Computation of the Schläfli Function

The Schläfli function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{n}(x)$ </tex-math></inline-formula> allows to compute volume of a regular <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n-1)$ </tex-math></inline-formula>-dimensional spherical simplex of the dihedral angle <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\alpha =$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f_{n}(x)$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\geq 4$ </tex-math></inline-formula> in the frequently used in applications interval <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x\in [n-1,n+1]$ </tex-math></inline-formula>. The computation is based on the Chebyshev approximation of the function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q_{n}(x)$ </tex-math></inline-formula>, 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.