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/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

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