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I have a question concerning the branch of successive iteration methods (Newton, Runge-Kutta). I definitely know (or can read in Wikipedia) the implementation of these methods. But I was wondering about the precision in special case.

I have a function $$ D(\omega)=det(L_{m,n}(\omega)), \omega = \omega'-i\omega'' $$ and need to solve the equation $$ D(\omega)=0 $$ For that task I want to use simple fixed-point iteration method $$ \omega_{0}, \omega_{1} \mathrm{- given}\\ \omega_{n+1}=\alpha \omega_{n}+(1-\alpha)[\omega_{n}-\frac{D(\omega_{n})}{dD/d\omega|_{\omega_{n}}}]\\ dD/d\omega|_{\omega_{n}}=\frac{D(\omega_{n})-D(\omega_{n-1})}{\omega_{n}-\omega_{n-1}} $$ So, could someone tell me if this method is well converging or not.

I am asking because previously I've used this method only for real-valued variables and now I have complex-valued. This may lead to component entanglement while calculating the derivative.

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    $\begingroup$ How is your operator $L$? I don't think that the question can be completely answered in general. $\endgroup$ – nicoguaro Oct 18 '17 at 16:59
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    $\begingroup$ This can also be called a nonlinear eigenvalue problem (en.wikipedia.org/wiki/Nonlinear_eigenproblem), and there are other, likely better, methods for it than just using root-finding on the determinant (see mat.tuhh.de/forschung/rep/rep164.pdf) $\endgroup$ – Kirill Oct 18 '17 at 17:30
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    $\begingroup$ The problem setup needs to be fleshed out. It would appear that $\omega$ is a complex scalar parameter, but you've said nothing about how matrix(?) $L_{m,n}(\omega)$ depends on that parameter. A simple case might be that all entries are polynomial functions of $\omega$, from which it would follow that $D(\omega)$ inherits that polynomial dependence. So one interpretation of your problem could lead to examining whether your iteration method necessarily converges to a root of $D(\omega)$. $\endgroup$ – hardmath Oct 19 '17 at 2:23

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