Associated matrix M_n(C) of a simplicial cone

Constructs the associated matrix $M_n(C)$ of Ribando (2006, Definition 1.4) for a simplicial cone with generator matrix V. This is the discriminant on which the convergence of the hypergeometric series rests: the series converges absolutely if and only if $M_n(C)$ is positive definite (Ribando 2006, Theorem 1.5).

Usage

compute_associated_matrix(V)

Arguments

V: A square $n \times n$ numeric matrix whose columns are the cone generators (need not be unit-normalized; the associated matrix is invariant under positive rescaling of each column).

Value

A symmetric $n \times n$ numeric matrix with ones on the diagonal and $-|v_i \cdot v_j|$ for the off-diagonal entries.

Details

The associated matrix is defined entry-wise as $$m_{ii} = 1, \qquad m_{ij} = -|\langle v_i, v_j \rangle|/(\|v_i\|\,\|v_j\|), \quad i \neq j.$$ Positive definiteness of this matrix is the necessary and sufficient condition for the absolute convergence of the Ribando series (Theorem 1.5). The dispatcher compute_solid_angle consults this criterion when selecting between the series, tridiagonal, and decomposition backends.

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

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

Examples

V <- cbind(c(1, 0), c(1, 1) / sqrt(2))
M <- compute_associated_matrix(V)
eigen(M)$values        # all positive: convergence guaranteed
#> [1] 1.7071068 0.2928932