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.) Lmatrices 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.

A Fixed-Point Policy-Iteration-Type Algorithm for Symmetric Nonzero-Sum Stochastic Impulse Control Games

“…To perform the analysis, we impose assumptions on the discretization scheme used on the system of QVIs and the discrete admissible strategies. These naturally generalize those of the impulse control case [4] and admit graph-theoretic interpretations in terms of weakly chained diagonally dominant (WCDD) matrices and their recently introduced matrix sequences counterpart [7]. We establish a clear parallel between these discrete type assumptions, the behaviour of the players and the Verification Theorem.…”

“…Briefly, (ϕ 1 , ϕ 2 ) is admissible if it gives well-defined payoffs for all x ∈ R, X ∞ has finite moments and, although each player can intervene immediately after the other, infinite simultaneous interventions are precluded. 7 As an example, if the running payoffs have polynomial growth, the "never intervene strategies" ϕ 1 = ϕ 2 = (∅, ∅ → R) are admissible and the game can be played.…”

“…index of contraction conA) by computing for each state the least length that needs to be walked on graphA to reach a non-trouble one, and then taking the maximum over all states (more details in Appendix A). This recently introduced concept gives an equivalent charaterization of the WCDD property for a WDD matrix as conA < +∞, and can be efficiently checked for sparse matrices in O(|G|) operations [7]. On the other hand, if A is substochastic then con A < ∞ if and only if its spectral radius verifies ρ(A) < 1 (Theorem A.1.6).…”

High Order Bellman Equations and Weakly Chained Diagonally Dominant Tensors

“…Let A be a weakly diagonally dominant Z-tensor with nonnegative diagonals. Then, the following are equivalent:( i) A is a strong M-tensor.( ii) Zero is not an eigenvalue of A.( iii) A is weakly chained diagonally dominant (Definition 15).An analogous equivalence result for matrices was recently proved in [1]. Since a weakly irreducibly diagonally dominant tensor is a weakly chained diagonally dominant tensor (Lemma 17), the following is an immediate consequence:…”

“…In this work, we introduce and study the nonlinear problem find u ∈ R n such that min P ∈P A(P )u m−1 − b(P ) = 0 (1) where A(P ) is an m-order and n-dimensional real tensor, b(P ) is a real vector, P is a nonempty compact set, and the minimum is taken with respect to the coordinatewise order on R n (see (iii) in Section 5). If m = 2, then A(P ) is a square matrix and A(P )u m−1 ≡ A(P )u is the ordinary matrixvector product.…”