format long e
% format short
n=2^20;
logn=log(n)/log(2)
a=0;
b=1;
dx=(b-a)/n;
xx=[a:dx:b]';
yy=exp(-xx.*xx);
[xx(1:10),yy(1:10)];
left=sum(yy(1:n-1))/n;
rite=sum(yy(2:n))/n;
ritesum=yy(n+1);
newind=[n/2+1];
midsum=yy(newind);
ritesum(2)=(ritesum+midsum)/2
for k = 2:logn
   newind=(newind+1)/2;
   newind=[newind,newind+n/2];
   midsum(k)=sum(yy(newind))/2^(k-1);  
   ritesum(k+1)=(ritesum(k)+midsum(k))/2;
end
ritesum=ritesum*(b-a);
nn=2.^[0:1:logn];
leftsum=ritesum+(yy(1)-yy(n+1))*(b-a)./nn;
trapsum=(ritesum+leftsum)/2;
xx=xx(1:n)+dx/2;
yy=exp(-xx.*xx);
midsum(logn+1)=sum(yy)/n;
simps=(trapsum+2*midsum)/3;
format short
nn'
format long g
leftsum'
ritesum'
trapsum'
midsum'
simps'
B=[' num(ints) ',' left sum',' rite sum ',' trapezoid ',' midpoint ', ' simpson ']
A=[nn;leftsum;ritesum;trapsum;midsum;simps]'