*Correspondence: [email protected] Department of Mathematics, National Taiwan Normal University, 88 Sec. 4, Ting Chou Road, Taipei, 11677, Taiwan Abstract Stability of general equilibrium is usually depicted by a dynamic process of price adjustment which makes the flow of prices eventually come to rest at certain prices, so that the supply and demand of every commodity tend to equal each ...

We define a matrix concept we call factor width. This gives a hierarchy of matrix classes for symmetric positive semidefinite matrices, or a set of nested cones. We prove that the set of symmetric matrices with factor width at most two is exactly the class of (possibly singular) symmetric H-matrices (also known as generalized diagonally dominant matrices) with positive diagonals, H+. We prove b...

We show that Laplacian and symmetric diagonally dominant (SDD) matrices can be well approximated by linear-sized sparse Cholesky factorizations. Specifically, n × n matrices of these types have constant-factor approximations of the form LL , where L is a lowertriangular matrix with O(n) non-zero entries. This factorization allows us to solve linear systems in such matrices in O(n) work and O(lo...

We examine stochastic dynamical systems where the transition matrix, Φ, and the system noise, ΓQΓ T , covariance are nearly block diagonal. When H TR−1H is also nearly block diagonal, where R is the observation noise covariance and H is the observation matrix, our suboptimal filter/smoothers are always positive semidefinite, and have improved numerical properties. Applications for distributed d...

This work introduces a new perturbation bound for the L factor of the LDU factorization of (row) diagonally dominant matrices computed via the column diagonal dominance pivoting strategy. This strategy yields L and U factors which are always well-conditioned and, so, the LDU factorization is guaranteed to be a rank-revealing decomposition. The new bound together with those for the D and U facto...

Semidefinite programs (SDPs) are standard convex problems that frequently found in control and optimization applications. Interior-point methods can solve SDPs polynomial time up to arbitrary accuracy, but scale poorly as the size of matrix variables number constraints increases. To improve scalability, be approximated with lower upper bounds through use structured subsets (e.g., diagonally-dom...

A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. ...

Generalized diagonal dominance, a specific property of the matrix, can be very useful in various fields of the applied linear algebra, as a tool for discovering and proving new results. First, it gives possibility to be able to define different subclasses of H-matrices (papers [3], [11]). Second, it suggests a way to deduce and prove new theorems on the eigenvalue localization (papers [2], [4],...

We consider linear systems of algebraic equations Su = f with tridiagonal interval matrix S and interval vector f An interval version of the sweep method allows us to find an interval vector u = ( u t , u 2 , . . . , u n ) T that contains the united set of solutions of the system. In the paper we present estimates of the absolute value and the width of the intervals ui, i = 1, 2 , . . . , ~z ti...

Standard preconditioners, like incomplete factorizations, perform well when the coeecient matrix is diagonally dominant, but often fail on general sparse matrices. We experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices. We target highly indeenite, nonsymmetric problems which cause dii...