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Conference Papers Year : 2023

Beating binary powering for polynomial matrices

Abstract

The $N$th power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in $N$. When Fast Fourier Transform (FFT) is available, the resulting arithmetic complexity is \emph{softly linear} in $N$, i.e. linear in $N$ with extra logarithmic factors. We show that it is possible to beat binary powering, by an algorithm whose complexity is \emph{purely linear} in $N$, even in absence of FFT. The key result making this improvement possible is that the entries of the $N$th power of a polynomial matrix satisfy linear differential equations with polynomial coefficients whose orders and degrees are independent of $N$. Similar algorithms are proposed for two related problems: computing the $N$th term of a C-recursive sequence of polynomials, and modular exponentiation to the power $N$ for bivariate polynomials.
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Dates and versions

hal-03979664 , version 1 (08-02-2023)
hal-03979664 , version 2 (26-05-2023)

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  • HAL Id : hal-03979664 , version 2

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Alin Bostan, Vincent Neiger, Sergey Yurkevich. Beating binary powering for polynomial matrices. ISSAC 2023 - 48th International Symposium on Symbolic and Algebraic Computation, Jul 2023, Tromso, Norway. ⟨hal-03979664v2⟩
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