We present a transformation for stochastic matrices and analyze the effects of using it in stochastic comparison with the strong stochastic (st) order. We show that unless the given stochastic matrix is row diagonally dominant, the transformed matrix provides better st bounds on the steady state probability distribution.

This paper analyzes a novel method for constructing preconditioners for diagonally-dominant symmetric positivedefinite matrices. The method discussed here is based on a simple idea: we construct M by simply dropping offdiagonal nonzeros from A and modifying the diagonal elements to maintain a certain row-sum property. The preconditioners are extensions of Vaidya’s augmented maximum-spanning-tre...

This work introduces new relative perturbation bounds for the LDU factorization of (row) diagonally dominant matrices under structure-preserving componentwise perturbations. These bounds establish that if (row) diagonally dominant matrices are parameterized via their diagonally dominant parts and off-diagonal entries, then tiny relative componentwise perturbations of these parameters produce ti...

A new criterion for identifying generalized diagonally dominant matrices (GDDM, or H -matrices) was given in [K. Ojiro, H. Niki, M. Usui, A note on a new criterion for the H -matrix property, J. Comput. Appl. Math. 150 (2003) 293–302], where the criterion is effective when the given matrix is irreducible. However, when the matrix is reducible, it may lead to a false answer. In this work, to ove...

The most important property of the class of H-matrices, in the sense of applications presented here, is the following: every H-matrix can be diagonally scaled into a strictly diagonally dominant form. When the scaling matrix is chosen to be a diagonal matrix of the special form, we are able to characterize the corresponding subclass of H-matrices by relatively simple conditions. Precisely this ...

In this paper we consider the parallel generalized SAOR iterative method based on the generalized AOR iterative method presented by James for solving large nonsingular system. We obtain some convergence theorems for the case when coefficient matrix is a block diagonally dominant matrix or a generalized block diagonal dominant matrix. A numerical example is given to illustrate to our results. Cr...

Although kernel approximation methods have been widely applied to mitigate the O(n3) cost of n×n matrix inverse in Gaussian process methods, they still face computational challenges. The ‘residual’ between covariance and approximating component is often discarded as it prevents reduction. In this paper, we propose a computationally efficient approach that achieves better efficiency, O(mn2), com...

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...

Given a matrix family, it is interesting to know whether some important properties or structures of the class of matrices are inherited by their submatrices or by the matrices associated with the original matrices. It is known that the principal submatrices and the Schur complements of positive semidefinite matrices are positive semidefinite matrices; the same is true of M-matrices, H-matrices,...