In this paper we examine the positivity of Rv where R∈R ^N×N, v∈R^N, v≥0 with R=r(τA), r is a given (rational) function, A∈R^N×N and τ∈(0,∞). Here we mean by positivity the ordering w.r.t. an arbitrary order cone, which includes the classical entrywise positivity of vectors. Since the requirement R≥0 leads to very severe restrictions on r and τ we construct a positive cone ℘=℘(A) and determine τ*=τ*(r,℘) such that r(τA)℘⊂℘ for all τ∈[0,τ*]. Finally we give an example arising from applications to partial differential equations where our results explain actual computations much better than the general theory on R≥0. © 2004 Elsevier Inc. All rights reserved.