Weingarten calculus and the IntHaar package for integrals over compact matrix groups

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 Maplein 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. (C) 2019 Elsevier Ltd. All rights reserved.