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I'm struggling to understand Newton-Horner's method and to find information about it so pardon me if I seem a bit lost in expressing my question, my goal is to implement it in racket as a learning exercise.

Given a polynomial $P(x)$ of $n$ degree and that has $\bar x_n$ roots it is used to find said roots.

$$ x_{i + 1} = x_i - \frac{\frac{f(x_i)}{f'(x_i)}}{1 - \frac{f(x_i)}{f'(x_i)} \sum_{j=1}^{n}(\frac{1}{x_i - \bar x_j})} $$

I can identify Newton's Method, however I'm a bit lost in how exactly the whole method is applied.

I know that for the first iteration it is the result if using Newton's method with a supplied first guess value (because the Sum part is $0$). I've evaluated a sample polynomial $P(x) = 4x^5 - 3x^4 - 2x^2 +12x + 4$ for $x_0 = 0$ but on subsequent iterations the sum part is always zero.

I've searched quite extensively but all I find are the descriptions of Newton's Method and Horner's Method used in an independent manner I've also found lots of old homework exercises implemented in Mathematica or matlab, but none that describes the method thoroughly. If anyone has links to resources or can explain it I'd appreciate it a lot.

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    Maybe you can work through this example (if you have mathematica, you can actually run the example and change it). http://math.fullerton.edu/mathews/n2003/horner/HornerMod/Links/HornerMod2_lnk_1.html2012-11-22
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    I correct the most glaring error in the formula so that it looks like Newtons method. The sum is still wrong, j=i needs to be excluded. If the sum is over the most recent approximations, under consideration of $j\ne i$, then this is the Aberth-Ehrlich method for the simultaneous approximation of the roots. https://en.wikipedia.org/wiki/Aberth_method2014-05-13

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