Powers of exponents

[Vladimir V. Kisil-2]

                Dear All,

                I noticed that exp() functions is not aware of power()
  method. As a results simple things like pow(exp(a), 2) are not
  simplified to exp(2*a). I am attaching the corresponding patch.

  Best wishes,
  Vladimir
--
Vladimir V. Kisil     email: kisilv@...
--                      www: http://maths.leeds.ac.uk/~kisilv/

diff --git a/ginac/inifcns_trans.cpp b/ginac/inifcns_trans.cpp
index e30f573..ab47531 100644
--- a/ginac/inifcns_trans.cpp
+++ b/ginac/inifcns_trans.cpp
@@ -99,9 +99,15 @@ static ex exp_imag_part(const ex & x)
  return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
 }
 
+static ex exp_power(const ex & x, const ex & a)
+{
+ return exp(x*a);
+}
+
 REGISTER_FUNCTION(exp, eval_func(exp_eval).
                        evalf_func(exp_evalf).
                        derivative_func(exp_deriv).
+                       power_func(exp_power).
                        real_part_func(exp_real_part).
                        imag_part_func(exp_imag_part).
                        latex_name("\\exp"));

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[Richard B. Kreckel]

Dear Vladimir,

Vladimir V. Kisil wrote:
> I noticed that exp() functions is not aware of power()
>   method. As a results simple things like pow(exp(a), 2) are not
>   simplified to exp(2*a). I am attaching the corresponding patch.

The rewriting rule pow(exp(x),a) -> exp(a*x) is not correct in the
general complex case. Consider a=1/2 and x=-I*Pi, for instance. This is
because, generally, log(exp(x)) -> x is not correct.

GiNaC is very careful about *not* doing such simplifications that are
not guaranteed to be correct, even in the general complex case. Maybe
you are unaware of this file
<http://www.ginac.de/ginac.git?a=blob;f=doc/powerlaws.tex;h=ade0547d048a51f65f414c08416a3979ea885ee8;hb=master>.

Can you fix your patch to only do this simplification when it cannot be
wrong?

Best wishes
    -richy.
--
Richard B. Kreckel
<http://www.ginac.de/~kreckel/>
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[Vladimir V. Kisil-2]

                Dear Richard,

>>>>> On Wed, 30 Sep 2009 23:39:25 +0200, "Richard B. Kreckel" <kreckel@...> said:

    RK> The rewriting rule pow(exp(x),a) -> exp(a*x) is not correct in
    RK> the general complex case. Consider a=1/2 and x=-I*Pi, for
    RK> instance.

        Thanks for pointing out this. I am attaching a patch which do the
  substitution only for real x and a.

  Best wishes,
  Vladimir
--
Vladimir V. Kisil     email: kisilv@...
--                      www: http://maths.leeds.ac.uk/~kisilv/

--- inifcns_trans.cpp 2009-02-17 13:39:22.000000000 +0000
+++ patches/inifcns_trans.cpp 2009-10-01 16:03:58.000000000 +0100
@@ -99,9 +99,18 @@
  return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
 }
 
+static ex exp_power(const ex & x, const ex & a)
+{
+ if (x.info(info_flags::real) && a.info(info_flags::real))
+ return exp(x*a);
+
+ return power(exp(x), a).hold();
+}
+
 REGISTER_FUNCTION(exp, eval_func(exp_eval).
                        evalf_func(exp_evalf).
                        derivative_func(exp_deriv).
+                       power_func(exp_power).
                        real_part_func(exp_real_part).
                        imag_part_func(exp_imag_part).
                        latex_name("\\exp"));

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[Stephen Montgomery-Smith-3]

Vladimir V. Kisil wrote:

> Dear Richard,
>
>>>>>> On Wed, 30 Sep 2009 23:39:25 +0200, "Richard B. Kreckel" <kreckel@...> said:
>
>     RK> The rewriting rule pow(exp(x),a) -> exp(a*x) is not correct in
>     RK> the general complex case. Consider a=1/2 and x=-I*Pi, for
>     RK> instance.
>
> Thanks for pointing out this. I am attaching a patch which do the
>   substitution only for real x and a.

It will also be true for arbitrary x if a is an integer.
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[Vladimir V. Kisil-2]

>>>>> On Thu, 01 Oct 2009 10:49:03 -0500, Stephen Montgomery-Smith <stephen@...> said:
    SMS> It will also be true for arbitrary x if a is an integer.

        Good point! The third iteration of the patch is attached.

--
Vladimir V. Kisil     email: kisilv@...
--                      www: http://maths.leeds.ac.uk/~kisilv/

--- inifcns_trans.cpp 2009-02-17 13:39:22.000000000 +0000
+++ patches/inifcns_trans.cpp 2009-10-01 21:04:09.000000000 +0100
@@ -99,9 +99,19 @@
  return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
 }
 
+static ex exp_power(const ex & x, const ex & a)
+{
+ if (x.info(info_flags::integer) || a.info(info_flags::integer)
+ || (x.info(info_flags::real) && a.info(info_flags::real)))
+ return exp(x*a);
+
+ return power(exp(x), a).hold();
+}
+
 REGISTER_FUNCTION(exp, eval_func(exp_eval).
                        evalf_func(exp_evalf).
                        derivative_func(exp_deriv).
+                       power_func(exp_power).
                        real_part_func(exp_real_part).
                        imag_part_func(exp_imag_part).
                        latex_name("\\exp"));

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[Richard B. Kreckel]

Vladimir V. Kisil wrote:
>>>>>> On Thu, 01 Oct 2009 10:49:03 -0500, Stephen Montgomery-Smith <stephen@...> said:
>     SMS> It will also be true for arbitrary x if a is an integer.
>
> Good point! The third iteration of the patch is attached.
[...]
> + if (x.info(info_flags::integer) || a.info(info_flags::integer)
> + || (x.info(info_flags::real) && a.info(info_flags::real)))
> + return exp(x*a);

Maybe it is just too late, but I don't see the motivation for the "is x
an integer" condition.

   -richy.
--
Richard B. Kreckel
<http://www.ginac.de/~kreckel/>
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[Vladimir V. Kisil-2]

>>>>> On Fri, 02 Oct 2009 00:00:13 +0200, "Richard B. Kreckel" <kreckel@...> said:
    RK> Maybe it is just too late, but I don't see the motivation for
    RK> the "is x an integer" condition.

        Why not, if it is a correct substitution. Who know, may be as a
  result of some evaluation power of the exponents will becomes 2.

  Best wishes,
  Vladimir  
--
Vladimir V. Kisil     email: kisilv@...
--                      www: http://maths.leeds.ac.uk/~kisilv/
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[Stephen Montgomery-Smith-3]

Vladimir V. Kisil wrote:
>>>>>> On Fri, 02 Oct 2009 00:00:13 +0200, "Richard B. Kreckel" <kreckel@...> said:
>     RK> Maybe it is just too late, but I don't see the motivation for
>     RK> the "is x an integer" condition.
>
> Why not, if it is a correct substitution. Who know, may be as a
>   result of some evaluation power of the exponents will becomes 2.

I think it is wrong to include "x is an integer."  Even 1^i (which is
exp(0)^i) is not well defined (it can be any of e^(2 pi n) for all
integers).


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[Vladimir V. Kisil-2]

>>>>> On Fri, 02 Oct 2009 07:11:51 -0500, Stephen Montgomery-Smith <stephen@...> said:
    SMS> I think it is wrong to include "x is an integer."  Even 1^i
    SMS> (which is exp(0)^i) is not well defined (it can be any of e^(2
    SMS> pi n) for all integers).

        Alright, here is the patch to stay on the safe side.

        Best wishes,
        Vladimir
--
Vladimir V. Kisil     email: kisilv@...
--                      www: http://maths.leeds.ac.uk/~kisilv/

--- inifcns_trans.cpp 2009-02-17 13:39:22.000000000 +0000
+++ patches/inifcns_trans.cpp 2009-10-02 15:45:17.000000000 +0100
@@ -99,9 +99,18 @@
  return exp(GiNaC::real_part(x))*sin(GiNaC::imag_part(x));
 }
 
+static ex exp_power(const ex & x, const ex & a)
+{
+ if (a.info(info_flags::integer) || (x.info(info_flags::real) && a.info(info_flags::real)))
+ return exp(x*a);
+
+ return power(exp(x), a).hold();
+}
+
 REGISTER_FUNCTION(exp, eval_func(exp_eval).
                        evalf_func(exp_evalf).
                        derivative_func(exp_deriv).
+                       power_func(exp_power).
                        real_part_func(exp_real_part).
                        imag_part_func(exp_imag_part).
                        latex_name("\\exp"));

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[Richard B. Kreckel]

Hi!

Stephen Montgomery-Smith wrote:
> Vladimir V. Kisil wrote:
>>>>>>> On Fri, 02 Oct 2009 00:00:13 +0200, "Richard B. Kreckel"
>>>>>>> <kreckel@...> said:
>>     RK> Maybe it is just too late, but I don't see the motivation for
>>     RK> the "is x an integer" condition.
>>
>>     Why not, if it is a correct substitution. Who know, may be as a
>>   result of some evaluation power of the exponents will becomes 2.

Actually, including "x is an integer" turns out to be correct. But we
can do even better: it is sufficient to test if a is real.

Here is a proof. With arbitrary complex b:

     pow(exp(a),b) == exp(b*log(exp(a))

Since a is real, we know that exp(a) is real and positive. So
log(exp(a))==a and exp(b*log(exp(a))==exp(b*a).  q.e.d.

I'll push a patch for the exp function and for doc/powerlaws.tex.

> I think it is wrong to include "x is an integer."  Even 1^i (which is
> exp(0)^i) is not well defined (it can be any of e^(2 pi n) for all
> integers).

That argument is confused. After all, we rewrite exp(2*Pi*I) -> 1.

   -richy.
--
Richard B. Kreckel
<http://www.ginac.de/~kreckel/>
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[Richard B. Kreckel]

Hi!

I wrote:
> I'll push a patch for the exp function and for doc/powerlaws.tex.

Oh, it is all there in doc/powerlaws.tex!

This is another fine example where we saved an hour or so reading the
documentation by spending a couple of days arguing back and forth.  :-)

   -richy.
--
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<http://www.ginac.de/~kreckel/>
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[Richard B. Kreckel]

Hi!

I wrote:
> I'll push a patch for the exp function and for doc/powerlaws.tex.

Well, I didn't push it yet because it turns out that it interferes in
surprising ways with mul::eval(). We used to have exp(x)/exp(x) -> 1 but
that doesn't work any more because exp(x)^(-1) evaluates to exp(-x)
first, resulting in exp(x)/exp(x) -> exp(x)*exp(-x).  :-(

Unless somebody comes up with a convincing idea how to fix eval() in the
presence of pow(exp(x),a) -> exp(a*x) I'm not going to apply this patch.

Bye
   -richy.
--
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<http://www.ginac.de/~kreckel/>
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[Francois Maltey]

Hello everybody,

I'm a new ginac user because I discover sage. So you might excuse my
point of view if my arguments are out of ginac purpose. And this message
isn't direct reponse for this patch.

// 1 //

I understand that ginac operates over complex analysis as the other
computer algebra systems.

In this case it's very curious to write exp(u)^v == exp(u*v).
Fine choices of "branch cuts" may allow this point of view for a local
study,
but it isn't usual mathematics for general purpose.

You understand I don't like the exp(u)^v == exp(u*v)

// 2 //

It' bad if exp(x)/exp(x) remains : sage reduces sin(x)/sin(x)  == 1 as
usual.
I don't know the inner algorithms of ginac but I suppose that all
function calls as sin(x) are seen as a new variable in the expression
which is a fraction with a lot of variables, even if someones as sin(x)
and cos(x) are linked together by cos(x)^2+sin(x)^2==1.

So exp(x)/exp(x) must be simplified in 1.

// 3 //

Look at sin(x) and cos(x).

Sometimes the user prefers the expanded formula with Tchebytchev polynomials
(cos(2x)+1)/(sin(2x)) == (2 cos(x)^2)/(2 sin(x) cos(x)) == cos(x)/sin(x)
== cotan(x).

Othertimes the user wants to combine 2 cos(x)^2 into the (almost) linear
form cos(2x)+1.
Computer algebra systems don't have any automatic transform
but the user calls the expand or the combine function for theses
opposite purposes.

Calculus are similar with exp : both transforms exp(x)^2 <==> exp(2*x)
are useful.

I observe that ginac respects algebraic user input :
by example there are very view transform with

x = 2*t/(1-t^2) ;
x.subs(t=x) == 4*t / (1 - (2*t/(1-t^2))^2)

Then the user calls expand, "simplify_fractions" or others functions if
he wants an other form of this input.

power and exp functions might be in the same case :
exp(x)^2 remains exp(x)^2 and exp(2*x) remains exp(2*x).
Then an expand call translates exp(2*x) to exp(x)^2
and a combine (or an other name) translates back exp(x)^2 to exp(2*x)

If it does so, ginac respects also the user choice for the exp function
as it does for the fractions.

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