Hi Jason,
thanks for your suggestion and your detailed answer,
but actually I did not start this thread for performance reasons,
I did start it to ask why "i^2" is not treated like an exact symbolic
expression in sage:
----------------------------------------------------------------------
| Sage Version 3.2, Release Date: 2008-11-20 |
| Type notebook() for the GUI, and license() for information. |
----------------------------------------------------------------------
sage: i^2
-1
sage: CDF(_)
-1.0 + 1.22460635382e-16*I
and I was wondering if this is a bug, obviously it is not, but I still
don't understand it :-)
Georg
On 3 Dez., 04:06, Jason Grout <[EMAIL PROTECTED]> wrote:
> ggrafendorfer wrote:
> > Hi Robert,
> > this was not a misunderstanding, there is an "n" missing :-), I
> > corrected it:
> > I wanted to write
>
> > then i^2 should also be of type CDF, but
>
> > rather then
>
> > the i^2 should also ....
>
> > as an answer to your statement, namely that i^2 is getting turned into
> > CDF(i)^CDF(2)
>
> > I hope this is clear now
>
> >> Generally in Sage we prefer exact arithmetic to numeric
> >> approximations, because once you start doing numeric things one has
> >> to deal with rounding errors, etc. and there's no going back.
>
> > and that's exactly the reason why I don't understand why "i^2" is not
> > be treated like an exact symbolic expression
>
> See the bottom of the message for my suggestion, which basically is to
> use the python complex i if you really want speed (i.e., at the top of
> your program, do "i = complex(I)"
>
> I'll take a try at explaining this. This may be more for the interested
> reader than for you specifically, so sorry if it's rehashing ground that
> has already been covered.
>
> In Sage, as you know, we have numerous ways of representing i. The
> default way is to represent it using a symbolic expression:
> sage: type(i)
> <class 'sage.functions.constants.I_class'>
> sage: type(i^2)
> <class 'sage.calculus.calculus.SymbolicArithmetic'>
>
> Of course, we can also treat i as a numeric quantity (basically, we
> treat complex numbers as double precision floating point numbers; one
> double for the real part, one for the imaginary part). That's using CDF:
>
> sage: cdf_i=CDF.0
> sage: cdf_i
> 1.0*I
> sage: cdf_i^2
> -1.0 + 1.22460635382e-16*I
> sage: cdf_i*cdf_i
> -1.0
>
> Unfortunately, as you see above, this representation is prone to
> roundoff error and gives slightly inaccurate results when squaring.
> That's what happens when you work with a numerical i (that's why the
> default "i" in Sage is symbolic).
>
> You could also declare i as the root of the polynomial x^2+1; that's
> what Robert did earlier, and is still fast, and what's more, it is exact.
>
> sage: K.<ii> = QuadraticField(-1)
> sage: ii
> ii
> sage: ii^2
> -1
>
> In fact, using the quadratic field is the fastest Sage solution from
> what we have above:
>
> sage: #Symbolic I
> sage: timeit('i^2')
> 625 loops, best of 3: 86.7 µs per loop
> sage: # CDF I
> sage: timeit('cdf_i^2')
> 625 loops, best of 3: 41.3 µs per loop
> sage: # Exact I as a root of a polynomial
> sage: timeit('ii^2')
> 625 loops, best of 3: 5.65 µs per loop
>
> However, for your purposes, maybe using the builtin python complex type
> is best; it is by far the fastest and it has the property that you want
> (namely, i^2=-1)
>
> sage: python_i = complex(I)
> sage: python_i
> 1j
> sage: python_i^2
> (-1+0j)
> sage: timeit('python_i^2')
> 625 loops, best of 3: 2.72 µs per loop
>
> Jason
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