Beating binary powering for polynomial matrices

TL;DR

This paper introduces a novel algorithm that computes the Nth power of polynomial matrices with purely linear complexity in N, surpassing traditional binary powering methods, even without FFT.

Contribution

It presents a new approach leveraging linear differential equations to achieve linear complexity for polynomial matrix powering, applicable to related polynomial and sequence problems.

Findings

Achieves purely linear complexity in N for polynomial matrix powering

Applicable to C-finite sequences and bivariate polynomial exponentiation

Provides algorithms that outperform binary powering without FFT

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.