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

[Antoine ELIAS]

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 ?
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. 
> http://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
>
>