Solving Integral Approximations with MatLab

[renolovexoxo]

I have this problem to do, involving estimating the error for the derivative as well as the Gaussian Approximation. I am having a lot of trouble figuring out if I can alter this code or if I have to write something new. This is the first time I've ever used MatLab, and I could use some help if anyone has any ideas. The assignment is attached. I used this, which was given in class, to do (ii)

% f(x), the function to integrate
% f= @(x) x^4-2*x ;
% f= @(x) exp(x);
f=@(x) sin(x);
% a, the lower limit of integration
a=0 ;
% b, the upper limit of integration
b=pi ;
% b=1.0;
% n, the number of segments. Note that this number must be even.
% n=20 ;
%************************************************* *********************
format long g
h=(b-a)/n ;
% Sum the odd index function values, and multiply by 4
sumOdd=0 ;
for i=1:2:n-1
sumOdd=sumOdd+f(a+i*h) ;
end
% Sum the even index function values, and multiply by 2
sumEven=0 ;
for i=2:2:n-2
sumEven=sumEven+f(a+i*h) ;
end
sum=4*sumOdd+2*sumEven+f(a)+f(b) ;
% Then multiply by h/3
approx=h/3*sum ;
%exact = quad(f,a,b) ;
%exact=exp(b)-exp(a);
exact=-cos(b)-(-cos(a));
error=abs(approx-exact);
disp(approx);
disp(exact);
disp(error);

 

[NateD]

</code>The problem I need to doA:So for the derivative, you want to calculate two things:The numerical derivative of your function at some point <code>x0</code>.The error in this derivative.To calculate the numerical derivative, we use the central difference formula:<code>f'(x0) = (f(x0+h) - f(x0-h)) / (2*h)</code>where <code>h</code> is a small number.To calculate the error, you can look at the formula itself. Note that the derivatives of the function around the point <code>x0</code> can be approximated using a Taylor series expansion, i.e.<code>f(x0+h) ~= f(x0) + h*f'(x0) + 1/2*h^2*f''(x0) + ...f(x0-h) ~= f(x0) - h*f'(x0) + 1/2*h^2*f''(x0) + ...</code>Substituting this back into the central difference formula, we get:<code>f'(x0) ~= 1/2*h*(f(x0+h) + f(x0-h)) + 1/8*h^3*f'''(x0) + ...</code>From here, you can use the error formula for numerical differentiation:<code>error ~= 1/12*h^2*f'''(x0)</code>So all you have to do is calculate <code>f'''(x0)</code> and plug it into the error formula.For the Gaussian approximation, you can use the same technique as before. First calculate the numerical integral using the trapezoidal rule, then use a Taylor series expansion around the point <code>x0</code> to approximate the function and its derivatives. Finally, use the error formula for numerical integration:<code>error ~= 1/12*h^2*f''''(x0)</code>where <code>h</code> is the step size