[Scilab-users] [EXTERNAL] Re: acos leads to complex values

[Stéphane Mottelet]

Le 29/01/2019 à 23:10, Federico Miyara a écrit :
>
> Stephane,
>
>>> However, I must say that acos(1-%eps) gives on my Scilab 6.0.1 
>>> (Windows 7, i7 processor) a result of
>> Federico, you have to try 1+%eps or -1-%eps !
>
> acos(x) with x > 1 is indeed imaginary, so there is no unexpected result.
>
> However, I see now that it hadn't been said that in-domain arguments 
> yielded complex results but -1 - %eps was the culprit.
>
> But, for instance,
>
> min([-1-%eps, -1-%eps*2/3, -1, -1+ %eps, -0.9]) >= -1
>
> is false (on my system), so the test performed by Paul Carrico would 
> not succeed if there were a -1-%eps among the values of the vector.
>
> Only when alpha*%eps  with alpha < 0.5 is the minimum the test is 
> successful, but in such case acos() yields a value close to %pi, not a 
> complex value
>
> --> acos(-1 - 0.499*%eps)
>  ans  =
>
>    3.141592653589793100000

logical, since

--> -1 - 0.499*%eps == -1
  ans  =

   T

--> acos(-1 - 0.499*%eps)==%pi
  ans  =

   T

S.

>
> When alpha > 0.5 the test is not successful nor is the acos() real:
>
> --> acos(-1 - 0.50000001*%eps)
>  ans  =
>
>    3.141592653589793100000 - 0.000000021073424255447i
>
> (this is a binary rounding situation since %eps is 2^-52)
>
> I think the original question is not completely solved. Why a number 
> that is not smaller to -1 (asuming the min test is successful) yields 
> a complex acos?
>
> An why it does in some systems and not in others? (Assuming Paul's 
> report is accurate)
>
> There is another situation where this happens, when an fft is 
> performed on a real signal, then filtered in the frequency domain and 
> then the ifft is applied. The result should be real, but sometimes 
> there are small imaginary parts.
>
> In this case there is a reason, complex exponentials are used in the 
> algorithm so arithmetioc errors may not completely cancel out because 
> of arithmetic and precision issues.
>
> Federico Miyara
>
>
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