On the numerical characterization of the reachability cone for an essentially nonnegative matrix

Given an n×n real matrix A with nonnegative off-diagonal entries, the solution to , x0=x(0), t⩾0 is x(t)=etAx0. The problem of identifying the initial points x0 for which x(t) becomes and remains entrywise nonnegative is considered. It is known that such x0 are exactly those vectors for which the iterates x(k)=(I+hA)kx0 become and remain nonnegative, where h is a positive, not necessarily small parameter that depends on the diagonal entries of A. In this paper, this characterization of initial points is extended to a numerical test when A is irreducible: if x(k) becomes and remains positive, then so does x(t); if x(t) fails to become and remain positive, then either x(k) becomes and remains negative or it always has a negative and a positive entry. Due to round-off errors, the latter case manifests itself numerically by x(k) converging with a relatively small convergence ratio to a positive or a negative vector. An algorithm implementing this test is provided, along with its numerical analysis and examples. The reducible case is also discussed and a similar test is described.