Numerical Verification Methods for Spherical $t$-Designs

Abstract

The construction of spherical $t$-designs with $(t+1)^2$ points on the unit sphere $S^2$ in $\mathbb{R}^3$ can be reformulated as an underdetermined system of nonlinear equations. This system is highly nonlinear and involves the evaluation of a degree $t$ polynomial in $(t+1)^4$ arguments. This paper reviews numerical verification methods using the Brouwer fixed point theorem and Krawczyk interval operator for solutions of the underdetermined system of nonlinear equations. Moreover, numerical verification methods for proving that a solution of the system is a spherical $t$-design are discussed.