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Topic: Least-square optimization with a complex residual function
Replies: 4   Last Post: Oct 30, 2010 11:16 AM

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elgen

Posts: 7
Registered: 8/27/10
Re: Least-square optimization with a complex residual function
Posted: Oct 30, 2010 11:16 AM
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On 10-10-29 02:44 AM, kym@kymhorsell.com wrote:
> elgen<sket16@no.spam.hotmail.com> wrote:
> ...

>> How would I proceed to calculate its gradient? Would you mind being more
>> specific? What is "SoS"?

>
> Sum of Squares. A quite usual term when working in least squares.
>
> If you have sum_i (x_i^2+y_i^2) you are saying you can not compute the
> partial derrivatives wrt all the x_i and y_i?
>
> If not, I'm sorry, it's sounding like some kind of assignment
> and I'm in the habit of only giving minimal hints.
>


I am working on a project to optimize the time-harmonic electromagnetic
field, which is a complex quantity. As I didn't have complex analysis in
the undergraduate years, differentiation and integration involving
complex numbers do not make me feel very comfortable. I am slowing
picking up things along the way.

I get the idea that the conjugated inner product makes the residual
real. So the problem is to find the gradient of the real function with
respect to its complex argument z, i.e. x_i(z) and y_i(z). My feeling is
that I need to take the differentiation with respect to the real(z) and
imag(z).

Wait a second ... as the residual is a real function of a complex
argument, would the residual be analytic? Could I directly take the
derivative with respect to z without heeding its real and complex part?

Thank you for the previous hints. That has made my way to complex
analysis a little easier.


elgen




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