive integer r is the interpolation factor, which means that the spline will be evaluated at r$1$ uniformly spaced points between each pair of adjacent data points. The spline values will appear in the column vector s$.$ Incorporate the trid function from the last homework to solve the tridiagonal matrix equation that arises in this problem. $($b$)$ Let y $=$ randn$(1000000,1),$ r $=3,$ x $=(1$:$3$:$2999998),$ xx $=(1$:$2999998).$ Check your myspline function by computing norm$($s$-$sm$)/$norm$($sm$),$ where

sm $=$ spline$($x$,$y$,$xx$)$

and spline is MATLAB$$s built$-$in spline interpolation function. $($c$)$ Using the same values as in $($b$),$ how many times faster is myspline compared to spline?

The function code used in last class is:

function x $=$ trid$($a$,$ b$,$ g$)$

n $=$ length$($g$)$;

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

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

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

D$(1)=$ a;

for i $=2$:n

L$($i$-1)=$ b $/$ D$($i$-1)$;

D$($i$)=$ a $-$ D$($i$-1)*($L$($i$-1)^2)$;

end

x$(1)=$ g$(1)/$a;

for i $=2$:n

x$($i$)=($g$($i$)-$b$*$x$($i$-1))/$D$($i$)$;

end

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

x$($i$)=$ x$($i$)-$ L$($i$)*$x$($i$+1)$;

end

end

The test code:

clear all;

format long;

n $=$ randi$([10,50],1,1)$;

a $=$ randi$([11,20],1,1)$;

b $=$ randi$([-5,5],1,1)$;

c $=[$a b zeros$(1,$ n$-2)]$;

r $=[$a b zeros$(1,$ n$-2)]$;

A $=$ toeplitz$($c$,$ r$)$;

g $=$ randi$([1,10],$ n$,1)$;

x $=$ trid$($a$,$ b$,$ g$)$;

x$\_$hat$\_1=$ A$\backslash$g; $\%$ backslash

opts.POSDEF $=$ true;

opts.SYM $=$ true;

x$\_$hat$\_2=$ linsolve$($A$,$ g$,$ opts$)$; $\%$ linsolve

error$1=$ sum$($abs$($x $-$ x$\_$hat$\_1))/$n;

error$2=$ sum$($abs$($x $-$ x$\_$hat$\_2))/$n;

fprintf$(\text{'}$The average difference between x and x$\_$hat$\_1$ is:$\text{'})$;

disp$($error$1)$;

fprintf$(\text{'}$The average difference between x and x$\_$hat$\_2$ is:$\text{'})$;

disp$($error$2)$; Please don't use AI $.$ I will give upvote if it is giving output and without using CHAT GPT