Hi please give a Matlab code such that it addresses ALL aspects of Question $3$ using only Newton Second$-$Order Method and need to determine a good estimates of the starting points in order for newton method to work $.$My idea is to use gradient descent to determine a good estimate of starting points then pass to newton method to converge? I kept getting wrong figures and distances for my for my code. Please help.

function $[$min$\_$distance, t$1\_$min, t$2\_$min$]=$ Distance$($r$1,$ e$1,$ e$2,$ a$1,$ b$1,$ n$1,$ r$2,$ e$1\_2,$ e$2\_2,$ a$2,$ b$2,$ n$2)$

$\%$ Define tolerance and maximum iterations for optimization

tol $=1$e$-6$;

max$\_$iter $=100$;

$\%$ Set initial values for t$1$ and t$2$

t$1\_$init $=5$; $\%$ Initial angle for superellipse $1$

t$2\_$init $=5$; $\%$ Initial angle for superellipse $2$

$\%$ Initial Phase: Gradient Descent to find a good estimate

$[$t$1\_$gd$,$ t$2\_$gd$]=$ gradient$\_$descent$($a$1,$ b$1,$ n$1,$ a$2,$ b$2,$ n$2,$ tol, max$\_$iter, t$1\_$init, t$2\_$init$)$;

$\%$ Refinement Phase: Newton's Method for precise convergence

$[$t$1\_$min, t$2\_$min, min$\_$distance$]=$ find$\_$min$\_$distance$($a$1,$ b$1,$ n$1,$ a$2,$ b$2,$ n$2,$ tol, max$\_$iter, t$1\_$gd$,$ t$2\_$gd$)$;

$\%$ Draw superellipses and closest points

draw$\_$superellipses$($r$1,$ e$1,$ e$2,$ a$1,$ b$1,$ n$1,$ r$2,$ e$1\_2,$ e$2\_2,$ a$2,$ b$2,$ n$2,$ t$1\_$min, t$2\_$min$)$;

end

function $[$t$1\_$gd$,$ t$2\_$gd$]=$ gradient$\_$descent$($a$1,$ b$1,$ n$1,$ a$2,$ b$2,$ n$2,$ tol, max$\_$iter, t$1\_$init, t$2\_$init$)$

$\%$ Initialize parameters for gradient descent

t$1=$ t$1\_$init;

t$2=$ t$2\_$init;

alpha $=0\mathrm{.}05$; $\%$ Adjusted learning rate for gradient descent

for iter $=1$:max$\_$iter

$[$~$,$ grad$]=$ distance$\_$and$\_$gradient$($t$1,$ t$2,$ a$1,$ b$1,$ n$1,$ a$2,$ b$2,$ n$2)$;

$\%$ Update parameters using gradient descent

t$1=$ t$1-$ alpha $*$ grad$(1)$;

t$2=$ t$2-$ alpha $*$ grad$(2)$;

$\%$ Check convergence

if norm$($grad$)<$ tol

break;

end

end

$\%$ Return the final parameters from gradient descent

t$1\_$gd $=$ t$1$;

t$2\_$gd $=$ t$2$;

end

function $[$t$1\_$min, t$2\_$min, min$\_$distance$]=$ find$\_$min$\_$distance$($a$1,$ b$1,$ n$1,$ a$2,$ b$2,$ n$2,$ tol, max$\_$iter, t$1\_$init, t$2\_$init$)$

$\%$ Initialize parameters for Newton's method

t$1=$ t$1\_$init;

t$2=$ t$2\_$init;

iter $=0$;

regularization$\_$factor $=1$e$-6$; $\%$ Regularization to prevent singular Hessian

while iter $<$ max$\_$iter

$[$d$,$ grad, hessian$]=$ distance$\_$and$\_$gradient$($t$1,$ t$2,$ a$1,$ b$1,$ n$1,$ a$2,$ b$2,$ n$2)$;

$\%$ Add regularization to Hessian

hessian $=$ hessian $+$ regularization$\_$factor $*$ eye$(2)$;

$\%$ Check if Hessian is valid

if all$($isfinite$($hessian$($:$)))$ && rank$($hessian$)==2$ && cond$($hessian$)<1$e$10$

$\%$ Calculate optimal update direction using Newton's method

delta $=-$hessian $\backslash$ grad; $\%$ Solve linear system

$\%$ Update parameters with the found step size

t$1=$ t$1+$ delta$(1)$;

t$2=$ t$2+$ delta$(2)$;

else

warning$(\text{'}$Hessian is singular or contains non$-$finite values, stopping optimization.$\text{'})$;

min$\_$distance $=$ d; $\%$ Return the last computed distance

t$1\_$min $=$ t$1$;

t$2\_$min $=$ t$2$;

return; $\%$ Exit the function

end

$\%$ Check convergence

if norm$($grad$)<$ tol

break;

end

iter $=$ iter $+1$; $\%$ Increment iteration counter

end

$\%$ Final minimum distance

min$\_$distance $=$ d; $\%$ This is the calculated distance

t$1\_$min $=$ t$1$;

t$2\_$min $=$ t$2$;

end. function $[$x$,$ y$]=$ superellipse$\_$point$($t$,$ a$,$ b$,$ n$)$

$\%$ Compute $($x$,$ y$)$ point on the superellipse given parameter t

x $=$ a $*$ sign$($cos$($t$)).*$ abs$($cos$($t$)).^(2/$n$)$;

y $=$ b $*$ sign$($sin$($t$)).*$ abs$($sin$($t$)).^(2/$n$)$;

end

function $[$x$,$ y$]=$ superellipse$\_$points$($r$,$ a$,$ b$,$ n$,$ t$)$

$\%$ Compute points on the superellipse given parameter t

$[$x$,$ y$]=$ arrayfun$($@$($ti$)$ superellipse$\_$point$($ti$,$ a$,$ b$,$ n$),$ t$)$;

x $=$ x $+$ r$(1)$;

y $=$ y $+$ r$(2)$;

end

function $[$d$,$ grad, hessian$]=$ distance$\_$and$\_$gradient$($t$1,$ t$2,$ a$1,$ b$1,$ n$1,$ a$2,$ b$2,$ n$2)$

$\%$ Calculate distance and gradient for Newton's method

$[$x$1,$ y$1]=$ superellipse$\_$point$($t$1,$ a$1,$ b$1,$ n$1)$;

$[$x$2,$ y$2]=$ superellipse$\_$point$($t$2,$ a$2,$ b$2,$ n$2)$;

$\%$ Calculate distance

delta$\_$x $=$ x$2-$ x$1$;

delta$\_$y $=$ y$2-$ y$1$;

d $=$ sqrt$($delta$\_$x$^2+$ delta$\_$y$^2)$;

$\%$ Regularization to avoid division by zero

if d $<1$e$-10$

d $=1$e$-10$; $\%$ Set a small distance to avoid NaN

end

$\%$ Compute gradients

dx$\_$dt$1=-($a$1*$ sin$($t$1)*$ abs$($cos$($t$1))^(2/$n$1-1))*$ delta$\_$x $/$ d;

dy$\_$dt$1=($b$1*$ cos$($t$1)*$ abs$($sin$($t$1))^(2/$n$1-1))*$ delta$\_$y $/$ d;

dx$\_$dt$2=($a$2*$ sin$($t$2)*$ abs$($cos$($t$2))^(2/$n$2-1))*$ delta$\_$x $/$ d;

dy$\_$dt$2=-($b$2*$ cos$($t$2)*$ abs$($sin$($t$2))^(2/$n$2-1))*$ delta$\_$y $/$ d;

grad $=[$dx$\_$dt$1+$ dy$\_$dt$1$; dx$\_$dt$2+$ dy$\_$dt$2]$; $\%$ Gradient vector

$\%$ Compute Hessian $($second derivatives$)$

hessian $=$ zeros$(2,2)$;

$\%$ For each element of Hessian, compute the appropriate second derivatives

hessian$(1,1)=($a$1^2*($cos$($t$1)^(2/$n$1-2)*(2/$n$1)*$ sin$($t$1)^2+$ sin$($t