python - Unexpected behavior of numpy.fft.fft with high precision numbers

Question

I have the following code...Note the two lines under # generate sine curve. One uses a higher precision value for 2pi than the other, they should still give near identical results though.

import numpy as np
import matplotlib.pyplot as plt


t1 = np.arange(0., 1., .01)

# generate sine curve
y1 = np.sin(6.28318*5.*t1)  
#y1 = np.sin(6.283185307179586*5.*t1) # equivalent to np.sin(2*np.pi*t1)

# calculate the fft (no averaging!) of the time series
ffty = np.fft.fft(y1)

fig, ax_list = plt.subplots(3,1)
ax_list[0].plot(t1,y1, '.-')

ax_list[1].plot(ffty.real, '.-', label='Real Part')
ax_list[1].legend()

ax_list[2].plot(ffty.imag, '.-', label='Imag Part')
ax_list[2].legend()


plt.show()

If you run the code with the lower precision 6.28318 you get the expected result for the fft...

However, if you run the code with the higher precision 6.283185307179586 which is equal to 2.*numpy.pi, you get the unexpected result below... the real part is drastically wrong...The amplitudes are way off, it's not symmetric, it doesn't make any sense.

I'm at a loss as to what is causing this. Anyone have any ideas?

Answer 1

This is totally expected behavior. Computers use floating-point computations, which are inherently imprecise.

Note the y-axis for your real result. If no numerical inaccuracy existed, the real component would be identically 0. With your "higher precision" result, the real part is almost identical to 0 (1e-14 is very close to the precision of double-precision floats). With a lower precision, the real part becomes much larger (though still much, much smaller than the imaginary part). Because of the larger numbers, there is more structure as well (i.e. the error is not given by rounding errors, but by an actual feature of your input data, a period that is slightly shorter than ideal).