Make the QuadN function suitable for up to n$=8$ Gauss quadrature rule $($that is$,$ it can change from n$=1$ to $8$ and find the relevant coefficient and certain xi values$)$ by revising our trapezoidal, rectangular, integral functions according to Simphson rules to a state that stops when a certain absolute relative error value is reached. test via a function: function $[$I$]=$ GausQuadn$($f$,$a$,$b$,$n$)$

$\%$ xi wi $\text{'}$s are the points and weights for calculation

$\%$  improper integrals with special points

f$=$inline $($f$)$

n

I$=0$;

if n$==2$

disp$(\text{'}$ I was here'$)$

w$(1)=1$;x$(1)=0\mathrm{.}5773502691896257$ 

w$(2)=1$;x$(2)=-0\mathrm{.}5773502691896257$

for i$=1$:n

I$=$I$+($b$-$a$)*0\mathrm{.}5*$w$($i$)*$f$(($b$-$a$)*0\mathrm{.}5*$x$($i$)+($b$+$a$)*0\mathrm{.}5)$;

end

elseif n$==3$

w$(1)=0\mathrm{.}5555555555555556$;x$(1)=0\mathrm{.}77459666924148347$ ;

w$(2)=0\mathrm{.}8888888888888888$;x$(2)=0$

w$(3)=0\mathrm{.}5555555555555556$;x$(3)=-0\mathrm{.}7745966692414834$

for i$=1$:n

I$=$I$+($b$-$a$)*0\mathrm{.}5*$w$($i$)*$f$(($b$-$a$)*0\mathrm{.}5*$x$($i$)+($b$+$a$)*0\mathrm{.}5)$;

end

elseif n$==4$

w$(1)=0\mathrm{.}3478548451374544$;x$(1)=0\mathrm{.}8611363115940525$  ;

w$(2)=0\mathrm{.}6521451548625460$;x$(2)=0\mathrm{.}3399810435848563$ ;

w$(3)=0\mathrm{.}6521451548625460$;x$(3)=-0\mathrm{.}3399810435848563$;

w$(4)=0\mathrm{.}3478548451374544$;x$(4)=-0\mathrm{.}8611363115940525$  ;

for i$=1$:n

I$=$I$+($b$-$a$)*0\mathrm{.}5*$w$($i$)*$f$(($b$-$a$)*0\mathrm{.}5*$x$($i$)+($b$+$a$)*0\mathrm{.}5)$;

end

else

disp$(\text{'}$This version work only up to $4$ points Gaussian quadrature'$)$

end

I

end