Extended diagnostic summary for a polyhedral cone

Computes spectral diagnostics beyond the basic print: condition number of the associated matrix, eigenvalue spread, a five-level stability classification, and a method-selection rationale that records the discriminating quantity (typically $\kappa(M_n(C))$) used by the dispatcher.

Usage

# S3 method for class 'cone_diagnosis'
summary(object, ...)

Arguments

object: An object of class cone_diagnosis returned by diagnose_cone.
...: Currently unused; present for S3 generic compatibility.

Value

An object of class summary.cone_diagnosis; a list with components

dimension: Ambient dimension of the cone.
eigenvalues: Eigenvalues of the associated matrix.
condition_number: Ratio of the largest to the smallest absolute eigenvalue, or Inf when the smallest is below .Machine$double.eps.
eigenvalue_spread: Difference between the largest and smallest eigenvalue.
stability: One of "excellent", "good", "moderate", "poor", "critical", derived from the condition number with thresholds at $10$, $10^2$, $10^6$, $10^{10}$.
method_rationale: Character string explaining the recommended backend in terms of the spectrum.
original: The original cone_diagnosis object.

Details

The condition number is computed on the absolute spectrum to remain meaningful when $M_n(C)$ is not positive semidefinite (in which case the dispatcher uses the decomposition route). The five stability bands are intentionally coarse and serve only to flag situations where the user should expect numerical sensitivity in the chosen backend.

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

Examples

if (FALSE) { # \dontrun{
V <- diag(4)
d <- diagnose_cone(V)
s <- summary(d)
print(s)
} # }