Why there is error between My DFT and built in Matlab FFT?

Question

I'm trying to implement DFT/IDFT and FFT/IFFT in Matlab and C.

I got wrong values while I implement IDFT/IFFT. So I'm trying to find a mistake.

Just for DFT, I used this algorithm to implement it in Matlab (from matlab's fft doc) But I really don't know why there is very small errors when I run this code. and even my Radix-2 FFT have same problems. Is there anything I'm missing or can I ignore this values? (mean of 256 values : -1.6881e-25)

Thank you for reading this.

for i=1:256
    x(i) = sin(2*pi*i*80) + sin(2*pi*i*46);
end

X = zeros(256,1);
for k=1:256
   for n=1:256
       X(k)=X(k) + x(n)*exp((-j*2*pi*(n-1)*(k-1))/256);
   end
end

built_in = fft(x);
error = abs(X) - abs(built_in)';

plot(abs(X)); hold on
plot(abs(built_in));
plot(error);
legend('My code', 'Matlab','error');

with result:

Answer 1

The reason why the two results are not exactly the same is because you compute the same result in two different ways. Direct implementation of the DFT is numerically different from the FFT algorithm, so you must expect different round-off behavior, resulting in (negligible) differences in the result.

Answer 2

Your error is the yellow line at the bottom. i see no difference between your DFT and the built-in FFT from MATLAB.