[Scilab-users] display of complex/not real numbers, again

[Stéphane Mottelet]

I prefer the display after applying 
https://codereview.scilab.org/#/c/20981/:

--> x0=%pi/4;h=%eps/2
  h  =

    1.110D-16

--> cos(x0+%i*h)
  ans  =

    0.7071068 - 7.850D-17i

However, we could discuss if the arbitrary switch to "e" mode is 
desirable or not, but since Scilab 6.0 we have got used to this display 
mixing "v" and "e" mode...

BTW, this patch also fixes the more general problem of ambiguous display 
of quantities like (below is the display after the patch)

--> 1+%eps
  ans  =

    1.0000000

Currently Scilab makes users believe that a complex/non real number is 
real by hiding small non-zero real parts, and makes users believe that a 
non-integer number is integer by hiding zeros in the fractional part. 
Each year I have to warn my students, and I am really getting upset 
about this. The aforementionned patch also fixes that.

S.


Le 12/09/2019 à 11:59, Stéphane Mottelet a écrit :
>
> Le 12/09/2019 à 11:55, Antoine ELIAS a écrit :
>> Hello Stéphane,
>>
>> In Scilab 6.0.2 without format("e", 24)
>>
>> --> h = %eps/128, x0=%pi/4
>>  h  =
>>    1.735D-18
>>
>>  x0  =
>>    0.7853982
>>
>> --> (cos(x0+h)-cos(x0-h))/2/h
>>  ans  =
>>    0.
>>
>> --> cos(x0+%i*h)
>>  ans  =
>>    0.7071068
>>
>> --> imag(cos(x0+%i*h))/h
>>  ans  =
>>   -0.7071068
>>
>> --> -sin(x0)
>>  ans  =
>>   -0.7071068
>>
>> It seems to be close of Matlab's outputs, no ?
>
> No, Scilab display is singularly different:
>
> --> cos(x0+%i*h)
>  ans  =
>    0.7071068
>
> the above has an imaginary part, which is quite small, but essential 
> in the computation. Matlab is quite explicit here:
>
> >> cos(x0+i*h)
> ans =
>    0.7071 - 0.0000i
>
>
>> I probably not understand your problem ...
>>
>> Antoine
>> Le 12/09/2019 à 10:26, Stéphane Mottelet a écrit :
>>> Hello all,
>>>
>>> The subject has been already discussed a lot but I would like it to 
>>> be discussed again because I now have a real rationale to promote a 
>>> change in the way complex numbers with small imaginary part are 
>>> displayed.
>>>
>>> I don't know if some of you were aware of the clever technique of 
>>> complex-step derivative approximation, but until yesterday I was not 
>>> (see e.g. 
>>> https://antispam.utc.fr/proxy/2/c3RlcGhhbmUubW90dGVsZXRAdXRjLmZy/antispam.utc.fr/proxy/1/c3RlcGhhbmUubW90dGVsZXRAdXRjLmZy/mdolab.engin.umich.edu/sites/default/files/Martins2003CSD.pdf). 
>>> Roughly speaking, using the extension of a real function x->f(x) to 
>>> the complex plane allows to compute an approximation of the 
>>> derivative f'(x0) at a real x0 without using a substraction, like in 
>>> the central difference formula (f(x0+h)-f(x0-h))/2/h which is 
>>> subject to substractive cancelation when h is small. In Scilab most 
>>> operators and elementary functions are already complex-aware so this 
>>> is easy to illustrate the technique. For example let us approximate 
>>> the derivative of x->cos(x) at x=%pi/4, first with the central 
>>> difference formula, then with the complex step technique:
>>>
>>> --> format("e",24)
>>>
>>> --> h=%eps/128, x0=%pi/4
>>>  h  =
>>>
>>>    1.73472347597680709D-18
>>>
>>>  x0  =
>>>
>>>    7.85398163397448279D-01
>>>
>>>
>>> --> (cos(x0+h)-cos(x0-h))/2/h
>>>  ans  =
>>>
>>>    0.00000000000000000D+00
>>>
>>>
>>> --> imag(cos(x0+%i*h))/h
>>>  ans  =
>>>
>>>   -7.07106781186547462D-01
>>>
>>>
>>> --> -sin(x0)
>>>  ans  =
>>>
>>>   -7.07106781186547462D-01
>>>
>>> You can see the pathological approximation with central difference 
>>> formula and the perfect (up to relative machine precision) 
>>> approximation of complex-step formula.
>>>
>>> However, the following is a pity:
>>>
>>>
>>> --> cos(x0+%i*h)
>>>  ans  =
>>>
>>>    7.07106781186547573D-01
>>>
>>> We cannot see the imaginary part although seeing the latter is 
>>> fundamental in the complex-step technique. We have to force the 
>>> display like this, and frankly I don't like having to do that with 
>>> my students:
>>>
>>> --> imag(cos(x0+%i*h))
>>>  ans  =
>>>
>>>   -1.22663473334669916D-18
>>>
>>> I hope that you will find that this example is a good rationale to 
>>> change the default display of Scilab. To feed the discussion, here 
>>> is how Matlab displays things, without having to change the default 
>>> settings:
>>>
>>>
>>> >> h=eps/128, x0=pi/4
>>> h =
>>>    1.7347e-18
>>> x0 =
>>>     0.7854
>>>
>>> >> (cos(x0+h)-cos(x0-h))/2/h
>>> ans =
>>>      0
>>>
>>> >> cos(x0+i*h)
>>> ans =
>>>    0.7071 - 0.0000i
>>>
>>> >> imag(cos(x0+i*h))/h
>>> ans =
>>>    -0.7071
>>>
>>> >> -sin(x0)
>>> ans =
>>>    -0.7071
>>>
>>>
>>
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>>
>
-- 
Stéphane Mottelet
Ingénieur de recherche
EA 4297 Transformations Intégrées de la Matière Renouvelable
Département Génie des Procédés Industriels
Sorbonne Universités - Université de Technologie de Compiègne
CS 60319, 60203 Compiègne cedex
Tel : +33(0)344234688
http://www.utc.fr/~mottelet