% Composite Trapezoid Rule and Composite Simpson's Rule 
% Input: f - integrant function, [a,b] - interval, m - number of nodes
% Output: tf - Trapezoidal formula result
%         sf - Simpson's rule result
function[tf,sf,h,h1]=trapez(f,a,b,m)
h=(b-a)/m; %Step size for Trapezoidal rule
h1=h/2;    %Step size for Simpson's rule
g=(h/2)*(f(a)+f(b));
s=(h1/3)*(f(a)+f(b));
for k=1:m-1;
    x=a+k*h; xs1=a+2*k*h1; xs2=a+(2*k-1)*h1;
    g=g+h*f(x);
    s=s+(2*h1/3)*f(xs1)+(4*h1/3)*f(xs2);
end
tf=g;
sf=s+(4*h1/3)*f(b-h1);
_____________________________________________________________________
clear;
a=1; b=6;
real=8.1834792077; % Exact inegration value.
f=inline('2+sin(2*sqrt(x))');
for j=1:5;
m=10+100*(j-1);
[tf sf h h1]=trapez(f,a,b,m);
A(j,1)=m; A(j,2)=h; A(j,3)=tf; A(j,4)=abs(real-tf);
B(j,1)=m; B(j,2)=h1; B(j,3)=sf; B(j,4)=abs(real-sf);
end
disp('Result from composite Trapezoid rule ')    
disp(' m           h                Q[f]           E[f] ')
disp('----------------------------------------------------------')
for k=1:5
  disp(sprintf('%2.0f    %7.3e    %7.15e  %7.3e',A(k,1),A(k,2),A(k,3),A(k,4)))
end
disp(' Result from composite Simpson rule')   
disp(' m           h                Q[f]           E[f] ')
disp('----------------------------------------------------------')
for k=1:5
  disp(sprintf('%2.0f    %7.3e    %7.15e  %7.3e',B(k,1),B(k,2),B(k,3),B(k,4)))
end
Result from composite Trapezoid rule 
 m           h                Q[f]           E[f] 
----------------------------------------------------------
10    5.000e-001    8.193854565172531e+000  1.038e-002
110    4.545e-002    8.183563906899268e+000  8.470e-005
210    2.381e-002    8.183502445342256e+000  2.324e-005
310    1.613e-002    8.183489871195299e+000  1.066e-005
410    1.220e-002    8.183485303793596e+000  6.096e-006
 Result from composite Simpson rule
 m           h                Q[f]           E[f] 
----------------------------------------------------------
10    2.500e-001    8.183447496636241e+000  3.171e-005
110    2.273e-002    8.183479205410587e+000  2.289e-009
210    1.190e-002    8.183479207493134e+000  2.069e-010
310    8.065e-003    8.183479207627007e+000  7.299e-011
410    6.098e-003    8.183479207651054e+000  4.895e-011