Re: [clisp-list] Re: reporting curious complex numbers appearance

[Pascal B. <pa...@af...>]

lin8080 wrote:

>
>Kaz Kylheku schrieb:
>
>
>
>This is what HyperSpec says in section SQRT, ISQRT (file
>/body/fun_sqrtcm_isqrt.html.) "... if the number is not a complex but is
>negative, then the result is a complex" (12.1. ff). This case seems to
>be clear. 
>
>The mathematicans also say that a negativ sqrt is complex (right?), but
>the same mathematicans say that (sqrt) is the reverse function of (*
>zahl zahl) (which will returned only positiv numbers) and they also
>describe for what kind of numbers this is valid. 
>

You have to take into account the domain of the function. To take the 
reverse function, you must consider the function only on a sub-domain 
where it's a bijection.

For example, if you consider: 

    f : ]-infinity, 0] ------> [0, +infinity[
                   x  |------> x*x

then the reverse is:

    f^-1 : [0, +infinity[ -----> ]-infinity, 0]
                   x     |-----> - sqrt(x)

Note that this other function:

    g : [0, +infinity[ ------> [0, +infinity[
                   x  |------> x*x

which looks a lot like f, has not the same reverse function!

And of course, a function square : R --> R, x |--> x*x is not a 
bijection, so it has no reverse.

So you see, you cannot expect the operators implemented in Lisp to 
correspond to the mathematical function you're manipulating.
In maths, there are a lot of square root function, all defined on a 
different domain, and with a different image set.

In computer languages we choose to implement one specific function, with 
a domain and image set considered useful. 

Since Lisp knows about complex, it naturally defines sqrt over the 
complexes and returns a complex (unless the imaginary part is null, in 
which case it's a real).
    

>
>
>The practical reason why there is +,- in trigononic functions, is to say
>where the point of the graph is in the xy-system. And for that no
>complex is needed, only 2 values.  
>
>So the question should be: is there a way to suppress the complex number
>representation in negative trigonomic values, when a sqrt is involved?  
>
Yes, define properly your functions, and implement them accordingly.

(defun my-square-over-R-plus (x)
  (assert (and (realp x) (<= 0 x)))
  (* x x))

(defun my-square-root-over-R-plus (x)
    (assert (and (realp x) (<= 0 x)))
    (sqrt x))



-- 
__Pascal Bourguignon__
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