Characterizations and stable tests for the Routh-Hurwitz conditions and for total positivity

Given a polynomial of degree n, a test of O(n(2)) elementary operations and growth factor 1 is presented in order to check the Routh-Hurwitz conditions. This optimal growth factor guarantees that the test presents better stability properties than other known tests. We also present a test of O(n(3)) elementary operations and growth factor q in order to check if a matrix is strictly totally positive. Finally, totally positive matrices are characterized by their symmetric-triangular decompositions. (C) 2003 Elsevier Inc. All rights reserved.