Middle and Ripple, fast simple O(lg n) algorithms for Lucas Numbers

TL;DR

This paper introduces efficient O(log n) algorithms for computing individual Lucas numbers, outperforming Fibonacci-based methods, and presents a recursive Lucas number algorithm with similar complexity.

Contribution

It provides novel fast iterative and recursive algorithms for Lucas numbers, leveraging their structure for improved computational efficiency.

Findings

O(log n) iterative Lucas number algorithm

Faster Fibonacci computation via Lucas numbers

Recursive Lucas number algorithm with O(log n) complexity

Abstract

A fast simple O(\log n) iteration algorithm for individual Lucas numbers is given. This is faster than using Fibonacci based methods because of the structure of Lucas numbers. Using a sqrt 5 conversion factor on Lucus numbers gives a faster Fibonacci algorithm. In addition, a fast simple recursive algorithm for individual Lucas numbers is given that is O(log n).