# Evaluating parametric holonomic sequences using rectangular splitting

Author: Fredrik Johansson

Published version (PDF)

Presentation slides (PDF)

*ISSAC 2014* pp. 256-263, DOI: 10.1145/2608628.2608629

arXiv preprint: http://arxiv.org/abs/1310.3741, published October 15, 2013.

## Abstract

We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the $n$-th term in a recurrent sequence of suitable type using $O(n^{1/2})$ "expensive" operations at the cost of an increased number of "cheap" operations.

Rectangular splitting has little overhead and can perform better than either naive evaluation or asymptotically faster algorithms for ranges of $n$ encountered in applications. As an example, fast numerical evaluation of the gamma function is investigated. Our work generalizes two previous algorithms of Smith.

## Implementation

Fast implementations of rising factorials and the gamma function are available as part of the open source Arb library. Some of the alternative implementations shown in the benchmarks were implemented using Arb but are not currently part of the library.

*Last updated July 31, 2014. Contact: fredrik.johansson@gmail.com.*

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