[SciPy-User] SciPy-User Digest, Vol 110, Issue 21
"Claas H. Köhler"
claas.koehler@dlr...
Fri Oct 12 11:39:03 CDT 2012
On 12/10/12 12:48, The Helmbolds wrote:
>> On 09/10/12 19:12, Pauli Virtanen wrote:
>>> 09.10.2012 19:28, "Claas H. K?hler" kirjoitti:
>>>> I have a question regarding the error function scipy.special.erf:
>>>>
>>>> Is it intended, that the erf of an imaginary argument yields a
>> non-vanishing real-part?
>>>>
>>>> I get e.g.
>>>> erf(1j)= 1.6504257587975431j
>>>> erf(5j)= (1+8298273879.8992386j)
>>>>
>>>> The first result is what I would expect in accordance with Wolfram
>> alpha. The second result, however,
>>>> has a real part of unity. As far as I know, the real part of erf should
>> always vanish for purely
>>>> imaginary numbers.
>>>>
>>>> Any support would be appreciated.
>>>
>>> The reason here is that the ye olde complex erf Fortran implementation
>>> that Scipy has uses the asymptotic expansion (Abramowitz & Stegun
>>> 7.1.23) to compute large-argument values. The asymptotic series is for
>>> erfc, and one always gets Re erf = 1 along the imaginary axis.
>>>
>>> Of course, this is somewhat naive. While it does produce reasonable
>>> relative accuracy as a complex number, the accuracy of the real and
>>> imaginary parts separately is not necessarily OK near the imaginary axis.
>>>
>>> The issue with Scipy here is twofold -- first, there are no better
>>> existing special function libraries we could use, or at least I'm not
>>> aware of them. Second, writing these from scratch takes time and
>>> expertise and nobody has so far volunteered to do any work in this
>>> direction.
>>>
>> Thanks for the quick response!
>>
>> The bottom line is that erf is actually not (correctly) implemented for complex
>> arguments, if I
>> understand you correctly.
>>
>> I suspect there are good reasons to provide a function which is known to yield
>> incorrect results, so
>> that throwing a type error is not an option? (This is what erfc does on my
>> machine)
>>
>> However, adding a warning when called with complex arguments could be helpful to
>> prevent naiive use
>> as in my case. Adding this important piece of information to the docs would not
>> harm either, from my
>> point of view.
>>
>> In any case, thanks for the quick support.
>>
>> Regards
>> Claas
>
> On my system, I get the correct answers if I'm careful about the call to erf.
> If I call erf with a single real value, I get the ordinary (not the complex) error function value.
> If I call erf with a NumPy array or a Python sequence, I get the complex er
ror function returned.
> I do not think SciPy's erf is supposed to be called with a complex number.
According to the docs it is. Otherwise I would expect to see a domain error, similar to erfc.
Regards
Claas
>
> For example:
>>>> special.erf(1j)
> 1.6504257587975431j # Wrong answer!
>>>> special.erf((0,1))
> array([ 0. , 0.84270079]) # Right answer.
>
> Two more examples:
>>>> for y in range(-10, 11):
> temp = special.erf((0,y))
> print y, temp # Calling with a sequence, returns a NumPy array
>
> -10 [ 0. -1.]
> -9 [ 0. -1.]
> -8 [ 0. -1.]
> -7 [ 0. -1.]
> -6 [ 0. -1.]
> -5 [ 0. -1.]
> -4 [ 0. -0.99999998]
> -3 [ 0. -0.99997791]
> -2 [ 0. -0.99532227]
> -1 [ 0. -0.84270079]
> 0 [ 0. 0.]
> 1 [ 0. 0.84270079]
> 2 [ 0. 0.99532227]
> 3 [ 0. 0.99997791]
> 4 [ 0. 0.99999998]
> 5 [ 0. 1.]
> 6 [ 0. 1.]
> 7 [ 0. 1.]
> 8 [ 0. 1.]
> 9 [ 0. 1.]
> 10 [ 0. 1.]
>
>
> OTOH--------------------------------------------------------------------------------------------
>>>> for y in range(-10, 11):
> temp = special.erf(y)
> print y, temp # Calling with a (scalar) real value returns a (scalar) real value.
>
> -10 -1.0
> -9 -1.0
> -8 -1.0
> -7 -1.0
> -6 -1.0
> -5 -0.999999999998
> -4 -0.999999984583
> -3 -0.999977909503
> -2 -0.995322265019
> -1 -0.84270079295
> 0 0.0
> 1 0.84270079295
> 2 0.995322265019
> 3 0.999977909503
> 4 0.999999984583
> 5 0.999999999998
> 6 1.0
> 7 1.0
> 8 1.0
> 9 1.0
> 10 1.0
>
> Bob and Paula H
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>
--
Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR)
Institut für Methodik der Fernerkundung | Experimentelle Verfahren | Münchner Str | 82234 Weßling
Claas H. Köhler
Telefon 08153 28-1274 | Telefax 08153 28-1337 | claas.koehler@dlr.de
www.DLR.de/EOC
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