The distance between regularity and strong regularity ∗

In this paper we give very sharp bounds for the distance between regularity and strong regularity. The solution of this problem uses a general matrix analogue [18] of the Perron-Frobenius Theory. This theory has been graded as a challenge problem for future research by the International Linear Algebra Society [7]. An interval matrix [A] ∈ IIMn(IR) is called regular, if every A ∈ [A] is nonsingular, whereas [A] is called strongly regular, if M := mid ( [A] ) ∈ Mn(IR) is nonsingular and ρ(M−1 ·rad ( [A] ) < 1. Strong regularity implies regularity. Consider a system of linear equations, the data of which are afflicted with tolerances. The solution complex Σ ( [A], [b] ) := {x ∈ IR ∣∣ ∃ A ∈ [A] ∃ b ∈ [b] : Ax = b } is bounded if [A] is regular. Self-validating algorithms provide methods to compute an inclusion of the solution complex. However, many of those, e.g. the methods based on preconditioning and the Krawczyk operator, require [A] to be strongly regular [9], [11], [16]. This raises the question: “How far is strong regularity from regularity?” More precisely, let matrices A and nonnegative ∆ be given and define ρsreg(A, ∆) := sup{ r ∈ IR ∣∣ [A− r ·∆, A + r ·∆] strongly regular } and ρreg(A, ∆) := sup{ r ∈ IR ∣∣ A− r ·∆, A + r ·∆] regular }, where the values may range within [0,∞]. Then the question is: are there finite bounds for the ratio ρreg(A, ∆)/ρsreg(A, ∆) independent on A and ∆ and only depending on the dimension? And if so, how sharp are the bounds? In this note we present an analysis of this question and, up to a small constant factor, a complete answer.