Fast computation of cubature formulae for the sphere

The near-uniform distribution of nodes on the surface of a sphere has found many uses in numerical integration, physics, chemistry, crystallography, and more recently in the capture, representation and reproduction of spatial audio. A popular solution posed by Fliege and Meyer treats nodes as charged particles that are constrained to lie on the surface of a sphere that undergo mutual repulsion. The potential energy contained within the system forms a cost function that is minimized by numerical optimization using simulated annealing followed by Limited memory-BFGS (L-BFGS). In this work the cost function's numerical gradient is replaced with an analytical gradient and it is shown that this is sufficient for a L-BFGS alone to achieve similar results. It is also proposed to limit iterations once the solution has converged beyond a threshold for use cases in mechanical designs with much higher error tolerances. A single-core C implementation on a modern machine shows up to 146 times faster convergence times compared with the Fliege implementation. In the limited precision case, all numbers of nodes under 1600 are calculated in under one minute and under 380 in under a second.