Question

MATLAB

Write an m-file capable of performing numerical integration for the equation

$f(x)=\int_{0}^{\pi /2} (6+3cos(x)) dx$

using the simpsons and trapezoidal functions:

function [value, error] = simpsons(func, a, b, n, I) %retuns the value and error

ret = 0;

h = (b-a)/(n+1); %step size

pts = a:h:b; % array of points

for i=2:(n+3)/2

a = 2*i-3;

b = 2*i-2;

c = 2*i-1;

ret = ret + (func(pts(a)) + 4*func(pts(b)) + func(pts(c)));

end

value = h*ret/3;

error = abs(I - value)*100/I; %error between value and simpsons estimate

end

------

%MATLAB Code for the function:

function [value, error] = trapezoidal(func, a, b, n, I) % returns the value and error

ret = 0;

h = (b-a)/(n+1); %step size

pts = a:h:b; %array of points

  

for i=2:n+2

ret = ret + (func(pts(i)) + func(pts(i-1)))*h/2;

end

value = ret;

error = abs(I - value)*100/I; %error between real value and trapezoidal estimate

end

The program is interactive and should include the following:

1. Ask the user to input which rule to use (e.g. `1’ for composite trapezoidal and `2’ for composite Simpson’s 1/3 rule).

2. Ask the user to input the number of points for which the integration is calculated. The program should notify the user if an invalid number of points are used.

3. Print the percentage error between the approximated solution and the solution obtained through the quad function.

Answer 1

Code in matlab:

Output:

%-----------------------Code---------------------

% test function takes func, a, b, I as parameters,
% n=number of points is taken as input when test() is called.
function [value, error]=test(func, a, b, I)
n=input('Enter number of points: '); % number of points
if(n<1) % checks the validity of n, if false calls the function again
disp('Please enter valid number of points: ')
test(func, a, b, I);
end
% option to select either 1. trapezoidal or 2. simpsons.
option=input('Enter an option: 1.> trapezoidal and 2.> simpsons:');
I_quad=quad(func, a, b); % integrates func from x=a to b using quad function
if(option==1)
[value, error]=trapezoidal(func, a, b,n, I);
% calculates percentage error for value and I_quad
fprintf('Percentage error when using quad function: %f', abs(value-I_quad)*100/value);
  
elseif(option==2)
[value, error]=simpsons(func, a, b,n, I);
% calculates percentage error for value and I_quad
fprintf('Percentage error when using quad function: %f', abs(value-I_quad)*100/value);
else % if option is neither 1 nor 2 then call function again
disp('Please select valid option');
test(func, a, b, I);
  
end
end
% simpsons method to integrate fuction func
function [value, error] = simpsons(func, a, b, n,I) %retuns the value and error
ret = 0;
h = (b-a)/(n+1); %step size
pts = a:h:b; % array of points
for i=2:(n+3)/2
a = 2*i-3;
b = 2*i-2;
c = 2*i-1;
ret = ret + (func(pts(a)) + 4*func(pts(b)) + func(pts(c)));
end
value = h*ret/3;
error = abs(I - value)*100/I; %error between value and simpsons estimate
  
end
function [value, error] = trapezoidal(func, a, b,n, I) % returns the value and error
ret = 0;
h = (b-a)/(n+1); %step size
pts = a:h:b; %array of points
for i=2:n+2
ret = ret + (func(pts(i)) + func(pts(i-1)))*h/2;
end
value = ret;
error = abs(I - value)*100/I; %error between real value and trapezoidal estimate
end