harold.cancellation_distance¶
-
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.
Parameters: - A (array_like) – Square input array (n x n)
- B (array_like) – Input array (n x m)
Returns: - 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 one_f
and whose radius is bounded by this output.
Notes
Implements the upper bounds given in [1]
References
[1] D. Boley, Estimating the Sensitivity of the Algebraic Structure of Pencils with Simple Eigenvalue Estimates, DOI:10.1137/0611046