Weingarten calculus and the IntHaar package for integrals over compact matrix groups. (March 2021)

• Record Type: Journal Article
• Authors: Ginory, Alejandro; Kim, Jongwon
• Is Part Of: Journal of symbolic computation. Volume 103(2021)
• Page Start: 178
• Page End: 200
• Publication Date: 2021-03
• Subjects: Weingarten calculus -- Haar integration -- Compact matrix groups -- Maple program; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.12.003
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this paper, we present a uniform formula for the integration of polynomials over the unitary, orthogonal, and symplectic groups using Weingarten calculus. From this description, we further simplify the integration formulas and give several optimizations for various cases. We implemented the optimized Weingarten calculus in Maple in a package called IntHaar for all three compact groups. Here we will discuss its functions, provide instructions for the package, and produce some examples of computed integrals.