Question

1. Approximate the integral,
exp(x), from 0 to 1,
using the composite midpoint rule, composite trapezoid rule, and composite Simpson’s method. Each method
should involve exactly n =( 2^k) + 1 integrand evaluations, k = 1 : 20. On the same plot, graph the absolute error
as a function of n.

Answer 1

%Matlab code for finding integration using different method
clear all
close all

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%function for which integration have to do
f1=@(x) exp(x);
fprintf('\tfunction for which integration have to do\n')
disp(f1)

%upper and lower limit
a=0; b=1;
fprintf('\tupper and lower limit of integration a=%f, b=%f.\n',a,b)

%exact integration value for this limit
ext_int=f1(1)-f1(0);
fprintf('\texact integration value for this limit is %f.\n',ext_int)

%all intervals
k=1:20;

%loop fpr all k
for i=1:length(k)
    n(i)=2^i;
    %finding integration using different method
    val_simp=Simpson_int(f1,a,b,n(i));
    val_trap=Trapizoidal_int(f1,a,b,n(i));
    val_mid=Midpoint_int(f1,a,b,n(i));
  
    %error in all method
    err_sim(i)=abs(ext_int-val_simp);
    err_tra(i)=abs(ext_int-val_trap);
    err_mid(i)=abs(ext_int-val_mid);

end

%loglog plot of error vs n
figure(1)
loglog(n,err_sim)
hold on
loglog(n,err_tra)
loglog(n,err_mid)
title('Loglog plot of n vs. error for y=exp(x)')
xlabel('Interval n')
ylabel('Error')
legend('Simpson','Trapizoidal','Mid point')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%Matlab function for Simpson Method
function val=Simpson_int(f,a,b,n)
%f=function for which integration have to do
%a=upper limit of integration
%b=lower limit of integration
%n=number of subintervals

    dx=(b-a)/n;     %interval length
    zs=f(a)+f(b);   %simpson integration
    %all x values for given subinterval
    xx=a:dx:b;
    %Simpson Algorithm for n equally spaced interval
    for i=2:n
        if mod(i,2)==0
            zs=zs+4*f(xx(i));
        else
            zs=zs+2*f(xx(i));
        end
    end
    %result using Simpson rule
    val=double((dx/3)*zs);
end

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%%Matlab function for Trapizoidal Method
function val=Trapizoidal_int(f,a,b,n)
    % f is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % n is the number of trapizoidal interval in [a,b]
    dx=(b-a)/n; %x interval
    val=0;
    %splits interval a to b into n+1 subintervals
    xx=linspace(a,b,n+1);
    %loop for trapizoidal integration
        for i=2:n
            val=val+2*double(f(xx(i)));
        end
        %Final integration value val for limit a to b
        val=(dx/2)*(val+f(a)+f(b));
end

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%%Matlab function for midpoint Method
function val=Midpoint_int(f,a,b,n)
    % f is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % n is the number of trapizoidal interval in [a,b]
    dx=(b-a)/n; %x interval
    val=0;
    %splits interval a to b into n+1 subintervals
    x=linspace(a,b,n+1);
    %loop for trapizoidal integration
        for i=1:n
            x_mid(i)=(x(i+1)+x(i))/2;
            val=val+double(f(x_mid(i)));
        end
    %result using midpoint integration method
    val=dx*val;
end
  
%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%