Timeline for Forcing a set of complex points to be closed under conjugation
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 23, 2013 at 14:12 | comment | added | Pierre Vuillemin | I evaluate $f(z)$, its gradient $\nabla f$ which opposite is used as a search direction $d = -\nabla f$ (in practice I estimate the hessian with BFGS). Then I look for the real step length $\alpha$ which minimizes $f(z + \alpha d)$ and will lead me to my next point in the algorithm. During this process, I might face points which are not composed of elements which are closed under conjugation, for instance I could have $3+i$ and $3-(1+\epsilon)i$. That is why I check each time the inputs of my function. | |
| Oct 23, 2013 at 13:49 | comment | added | Steven Landsburg | So suppose you evaluate $f(z,\overline{z})$ at the point where $z=3+i$ and $\overline{z}=3-i$. Then you go back and check to see whether $3+i$ and $3-i$ are still conjugate. But I can tell you in advance that $3+i$ and $3-i$ will be conjugate till the end of time. So what are you actually checking? | |
| Oct 23, 2013 at 13:39 | comment | added | Pierre Vuillemin | Maybe it would be more clear with the function itself. The function $f$, is given by $ f(z_1,\ldots,z_n,c_1,\ldots,c_n) = \sum_{i=1}^n \sum_{j=1}^n \frac{c_i c_j}{z_i+z_j}$. The $c_i$, $i=1,\ldots,n$ are also complex variables which come with their conjugate, if $z_i = \bar{z_j}$ then $c_i = \bar{c_j}$, $i\neq j$. | |
| Oct 23, 2013 at 13:26 | comment | added | Steven Landsburg | I'm afraid this makes even less sense now than it did before your clarification. | |
| Oct 23, 2013 at 13:12 | comment | added | Pierre Vuillemin | In fact, I meant that the function $f$ is a real valued function (by definition) of some complex variables $z$ and their complex conjugate $\bar{z}$, i.e. $f(z) = f(z,\bar{z})$. But in practice, $z$ and $\bar{z}$ are not separated entities and whenever the complex variables does not come with their complex conjugate, $f$ does not have the right meaning anymore, so I would like to force my variables to remain closed under conjugation. | |
| Oct 23, 2013 at 13:07 | review | Close votes | |||
| Oct 23, 2013 at 14:23 | |||||
| Oct 23, 2013 at 12:50 | comment | added | Steven Landsburg | Obviously, the fact that for each $i$ there exists a $j$ with $z_i=\overline{z}_j$ doesn't imply that the function is real valued. So I'm not at all sure what you mean to say. | |
| Oct 23, 2013 at 11:49 | history | edited | Pierre Vuillemin | CC BY-SA 3.0 |
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| Oct 23, 2013 at 11:38 | history | edited | Pierre Vuillemin | CC BY-SA 3.0 |
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| Oct 23, 2013 at 9:41 | review | First posts | |||
| Oct 23, 2013 at 9:47 | |||||
| Oct 23, 2013 at 9:25 | history | asked | Pierre Vuillemin | CC BY-SA 3.0 |