Efficient computation of the Zassenhaus formula

TL;DR

This paper introduces an efficient recursive method for computing the Zassenhaus formula, minimizing terms and computational resources, and extends its convergence domain in Banach algebras.

Contribution

A novel recursive procedure for high-order Zassenhaus formula computation that simplifies implementation and improves convergence in Banach algebras.

Findings

Reduces computational effort compared to previous algorithms

Provides a minimal-term linear combination of commutators

Extends convergence domain in Banach algebra settings

Abstract

A new recursive procedure to compute the Zassenhaus formula up to high order is presented, providing each exponent in the factorization directly as a linear combination of independent commutators and thus containing the minimum number of terms. The recursion can be easily implemented in a symbolic algebra package and requires much less computational effort, both in time and memory resources, than previous algorithms. In addition, by bounding appropriately each term in the recursion, it is possible to get a larger convergence domain of the Zassenhaus formula when it is formulated in a Banach algebra.