Golden and Alternating, fast simple O(lg n) algorithms for Fibonacci

TL;DR

This paper introduces two efficient O(log n) algorithms for computing Fibonacci numbers, compares them with existing methods, and discusses their applicability to Lucas numbers and data type overflow issues.

Contribution

It presents new simple recursive algorithms for Fibonacci and Lucas numbers with O(log n) complexity and provides a formula to estimate overflow limits.

Findings

Algorithms outperform previous methods in speed

Effective for large Fibonacci number computations

Highlights overflow risks in timing experiments

Abstract

Two very fast and simple O(lg n) algorithms for individual Fibonacci numbers are given and compared to competing algorithms. A simple O(lg n) recursion is derived that can also be applied to Lucas. A formula is given to estimate the largest n, where F_n does not overflow the implementation's data type. The danger of timing runs on input that is too large for the computer representation leads to false research results.