On Tue, Nov 18, 2014 at 5:57 AM, Bill Page <bill.p...@newsynthesis.org> wrote:
> On 17 November 2014 23:16, Ondřej Čertík <ondrej.cer...@gmail.com> wrote:
>> Hi Bill,
>>
>> Thanks for the clarification. So your point is that 2) is not
>> sufficient, that we really need two Wirtinger derivatives --- it's
>> just that one can be expressed using the other and a conjugate,
>> so perhaps CAS can only return one, but a chain rule needs
>> modification and probably some other derivatives handling as
>> well. I need to think about this harder.
>>
>
> Yes, that is a good summary. My tentative conclusion was that we
> could implement just one (Wirtinger) derivative, a modified chain rule
> and a sufficiently strong conjugate operation. This derivative is the
> same as the usual derivative in the case of analytic functions but we
> would have to live with the fact that it is slightly different (factor
> of 1/2) for the case of common real derivatives of non-analytic
> functions such as abs. Introducing a factor of 2, such as in the case
> of the definition of the sign function seems like a small price to
> pay.
>
>> Here is a relation that I found today in [1] (see also the references
>> there), I don't know if you are aware of it:
>>
>> D f / D z = df/dz + df/d conjugate(z) * e^{-2*i*theta}
>>
>> Where Df/Dz is the derivative in a complex plane along the direction
>> theta (the angle between the direction and the x-axis) and df/dz and
>> df/d conjugate(z) are the two Wirtinger derivatives. This formula
>> holds for any function. So all the derivatives no matter which
>> direction lie on a circle of radius df/d conjugate(z) and center
>> df/dz.
>>
>> [1] Pyle, H. R., & Barker, B. M. (1946). A Vector Interpretation of
>> the Derivative Circle. The American Mathematical Monthly, 53(2), 79.
>> doi:10.2307/2305454
>
> http://phdtree.org/pdf/36421281-a-vector-interpretation-of-the-derivative-circle/
>
> Thank you. I was not aware of that specific publication. I think
> their geometric interpretation is useful.
>
>>
>> For CAS, one could probably just say that theta=0 in our definition,
>> and then everything is consistent, and we only have one derivative,
>> 2). The other option is to return both derivatives and make the
>> derivative Df/Dz of non-analytic function equal to the above formula,
>> i.e. depending on df/dz, df/d conjugate(z) and theta.
>
> I think you are overly focused on trying to define a derivative that
> reduces to the conventional derivative of non-analytic functions over
> the reals.
I've just been casually following this conversation, but I think it's
important that the derivative of abs(x) be sign(x) not 2*sign(x) or
1/2*sign(x).
If you use a different function, like f.wirtinger_derivative(), then
it doesn't matter so much.
David
>
>>
>> I need to think about the chain rule. I would simply introduce the
>> theta dependence into all formulas, as that gives all possible
>> derivatives and gives the exact functional dependence of all
>> possibilities. And then see whether we need to keep all formulas
>> in terms of theta, or perhaps if we can set theta = 0 for everything.
>>
>
> It is not clear to me how to use such as "generic" derivative in the
> application of the chain rule.
>
> Bill.
>
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