Evaluating parametric holonomic sequences using rectangular splitting

TL;DR

This paper adapts the rectangular splitting technique to efficiently evaluate parametric holonomic sequences, reducing computational complexity and improving performance in applications like gamma function evaluation.

Contribution

It introduces a novel adaptation of rectangular splitting for holonomic sequences with parameters, enhancing computational efficiency over naive methods.

Findings

Reduces evaluation complexity to O(n^{1/2}) operations

Performs better than naive and some faster algorithms in practical ranges

Demonstrates effectiveness with gamma function evaluation

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.