Computation of the Schläfli Function

The Schläfli function f_{n}(x) allows to compute volume of a regular (n-1) -dimensional spherical simplex of the dihedral angle 2\alpha = arcsec(x) and it has many applications. For example, it defines conjectured upper bounds on the sphere packing problem and the kissing number problem, and a lower bound on the mean-squared error in the quantizing problem. The function is defined recursively via a first-order non-linear differential relation, that makes it difficult to compute, especially for large values of n. Here we present a method for an accurate numerical computation of the Schläfli function f_{n}(x) for n\geq 4 in the frequently used in applications interval x\in [n-1,n+1] . The computation is based on the Chebyshev approximation of the function q_{n}(x) , which is related to the Schläfli function via a simple factor of an algebraic expression and regular in the interval. We also present the computation algorithm based on the method.