Maybe some of these are bugs in GCL. I tried (sqrt -1.0) in
Allegro CL and got #c(0.0 1.0).
Lisp presumably does a^b as exp (b*log(a)) for the "hard" cases
of a and b.
If the only time that lisp is asked to do this, it is
appropriate to use this formula, then maybe we are OK.
It would be a mistake, I think, to convert 1/2+sqrt(-1) to a lisp
complex, since I think it would end up as floating point. But it
is possible to create exact complexes e.g. #c(1/2 1).
What about just replacing %i internally with #c(0 1)? It would
automatically be a number, a constant, complex, ... etc.
The use of lisp rational numbers for macsyma rational numbers
seems less problematical.
Stavros Macrakis wrote:
> Barton (WillisB) suggested using the native complex numbers from Lisp in
> Maxima. We've discussed this a bit in private mail, and I thought that
> it might be of general interest.
>
> Sometimes Maxima conventions are different from native conventions:
>
> sqrt(-1) == (-1)^(1/2) => %i
>
> (-1)^(1/3) => -1
>
> BUT in Lisp:
>
> (sqrt -1) == (expt -1 1/2) == (expt -1 0.5) => #c(6.1E-17 1.0)
> note non-exact (floating) result
>
> (expt -1 1/3) == (expt (float -1) (float 1/3)) => #C(0.50 0.87)
>
> Then again, Maxima conventions are not necessarily consistent:
>
> subst(-1,x,rectform(x^(1/3))) (answer x<0) gives 1/2+sqrt(3)/2*%i
> (not surprising of course)
>
> Sometimes Maxima seems more accurate than native complex floats:
>
> (-1.0)^(0.5),numer => %i
>
> (-1.0)^float(1/3),numer => -1.0
>
> but in fact this is a bug... compare:
>
> (-1.0)^(0.50000001),numer => %i
>
> ($ratepsilon is being used here, which it shouldn't be, and anyway its
> default value is much much too large)
>
> and there is an even more egregious bug lurking in there:
>
> (-1.0)^(0.50001),numer => 1.0 (!!!)
>
> (setting ratepsilon:1.0E-15 does not fix this!!)
>
> In some cases, Maxima is LESS accurate than native complex floats, and
> of course much slower:
>
> rt: rectform((-1)^(1/4))
>
> rectform(rt^200) => 1 -- 0.23 sec
>
> rectform(float(rt)^200) => -1.5E13 - 1.6E13 (!!!) -- 0.70 sec
>
> BUT with native complex floats
>
> (expt (expt -1 1/4) 200) => #C(1.0 -2.2E-14) -- 0.00 sec
>
> There is a trick, though...:
>
> rectform(float(rt)^200), maxposex:10; =>
> 1.000 + 9.8E-16 * %I -- 0.00 sec
>
> MaxPosEx (the largest positive exponent which will be expanded by the
> EXPAND command) controls whether rectform's use the trigometric form:
>
> rectform((a+b*%i)^2) => a^2-b^2 + 2*%I*a*c
>
> rectform((a+b*%i)^2),maxposex:0
> => (a^2+c^2) * cos(2 * atan2(c,a)) etc.
>
> Though I hesitate to add yet another global variable controlling
> behavior, it looks as though rectform needs its own MaxPosEx.... The
> algebraic form is usually easier to simplify, while the trigonometric
> form is smaller for exponent > 5 and is consistent with the form used
> for non-constant exponents.
>
> In the end, I think Barton is right that complex numbers should be
> represented specially, whether it's using something similar to RAT or
> using machine numbers. But it would take a lot of work in many parts of
> Maxima to make this work properly, I suspect.
>
> -s
>
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