Diagnose a polyhedral cone and recommend a computation method

Inspects the geometric and spectral structure of a polyhedral cone and returns a diagnostic S3 object summarising linear independence, positive definiteness of the associated matrix, tridiagonal structure of the Gram matrix, eigenvalue spectrum, and the recommended backend for compute_solid_angle.

Usage

diagnose_cone(V)

Arguments

V

A numeric matrix whose columns are the cone generators.

Value

An object of class cone_diagnosis; a list with components

dimension

Integer ambient dimension.

linearly_independent

Logical, TRUE when the columns of V are linearly independent.

associated_matrix_PD

Logical, positive definiteness of the associated matrix \(M_n(C)\).

is_tridiagonal

Logical, tridiagonal structure of \(V^\top V\).

eigenvalues

Numeric vector of eigenvalues of \(M_n(C)\) in ascending order.

min_eigenvalue

Smallest eigenvalue.

suggested_method

Character string identifying the recommended backend for compute_solid_angle.

Details

The diagnostic logic encodes the dispatcher's decision tree explicitly so that the user can audit method selection without running the computation. For \(n \leq 3\) the closed-form formulas are always recommended. For \(n \geq 4\) the recommendation depends on the spectrum of the associated matrix \(M_n(C) = (m_{ij})\) with $$m_{ij} = \langle v_i, v_j \rangle / (\|v_i\|\, \|v_j\|),$$ positive definiteness of which is required for the convergence of Ribando's hypergeometric series (Ribando 2006, Theorem 1.5). When \(V^\top V\) is additionally tridiagonal the simplified series of Fitisone & Zhou (2023, Theorem 4.1) reduces the per-term cost from \(O(n^2)\) to \(O(n)\).

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

See also

print.cone_diagnosis, summary.cone_diagnosis, plot.cone_diagnosis for the S3 method triad; compute_solid_angle as the dispatcher this diagnostic advises; compute_associated_matrix, is_positive_definite, is_tridiagonal for the underlying spectral tests.

Examples

if (FALSE) { # \dontrun{
# Full lifecycle: constructor -> print -> summary -> plot
V <- diag(4)
d <- diagnose_cone(V)
print(d)
s <- summary(d)
print(s)
plot(d)
} # }