Title: A fast and stable test to check if a weakly diagonally dominant matrix is a nonsingular M-matrix,
Abstract: We present a test for determining if a substochastic matrix is convergent. By
establishing a duality between weakly chained diagonally dominant (w.c.d.d.)
L-matrices and convergent substochastic matrices, we show that this test can be
trivially extended to determine whether a weakly diagonally dominant (w.d.d.)
matrix is a nonsingular M-matrix. The test's runtime is linear in the order of
the input matrix if it is sparse and quadratic if it is dense. This is a
partial strengthening of the cubic test in [J. M. Peña., A stable test to
check if a matrix is a nonsingular M-matrix, Math. Comp., 247, 1385-1392,
2004]. As a by-product of our analysis, we prove that a nonsingular w.d.d.
M-matrix is a w.c.d.d. L-matrix, a fact whose converse has been known since at
least 1964. We point out that this strengthens some recent results on
M-matrices in the literature.