Solid angle via the tridiagonal Fitisone-Zhou series

Computes the normalized solid angle of a simplicial cone whose Gram matrix \(V^\top V\) is tridiagonal, using the simplified series of Fitisone & Zhou (2023, Theorem 4.1, equation 23). The reduction from \(\binom{n}{2}\) to \(n - 1\) multi-index variables makes this backend asymptotically faster than the general hypergeometric_series when the structure is available.

Usage

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

Arguments

V

A square \(n \times n\) numeric matrix whose columns are the cone generators. Need not be unit-normalized; columns are rescaled internally.

max_terms

Integer. Hard cap on the number of series terms accumulated before truncation. Default 1000.

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, TRUE when the per-degree contribution fell below tol before max_terms was reached.

beta

Numeric vector of length \(n - 1\) with the consecutive dot products \(\beta_i = v_i \cdot v_{i+1}\).

is_tridiagonal

Logical flag, TRUE when \(V^\top V\) passed the tridiagonal test inside the function.

Details

For tridiagonal \(V^\top V\) the Ribando series collapses to $$\Omega(C) = \frac{|\det V|}{(4\pi)^{n/2}} \sum_{b \in \mathbb{N}_0^{n-1}} \frac{(-2)^{|b|}}{b!}\, \Gamma\!\left(\frac{1 + b_1}{2}\right) \prod_{i=2}^{n-1} \Gamma\!\left(\frac{1 + b_{i-1} + b_i}{2}\right) \Gamma\!\left(\frac{1 + b_{n-1}}{2}\right)\, \beta^{b},$$ with \(\beta_i = v_i \cdot v_{i+1}\). By Theorem 4.1 of Fitisone & Zhou (2023), tridiagonality of \(V^\top V\) implies positive definiteness of the associated matrix \(M_n(C)\), so the series always converges absolutely.

If the input does not satisfy the tridiagonal test, the function emits a warning but still iterates; the result is then a controlled approximation rather than the true solid angle and should not be trusted.

References

Fitisone, A., & Zhou, Y. (2023). Solid angle measure of polyhedral cones. arXiv:2304.11102 (math.CO). Theorem 4.1 and equation (23). https://arxiv.org/abs/2304.11102

See also

is_tridiagonal for the tridiagonal predicate; create_tridiagonal_cone for a constructor of tridiagonal generators from given consecutive angles; hypergeometric_series, hypergeometric_series_nd for the general series; solid_angle_decomposition for the decomposition route; compute_solid_angle for the dispatcher.

Examples

if (FALSE) { # \dontrun{
# 4D cone with only consecutive dot products non-zero
v1 <- c(1, 0, 0, 0)
v2 <- c(0.8, 0.6, 0, 0)
v3 <- c(0, 0.6, 0.8, 0)
v4 <- c(0, 0, 0.7, 0.7) / sqrt(0.98)
V  <- cbind(v1, v2, v3, v4)
res <- tridiagonal_series(V, max_terms = 500)
res$solid_angle
res$is_tridiagonal     # TRUE
} # }