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arXiv · 2302.04299

Beating binary powering for polynomial matrices

Alin Bostan·Vincent Neiger·Sergey Yurkevich
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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 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-finite sequence of polynomials, and modular exponentiation to the power $N$ for bivariate polynomials.

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Alin Bostan, Vincent Neiger, Sergey Yurkevich. 2023-02-08. Beating binary powering for polynomial matrices. https://arxiv.org/abs/2302.04299

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