Time- and space-efficient evaluation of the complex exponential function using series expansion

TL;DR

This paper introduces a new algorithm for efficiently computing the complex exponential function using series expansion, combining modified binary splitting and Karatsuba methods for improved time and space performance.

Contribution

It presents a quasi-linear time and linear space algorithm for evaluating the complex exponential function, enhancing computational efficiency over traditional methods.

Findings

Achieves quasi-linear time complexity O(M(n)log(n)^2)

Uses modified binary splitting for hypergeometric series

Employs modified Karatsuba method for exponential evaluation

Abstract

An algorithm for the evaluation of the complex exponential function is proposed which is quasi-linear in time and linear in space. This algorithm is based on a modified binary splitting method for the hypergeometric series and a modified Karatsuba method for the fast evaluation of the exponential function. The time complexity of this algorithm is equal to that of the ordinary algorithm for the evaluation of the exponential function based on the series expansion: O(M(n)log(n)^2).