HyperbolicInverseLangevin Method

High-accuracy inverse Langevin function: starts from Kröger's rational approximation (max. relative error ~0.28%) and refines it with a fixed number of Newton-Raphson steps on the exact equation L(x) = y, where L is the true Langevin function (not an approximation). Each Newton step roughly doubles the number of correct digits, so starting from ~0.28% (about 2-3 correct digits), 3 steps reach full double precision (~15-16 digits) everywhere except in the immediate vicinity of y = +-1, where L'(x) -> 0 and the problem becomes ill-conditioned regardless of solver.

public static double InverseLangevin(
	double y,
	int newtonSteps = 3
)