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Let $f, g \in \mathbb{R}[x]$ and $\deg f > \deg g$. I am looking for asymptotically fast and numerically stable algorithms for computing $f \bmod g$. In the applications intended, both $f, g$ are dense polynomials with double precision floating-point coefficients. But, for now, I'm more interested in the algorithms rather than the implementation. References for algorithms for computing GCD of numerical polynomials are also appreciated.

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Check the book by Dario Bini & Victor Pan: "Polynomial and Matrix computations, Volume 1: Fundamental algorithms", ISBN 0-8176-3786-9, Birkhäuser, 1994.

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