harold.cancellation_distance(F, G)

Computes the upper and lower bounds of the perturbation needed to render the pencil \([F-pI | G]\) rank deficient. It is used for assessing the controllability/observability degeneracy distance and hence for minimality assessment.

  • A (array_like) – Square input array (n x n)
  • B (array_like) – Input array (n x m)

  • upper2 (float) – Upper bound on the norm of the perturbation [dF | dG] such that [F+dF-pI | G+dG]` is rank deficient for some ``p.
  • upper1 (float) – A theoretically softer upper bound than upper2 for the same norm.
  • lower0 (float) – Lower bound on the norm given in upper2
  • e_f (complex) – Indicates the eigenvalue that renders [F+dF-pI | G+dG ] rank deficient
  • radius (float) – The perturbation with the norm bound upper2 is located within a disk in the complex plane whose center is on e_f and whose radius is bounded by this output.


Implements the upper bounds given in [1]


[1]D. Boley, Estimating the Sensitivity of the Algebraic Structure of Pencils with Simple Eigenvalue Estimates, DOI:10.1137/0611046