[GiNaC-devel] Power laws

[Vladimir V. Kisil]

	Dear All,

	Coming back to the previous discussion on exponent/power
  functions I am sending the collection of three patches. The first two
  are corrected/polished versions of two patches described in my
  previous email (see forwarded below). Briefly:

  1. First patch adds the rule (e^a)^b = e^(ab) to automatic evaluation
  in safe cases.

  2. Second improves normalisation method for exponents, it is able to
  reduce (exp(2*x)-1)/(exp(x)-1) to exp(x)+1.

  3. Third patch add the similar functionality for powers, it is able to
  reduce (x-1)/(sqrt(x)-1) to sqrt(x)+1.

  Patches have check components which illustrate further examples of
  normalisation.

  Best wishes,
  Vladimir
-- 
Vladimir V. Kisil                 http://www.maths.leeds.ac.uk/~kisilv/
  Book:     Geometry of Mobius Transformations     http://goo.gl/EaG2Vu
  Software: Geometry of cycles          http://moebinv.sourceforge.net/
  Jupyter: https://github.com/vvkisil/MoebInv-notebooks
>>>>> On Fri, 10 Apr 2020 08:43:52 +0100, "Vladimir V. Kisil" <V.Kisil at leeds.ac.uk> said:

    VVK> 	Dear Richard,

    VVK> 	Thank you for pointing out an issue with notmalisation
    VVK> of expressions. What about the attached _draft_ of the patch?
    VVK> It allows to reduce all suitable exponents with arguments
    VVK> different by a rational numeric factor to monomials of the same
    VVK> temporary variable. Thus it reduces

    VVK>   (exp(2*x)-1)/(exp(x)-1) to exp(x)+1

    VVK>   as well as other more complicated cases like

    VVK>   (exp(15*x)+exp(12*x)+2*exp(10*x)+2*exp(7*x))/(exp(5*x)+exp(2*x))
    VVK> to exp(5*x)^2+2*exp(5*x)

    VVK>   The patch modifies some signatures of functions, which
    VVK> however are not advertised as user interface. A footprint on
    VVK> the performance with expressions without exponents shall not be
    VVK> really noticeable.

    VVK>   If the approach is suitable I can add a similar behaviour for
    VVK> powers, then the simplification

    VVK>   (a^(2x)-1)/(a^x-1) to a^x+1

    VVK>   will work as well.

    VVK>   Some tests shall be added for the final version of the patch.

    VVK>   Once this will be working, shall we add the automatic
    VVK> simplification (a^b)^c=a^(b*c) for suitable cases as well?

    VVK>   Best wishes, Vladimir -- Vladimir V. Kisil
    VVK> http://www.maths.leeds.ac.uk/~kisilv/ Book: Geometry of Mobius
    VVK> Transformations http://goo.gl/EaG2Vu Software: Geometry of
    VVK> cycles http://moebinv.sourceforge.net/ Jupyter:
    VVK> https://github.com/vvkisil/MoebInv-notebooks
>>>>> On Thu, 9 Apr 2020 01:51:09 +0200, "Richard B. Kreckel"
    VVK> <kreckel at in.terlu.de> said:

    RK> Hi Vladimir!

    RK> On 06.04.20 14:34, Vladimir V. Kisil wrote:
    >>> Coming back to our previous discussion (with a long history) on
    >>> the power law (e^x)^a=e^(x*a). I am attaching a patch which does
    >>> not break the automatic simplification exp(x)/exp(x)=1.

    RK> Your new patch is much better since it doesn't break any
    RK> existing test suite.

    RK> Playing around with it, it still seems to raise some fundamental
    RK> questions: What justifies treating exp(x)^a fundamentally
    RK> different than any other (b^x)^a with a (positive) base b? With
    RK> the patch, there seems to be this discrimination: exp(x)^5 is
    RK> rewritten to exp(5*x) but (b^x)^5 is _not_ rewritten to b^(5*x).

    RK> It's a nice pastime to fancy consequences of this. Let y=b^x,
    RK> then normal((y^2-1)/(y+1)) returns b^x-1. But if y=exp(x), the
    RK> patch prevents the normalization to exp(x)-1. Ugh.

    RK> Or, consider this gedankenexperiment: If we didn't have exp(x)
    RK> as a function but instead a symbol e, would it be justified to
    RK> have special re-writing rules for (e^x)^a but not for (b^x)^a?
    RK> I'm not sure...

    RK> Best wishes, -richy.  -- Richard B. Kreckel
    RK> <https://in.terlu.de/~kreckel/>

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