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29 March 2024
 
  » arxiv » 1112.4014

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Note on fast division algorithm for polynomials using Newton iteration
Zhengjun Cao ; Hanyue Cao ;
Date 17 Dec 2011
AbstractThe classical division algorithm for polynomials requires $O(n^2)$ operations for inputs of size $n$. Using reversal technique and Newton iteration, it can be improved to $O({M}(n))$, where ${M}$ is a multiplication time. But the method requires that the degree of the modulo, $x^l$, should be the power of 2. If $l$ is not a power of 2 and $f(0)=1$, Gathen and Gerhard suggest to compute the inverse,$f^{-1}$, modulo $x^{lceil l/2^r ceil}, x^{lceil l/2^{r-1} ceil},..., x^{lceil l/2 ceil}, x^l$, separately. But they did not specify the iterative step. In this note, we show that the original Newton iteration formula can be directly used to compute $f^{-1},{mod},x^{l}$ without any additional cost, when $l$ is not a power of 2.
Source arXiv, 1112.4014
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