General n-dimensional Ribando series for solid angle

Computes the normalized solid angle of a simplicial cone in arbitrary dimension by enumerating the multi-indices of the Ribando series via a recursive weak-composition generator. This is the engine that powers hypergeometric_series for $n \geq 4$.

Usage

hypergeometric_series_nd(V, max_terms = 1000, tol = 1e-10)

Arguments

V: A square $n \times n$ numeric matrix whose columns are the cone generators.
max_terms: Integer. Hard cap on the number of series terms accumulated before truncation. Default 1000. The number of terms at total degree $d$ grows as $\binom{d + N - 1}{N - 1}$ with $N = \binom{n}{2}$, so the cap binds quickly for large $n$.
tol: Numeric. Per-degree convergence tolerance. Default 1e-10.

Value

A list with components

solid_angle: Normalized solid angle in $[0, 1]$.
n_terms: Total number of terms accumulated.
converged: Logical flag.
associated_matrix: The associated matrix $M_n(C)$.

Details

Mathematical Formulation: The function computes the series: $$T_{\alpha} = \frac{|\det V|}{(4\pi)^{n/2}} \sum_{\mathbf{a} \in \mathbb{N}_0^N} \left[ \frac{(-2)^{|\mathbf{a}|}}{\mathbf{a}!} \prod_{i=1}^n \Gamma\left(\frac{1 + \text{deg}_i(\mathbf{a})}{2}\right) \right] \alpha^{\mathbf{a}}$$

Here $N = \binom{n}{2}$ is the number of pairwise dot products; $\mathbf{a} = (a_{12}, a_{13}, \dots, a_{n-1,n})$ is a multi-index of exponents; $|\mathbf{a}| = \sum_{j<k} a_{jk}$ is the total degree of the term; and $\text{deg}_i(\mathbf{a}) = \sum_{k \ne i} a_{ik}$ is the "degree" of vertex $i$ in the multigraph defined by $\mathbf{a}$.

Algorithmic Strategy: Unlike hardcoded implementations for $n=3$, this function dynamically generates the multi-indices $\mathbf{a}$ for increasing total degrees $d = 0, 1, 2, \dots$. It uses a recursive backtracking algorithm to find all weak compositions of $d$ into $N$ parts.

Performance and Limitations: Regarding convergence, the series converges absolutely if and only if the associated matrix $M_n(C)$ is positive definite, which typically means the cone vectors must not be too close to each other (the cone must not be too narrow); if this condition is violated, the series may produce meaningless results or fail to converge. In terms of computational complexity, the number of terms grows combinatorially with $n$ and the degree $d$, so for $n \ge 5$, calculating high-degree terms becomes computationally expensive. Convergence speed varies substantially: for cones that are nearly orthogonal (dot products close to 0), convergence is very fast, whereas for cones with high correlation between generators (even if positive definite), convergence can be slow and may require many terms to achieve high precision; in such cases, Monte Carlo methods might be more efficient.

References

Ribando, J. M. (2006). Measuring solid angles beyond dimension three. Discrete & Computational Geometry, 36(3), 479-487. doi:10.1007/s00454-006-1253-4

Aomoto, K. (1977). Analytic structure of Schlafli function. Nagoya Mathematical Journal, 68, 1-16. https://projecteuclid.org/euclid.nmj/1118786429

Examples

if (FALSE) { # \dontrun{
# 4D orthant: should yield 1/16 = 0.0625
hypergeometric_series_nd(diag(4))$solid_angle

# Cross-check the closed-form 3D formula against the general enumerator
V3 <- normalize_vectors(matrix(c(1, 0.1, 0.1,
                                 0.1, 1,   0.1,
                                 0.1, 0.1, 1), nrow = 3))
omega_formula <- solid_angle_3d(V3[, 1], V3[, 2], V3[, 3])
omega_series  <- hypergeometric_series_nd(V3)$solid_angle
c(formula = omega_formula, series = omega_series)
} # }