We prove tight bounds for the ∞-norm of the inverse of a symmetric, diagonally dominant positive matrix J ; in particular, we show that ‖J‖∞ is uniquely maximized among all such J . We also prove a new lower-bound form of Hadamard’s inequality for the determinant of diagonally dominant positive matrices and an improved upper bound for diagonally balanced positive matrices. Applications of our r...

The explicit SPIKE algorithm applies to narrow banded linear systems which are strictly diagonally dominant by rows. The parallel bottleneck is the solution of the so-called reduced system which is block tridiagonal and strictly diagonally dominant by rows. The reduced system can be solved iteratively using the truncated reduced system matrix as a preconditioner. In this paper we derive a tight...

A method is presented to make a given matrix strictly diagonally dominant as much as possible based on Jacobi rotations in this paper. The strictly diagonally dominant rows are used to build a preconditioner for some iterative method. Roundoff error analysis of the method is also given. The numerical tests illustrate that the method works very well even for very ill-conditioned linear systems. ...

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 in...

We present a linear-system solver that, given an n-by-n symmetric positive semi-definite, diagonally dominant matrix A with m non-zero entries and an n-vector b, produces a vector ~x within relative distance of the solution to Ax = b in time O(m log(n f (A)= )), where f (A) is the log of the ratio of the largest to smallest non-zero eigenvalue of A. In particular, log( f (A)) = O(b log n), wher...

Systems of linear differential equations with constant coefficients, as well as Lotka–Volterra equations, with delays in the off–diagonal terms are considered. Such systems are shown to be asymptotically stable for any choice of delays if and only if the matrix has a negative weakly dominant diagonal.

This paper presents an indirect adaptive control scheme of continuous-time systems. The estimated plant model is controllable and then the adaptive scheme is free from singularities. Such singularities are avoided through a modification of the estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be nonsingular. That property is achieved by ensuring that the ...

We present a new O(n 3) algorithm which computes the SVD of a weakly diagonally dominant M-matrix to high relative accuracy. The algorithm takes as an input the offdiagonal entries of the matrix and its row sums.