Many of the currently popular ‘block algorithms’ are scalar algorithms in which the operations have been grouped and reordered into matrix operations. One genuine block algorithm in practical use is block LU factorization, and this has recently been shown by Demmel and Higham to be unstable in general. It is shown here that block LU factorization is stable if A is block diagonally dominant by c...

respectively. There are many iterative methods such as GMRES [7] and Bi-CGSTAB [9] algorithms for solving Eq. (1.1) which are more efficient than the Jacobi and Gauss-Seidel methods. However, when these methods are combined with the more efficient methods, for example as a preconditioner, can be quite successful. For example see [4, 6]. It has been proved that if A is a strictly diagonally domi...

We prove tight bounds for the ∞-norm of the inverse of symmetric diagonally dominant positive matrices. Applications include numerical stability for linear systems, bounds on inverses of differentiable functions, and the consistency of maximum entropy graph distributions from single samples.

Let A be a nonsingular M -matrix, and τ(A) denote its minimum eigenvalue. Shivakumar et al. [SIAM J. Matrix Anal. Appl., 17(2):298-312, 1996] presented some bounds of τ(A) when A is a weakly chained diagonally dominant M -matrix. The present paper establishes some new bounds of τ(A) for a general nonsingular M -matrix A. Numerical examples show that the results obtained are an improvement over ...

S-SDD class of matrices has been introduced in [1] as a generalization of Ostrowski class, known in the literature also as doubly diagonally dominant matrices. As it was shown in [2], this class can be characterized by the special form of its “scaling” into an SDD matrix. Due to this fact, we can use S-SDD matrices as a tool for measuring and exploiting the magnitude of diagonal dominance. How ...

Let A be a nonsingular M -matrix, and τ(A) denote its minimum eigenvalue. Shivakumar et al. [SIAM J. Matrix Anal. Appl., 17(2):298-312, 1996] presented some bounds of τ(A) when A is a weakly chained diagonally dominant M -matrix. The present paper establishes some new bounds of τ(A) for a general nonsingular M -matrix A. Numerical examples show that the results obtained are an improvement over ...

The aim of the presentation is to introduce some new localization techniques for generalized eigenvalues of a matrix pair obtained via famous Geršgorin theorem and its different generalizations. Special attention will be given to the techniques of computation and visual presentation of the obtained localization areas using MATLAB. Since the work that will follow involves much about nonnegative ...

Block Relative Gain (BRG) is a useful method for finding suitable pairings for block decentralized control. In this paper, we present some new algebraic properties of BRG and establish its relation with closed loop stability, controllability, block diagonal dominance and interactions. Contrary to published conjecture, we show that strongly interacting systems can have BRG that is nearly the ide...