  The radius of regularity sometimes spelled as the radius of nonsingularity is
a measure providing the distance of a given matrix to the nearest singular one.
Despite its possible application strength this measure is still far from being
handled in an efficient way also due to findings of Poljak and Rohn providing
proof that checking this property is NP-hard for a general matrix. To handle
this we can either find approximation algorithms or making known bounds for
radius of regularity tighter. Improvements of both have been recently shown by
Hartman and Hladik (doi:10.1007/978-3-319-31769-4\_9) utilizing relaxation to
semidefinite programming. These approaches consider general matrices without or
with just mild assumptions about the original matrix. This work explores a
process of regularity radius analysis and identifies useful properties enabling
easier estimation of the corresponding radius values based on utilization of
properties of special class of considered matrices. At first, checking
finiteness of regularity radius is shown to be a polynomial problem along with
determining a maximal bound on number of nonzero elements of the matrix to
obtain infinite radius. Further, relationship between maximum (Chebyshev) norm
and spectral norm is used to construct new bounds for the radius of regularity.
A new method based on Jansson-Rohn algorithm for testing regularity of an
interval matrix is presented which is not a priory exponential along with
numerical experiments. For a situation where an input matrix has a special
form, several results are provided such as exact formulas for several special
classes of matrices, e.g., for totally positive and inverse non-negative, or
approximation algorithms, e.g., rank-one radius matrices. For tridiagonal
matrices, an algorithm by Bar-On, Codenotti and Leoncini is utilized to design
a polynomial algorithm to compute the regularity radius.
