An O(n squared) Algorithm for Testing the Sign Stability of an n x n Matrix.

An n x n real matrix A a sub ij is stable if each eigenvalue has negative real part, and sign stable or qualitatively stable if each matrix B having the same sign pattern as A is stable, regardless of the magnitudes of Bs entries. Sign stability is of special interest when A is the inter-action matrix of an ecological system, for then the magnitudes of the a sub ijs may be virtually impossible to determine. Starting from a characterization due to Quirk and Ruppert, and to Jeffries, an On squared algorithm is developed for testing the sign stability of A, and when A is properly presented that is reduced to Omaxn, number of nonzero entries of A. Part of the algorithm is a matching procedure whose extensions are of independent interest. An ALGOL program is included.