No points without MATLAB setup and run well, please: Write ONE MATLAB script tha

Question

No points without MATLAB setup and run well, please:

Write ONE MATLAB script that does the recursive trapezoid rule using the function: f(x) = 2sin(x)cos(x), evaluating the integral between 0 and pi/2. Iterate until you achieve a relative error of 10^-4.

1) Perform the exact integration result analytically, and script produces the following:.

2) The resulting integral, with the final relative error in two ways, the final trapezoidal result relative to the exact result and the final trapezoidal result relative to the previous iteration

Explanation / Answer

e=exact value of integral= (-1/2)*cos(2*x)|0pi/2 = 1

re1=relative error w.r.t exact value

re2=relative error w.r.t previous iteration value

clc
clear all
e=1;
re1=.00001;
re2=1;
sum(1)=0;
i=2;
while abs(re2)>.0001
x=zeros(i,1);
y=zeros(i,1);
  
for j=2:i
x(j)=((j-1)*(pi/2))/(i-1);
y(j)=sin(2*x(j));
end
  
sum(i)=0;
for k=2:i-1
sum(i)=sum(i)+((x(2)-x(1))*y(k));
sum(i)=vpa(sum(i));
end
  
if i>=3
re2=(sum(i)-sum(i-1))/sum(i);
else
re2=1;
end
  
re1=(e-sum(i))/e;
i=i+1;
end
fprintf('Number of iterations = ')
disp(i-2)
fprintf('Number of terms in final sum = ')
disp(i-1)
fprintf('Value of integral = ')
disp(vpa(sum(i-1)))

output

Number of iterations =
26

Number of terms in final sum =
27

Value of integral =
0.99878303663202083750860538202687

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