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Proceedings
Symposium on Simplicity in Algorithms (SOSA)

A Simple and Fast Algorithm for Computing the N-th Term of a Linearly Recurrent Sequence

Abstract

We present a simple and fast algorithm for computing the N-th term of a given linearly recurrent sequence. Our new algorithm uses O(M(d) log N) arithmetic operations, where d is the order of the recurrence, and M(d) denotes the number of arithmetic operations for computing the product of two polynomials of degree d. The state-of-the-art algorithm, due to Fiduccia (1985), has the same arithmetic complexity up to a constant factor. Our algorithm is simpler, faster and obtained by a totally different method. We also discuss several algorithmic applications, notably to polynomial modular exponentiation and powering of matrices.

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cover image Proceedings
Symposium on Simplicity in Algorithms (SOSA)
Pages: 118 - 132
Editors: King Valerie, University of Victoria, Canada and Le Hung Viet, University of Massachusetts, Amherst, Massachusetts, USA
ISBN (Online): 978-1-61197-649-6

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Published online: 7 January 2021

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