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[求助] 为什么牛顿迭代法对复数初始值也有效?

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发表于 2019-8-8 13:26:17 | 显示全部楼层 |阅读模式

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我只知道泰勒展开,复数有洛朗展开,
有没有别的办法能够解释清楚,毕竟级数展开并不直观,
泰勒展开对于实数来说有几何意义,
但是洛朗展开有没有几何意义呢?
还有,维基百科似乎也提到了牛顿迭代法的复数迭代,但是我没看懂
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2019-8-8 13:34:56 | 显示全部楼层
Complex functions[edit]

Basins of attraction for x5 − 1 = 0; darker means more iterations to converge.
Main article: Newton fractal
When dealing with complex functions, Newton's method can be directly applied to find their zeroes.[8] Each zero has a basin of attraction in the complex plane, the set of all starting values that cause the method to converge to that particular zero. These sets can be mapped as in the image shown. For many complex functions, the boundaries of the basins of attraction are fractals.

In some cases there are regions in the complex plane which are not in any of these basins of attraction, meaning the iterates do not converge. For example,[9] if one uses a real initial condition to seek a root of x2 + 1, all subsequent iterates will be real numbers and so the iterations cannot converge to either root, since both roots are non-real. In this case almost all real initial conditions lead to chaotic behavior, while some initial conditions iterate either to infinity or to repeating cycles of any finite length.

Curt McMullen has shown that for any possible purely iterative algorithm similar to Newton's Method, the algorithm will diverge on some open regions of the complex plane when applied to some polynomial of degree 4 or higher. However, McMullen gave a generally convergent algorithm for polynomials of degree 3.[10]

https://en.wikipedia.org/wiki/Newton%27s_method
维基百科上面的这个是什么意思呢?
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2019-8-9 10:30:11 | 显示全部楼层
mathematica 发表于 2019-8-8 13:34
Complex functions[edit]

Basins of attraction for x5 − 1 = 0; darker means more iterations t ...

牛顿迭代法,为什么对复数初始值也有效果呢?
有什么几何意义呢?
我不能理解!
我感觉应该有几何意义的,但是
感觉没实数初始值迭代那么明显的几何意义!
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2019-8-9 12:33:35 | 显示全部楼层
mathematica 发表于 2019-8-9 10:30
牛顿迭代法,为什么对复数初始值也有效果呢?
有什么几何意义呢?
我不能理解!

话说
https://www.zhihu.com/question/31700172
这个不会是你吧
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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