Transcendental Functions on Continued Fractions

TL;DR

This paper presents a modified algorithm for continued fraction arithmetic that avoids infinite loops and combines it with the spigot algorithm to compute CF expansions of exponential, logarithmic, and trigonometric functions, implemented in Haskell.

Contribution

It introduces a new, loop-free continued fraction arithmetic algorithm and integrates it with the spigot algorithm for computing special function expansions.

Findings

The modified algorithm avoids infinite loops in CF arithmetic.

Successful implementation of CF expansions for exponential, logarithmic, and trigonometric functions.

Demonstrated effectiveness of the combined approach in Haskell.

Abstract

Gosper developed an algorithm for performing arithmetic operations on continued fractions (CFs), getting a CF as the result. Straightforward implementation of the algorithm leads to infinite loops on some inputs. Here we present a modified version of the arithmetic algorithm and prove that it avoids all difficulties with infinite loops. We then combine CF arithmetic with the spigot algorithm to compute the CF expansions of exponential, logarithmic, and trigonometric functions of CFs. We have implemented these algorithms in Haskell.