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

```
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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