# [SciPy-user] fft numerical precision

Robert Kern robert.kern at gmail.com
Wed Feb 15 16:45:27 CST 2006

```Ryan Krauss wrote:
> Sorry to bring this back up again (this thread has been dormant since
> 12/21/05), but I think this is the key road block between me and
> really understanding this problem (and I think it went over my head
> when it was originally posted).  If double precision numbers use 52
> bits for the mantissa, what does it mean to say that 2**-52 is the
> approximate fractional precision of double precision floats?  How does
> this affect optimization routines like fzero and fmin so that their
> accuracy seems limited to 2**-52 when the smallest possible floating
> point number should be more like 2**-1074.  Does this somehow relate
> to how floating point subtraction is done (as relates to my post from
> earlier today).

I don't have time right now to go into a lecture on condition numbers and other
such stuff, but let give you an example to chew on and a good reference:

In [23]: a = rand(1024)

In [24]: ifft(fft(a))
Out[24]:
array([ 0.27261246 -4.33680869e-17j,  0.08520686 -1.04083409e-17j,
0.66654776 +6.60887690e-17j, ...,  0.05895311 +8.37004077e-17j,
0.40454606 -2.34796237e-17j,  0.38051768 +8.63024929e-17j])

In [25]: a
Out[25]:
array([ 0.27261246,  0.08520686,  0.66654776, ...,  0.05895311,
0.40454606,  0.38051768])

In [26]: a *= 1e16

In [27]: ifft(fft(a))
Out[27]:
array([  2.72612457e+15+0.21875j   ,   8.52068614e+14-0.359375j  ,
6.66547756e+15-0.11579423j, ...,   5.89531079e+14+1.16015625j,
4.04546057e+15+0.7886458j ,   3.80517684e+15+0.71875j   ])

The first volume of _Numerical Computation_ by Christoph W. Ueberhuber
(_Computer-Numerik 1_ in the original German) covers these topics quite well and
extensively.

--
Robert Kern
robert.kern at gmail.com

"In the fields of hell where the grass grows high
Are the graves of dreams allowed to die."
-- Richard Harter

```