In this paper, we present parallel alternating two-stage methods for solving linear system Ax=b, where A is a symmetric positive definite matrix. And we give some convergence results of these methods for nonsingular linear system. Keywords—alternating two-stage, convergence, linear system, parallel.

We prove that it is NP-hard to compute the exact componentwise bounds on solutions of all the linear systems which can be obtained from a given linear system with a nonsingular matrix by perturbing all the data independently of each other within prescribed tolerances.

Convergence properties of additive and multiplicative Schwarz iterations for solving linear systems of equations with a symmetric positive semidefinite matrix are analyzed. The analysis presented applies to matrices whose principal submatrices are nonsingular, i.e., positive definite. These matrices appear in discretizations of some elliptic partial differential equations, e.g., those with Neum...

In this paper, we analyze matrix dynamics for online linear discriminant analysis (online LDA). Convergence of the dynamics have been studied for nonsingular cases; our main contribution is an analysis of singular cases, that is a key for efficient calculation without full-size square matrices. All fixed points of the dynamics are identified and their stability is examined. © 2010 Elsevier Inc....

Let e and n be positive integers and S = {x1, . . . , xn} be a set of n distinct positive integers. The n × n matrix having eth power [xi, xj ] of the least common multiple of xi and xj as its (i, j)-entry is called the eth power least common multiple (LCM) matrix on S, denoted by ([S]). The set S is said to be gcd closed (respectively, lcm closed) if (xi, xj) ∈ S (respectively, [xi, xj ] ∈ S) ...

Let e and n be positive integers and S = {x1, . . . , xn} be a set of n distinct positive integers. The n × n matrix having eth power [xi, xj ] of the least common multiple of xi and xj as its (i, j)-entry is called the eth power least common multiple (LCM) matrix on S, denoted by ([S]). The set S is said to be gcd closed (respectively, lcm closed) if (xi, xj) ∈ S (respectively, [xi, xj ] ∈ S) ...

In this note, we bound the inverse of nonsingular diagonal dominant matrices under the infinity norm. This bound is always sharper than the one in [P.N. Shivakumar, et al., On two-sided bounds related to weakly diagonally dominant M-matrices with application to digital dynamics, SIAM J. Matrix Anal. Appl. 17 (2) (1996) 298–312]. c © 2008 Published by Elsevier Ltd

In this paper we compare two recently proposed methods, FGMRES 5] and GMRESR 7], for the iterative solution of sparse linear systems with an unsymmetric nonsingular matrix. Both methods compute minimal residual approximations using preconditioners, which may be diierent from step to step. The insights resulting from this comparison lead to better variants of both methods.

Redu tions to polynomial matrix multipli ation are given for some lassi al problems involving a nonsingular input matrix over the ring of univariate polynomials with oeÆ ients from a eld. High-order lifting is used to ompute the determinant, the Smith form, and a rational system solution with about the same number of eld operations as required to multiply together two matri es having the same d...

The doubling algorithms are very efficient iterative methods for computing the unique minimal nonnegative solution to anM -matrix algebraic Riccati equation (MARE). They are globally and quadratically convergent, except for MARE in the critical case where convergence is linear with the linear rate 1/2. However, the initialization phase and the doubling iteration kernel of any doubling algorithm...