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For many applications, it is convenient to have good upper bounds for the norm of the inverse of a given matrix. In this paper, we obtain such bounds when A is a Nekrasov matrix, by means of a scaling matrix transforming A into a strictly diagonally dominant matrix. Numerical examples and comparisons with other bounds are included. The scaling matrices are also used to derive new error bounds for the linear complementarity problems when the involved matrix is a Nekrasov matrix. These error bounds can improve considerably other previous bounds.

Since nonlinear eigenvalue problems appear in many applications, the research on their proper treatment has drawn a lot of attention lately. Therefore, there is a need to develop computationally inexpensive ways to localize eigenvalues of nonlinear matrix-valued functions in the complex plane, especially eigenvalues of quadratic matrix polynomials. Recently, few variants of the Gersgorin localization set for more general eigenvalue problems, matrix pencils and nonlinear ones, were developed and investigated. Here, we introduce a more general approach to Gersgorin-type sets for nonlinear case using diagonal dominance, prove some properties of such sets and show how they perform on several problems in engineering.

In this paper, we estimate the Nekrasov diagonally dominant degree on the Schur complement of Nekrasov matrices. As an application, we offer new bounds of the determinant for several special matrices, which improve the related results in certain case. Further, we give an estimation on the infinity norm bounds for the inverse of Schur complement of Nekrasov matrices. Finally, we introduce new methods called Schur-based super relaxation iteration (SSSOR) method and Schur-based conjugate gradient (SCG) method to solve the linear equation by reducing order. The numerical examples illustrate the effectiveness of the derived result.

The Poisson Boltzmann equation (PBE) is a well-established implicit solvent continuum model for the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is still a challenge due to its exponential nonlinear term, strong singularity by the source terms, and distinct dielectric regions. In this paper, a new pseudo-transient approach is proposed, which combines an analytical treatment of singular charges in a two-component regularization, with an analytical integration of nonlinear term in pseudo-time solution. To ensure efficiency, both fully implicit alternating direction implicit (ADI) and unconditionally stable locally one-dimensional (LOD) methods have been constructed to decompose three-dimensional linear systems into one-dimensional (1D) ones in each pseudo-time step. Moreover, to accommodate the nonzero function and flux jumps across the dielectric interface, a modified ghost fluid method (GFM) has been introduced as a first order accurate sharp interface method in 1D style, which minimizes the information needed for the molecular surface. The 1D finite-difference matrix generated by the GFM is symmetric and diagonally dominant, so that the stability of ADI and LOD methods is boosted. The proposed pseudo-transient GFM schemes have been numerically validated by calculating solvation free energy, binding energy, and salt effect of various proteins. It has been found that with the augmentation of regularization and GFM interface treatment, the ADI method not only enhances the accuracy dramatically, but also improves the stability significantly. By using a large time increment, an efficient protein simulation can be realized in steady-state solutions. Therefore, the proposed GFM-ADI and GFM-LOD methods provide accurate, stable, and efficient tools for biomolecular simulations.

Recently, some error bounds for the linear complementarity problem with a weakly chained diagonally dominant B-matrix have been given under certain conditions. In this paper we provide a new and even optimal error bound for such linear complementarity problems under weaker conditions than those there, greatly improving the existing ones. Some numerical examples are performed to illustrate the sharpness and optimality of our new bound.

In this paper we start from the known lower bounds for minimal singular value of the matrices possessing certain kind of the diagonal dominance property, and derive Euclidean norm estimates of the inverses of several new subclasses of the block H-matrices. The motivation comes from applications where the matrix in question has distinguished block structure, which can be exploited to obtain useful information. An example arising from ecological modeling illustrates the benefits of the presented approach.

We investigate localization of spectra of infinite matrices that can act as linear operators from the Banach space l p , 1 ? p ? ∞ , to itself. To that end, we propose a method that is based on generalization of a strict diagonal dominance of infinite matrices that is adapted to l p and l q norms. Results are illustrated with several examples, and some interesting applications are discussed.

Absolute stability of Lurie direct control systems with time-varying coefficients and multiple nonlinearities is studied in this paper. Depending on the related methods, relative magnitude of the norm-unbounded coefficients was estimated. By knowledge of nonsingular M-matrix, the Lyapunov function was constructed, and some absolute stability criteria of this kind of systems were obtained. In addition, some simple and practical corollaries were derived from the theorem of Taussky. The main contribution of this paper is that the criteria which we introduced allow for the situation that the norms of coefficient matrices are unbounded. At last, the example shows the availability of the criteria.

In this paper, we obtain bounds for the spectral radius of the matrix l ω , r which is the iterative matrix of the generalized accelerated overrelaxation (GAOR) iterative method. Moreover, we present one convergence theorem of the GAOR method. Finally, we present two numerical examples.

The paper proves that Kogbetliantz method computes all singular values of a scaled diagonally dominant triangular matrix, which can be well scaled from both sides symmetrically, to high relative accuracy. Special attention is paid to deriving sharp accuracy bounds for one step, one batch and one sweep of the method. By a simple numerical test it is shown that the methods based on bidiagonalization are generally not accurate on that class of well-behaved matrices.

This paper addresses the problem of delay-dependent stability of 2D systems with time-varying delay subject to state saturation in the Roesser model. By introducing diagonally dominant matrices, new delay-dependent conditions are obtained in terms of linear matrix inequalities (LMIs) where the lower and upper delay bounds along horizontal and vertical directions, respectively, are known. numerical examples are provided to demonstrate the proposed results.

It is known that the diagonal-Schur complements of strictly diagonally dominant matrices are strictly diagonally dominant matrices [J.Z. Liu, Y.Q. Huang, Some properties on Schur complements of H-matrices and diagonally dominant matrices, Linear Algebra Appl. 389 (2004) 365-380], and the same is true for nonsingular H-matrices [J.Z. Liu, J.C. Li, Z.T. Huang, X. Kong, Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices, Linear Algebra Appl. 428 (2008) 1009-1030]. In this paper, we research the properties on diagonal-Schur complements of block diagonally dominant matrices and prove that the diagonal-Schur complements of block strictly diagonally dominant matrices are block strictly diagonally dominant matrices, and the same holds for generalized block strictly diagonally dominant matrices.

Recently, Behforooz [1], has introduced a new approach to construct cubic splines by using the integral values, rather than the usual function values at the knots. Also he has established different sets of end conditions for cubic and quintic splines by using the integral values, see Behforooz [2-4]. In this paper, we will use the same techniques of [1] to construct integro quintic splines. Although by using the integral values we expected to face a more complicated process for our construction, it turned out that the matrix of the system of linear equations that produces the parameters became a diagonally dominant matrix and the process became very simple. The selection of the required end conditions for our integro quintic splines will be discussed. The numerical examples and computational results illustrate and guarantee a higher accuracy for this approximation.

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.

In this paper, we present more investigations of the numerical solution of the 2D Bratu equation to obtain the second solution on the upper branch by Multigrid. Classical smoothers such as Gauss–Seidel and weighted Jacobi have proven ineffective for obtaining the second solution due to the loss of diagonal dominance and the presence of indefinite Jacobian system at some parameter values. In this paper, we modify the Multigrid algorithms by adding and combining some Krylov methods as smoothers to enhance the multigrid efficiency. Though the idea is not new but we could get new enhanced results compared to that presented by Hackbusch [W. Hackbusch, Comparison of different multi-grid variants for nonlinear equations, ZAMM Z. Angew. Math. Mech. 72 (1992) 148–151] and Washio and Oosterlee [T. Washio, C.W. Oosterlee, Krylov subspace acceleration for nonlinear multigrid schemes, Electron. Trans. Numer. Anal. 6 (1997) 271–290].

It is well-known [D. Carlson, T. Markham, Schur complements of diagonally dominant matrices, Czech. Math. J. 29 (104) (1979) 246–251, [1] ] that the Schur complement of a strictly diagonally dominant matrix is strictly diagonally dominant. Also, if a matrix is an H -matrix, then its Schur complement is an H -matrix, too [J. Liu, Y. Huang, Some properties on Schur complements of H -matrices and diagonally dominant matrices, Linear Algebra Appl. 389 (2004) 365–380, [8] ]. Recent research showed that the same type of statement holds for some special subclasses of H -matrices, see, for example, Liu et al. [J. Liu, Y. Huang, F. Zhang, The Schur complements of generalized doubly diagonally dominant matrices, Linear Algebra Appl. 378 (2004) 231–244]. The aim of this paper is to show that the proof of these results can be significantly simplified by using “scaling” approach as in Zhang et al. [F. Zhang et al., The Schur Complement and its Applications, Springer, New York, 2005] and to give another invariance result of this type.

This paper introduces a new efficient algorithm to obtain the accurate inverse of a given nonsingular matrix with maximal rate of convergence. The method is performed efficiently based on accurate inversion of the diagonally dominant matrix which is made from the given matrix, where it should be inverted. The presented algorithm can be employed in any numerical method.

In this paper, we present a novel Monte Carlo algorithm for obtaining the inverse of a given nonsingular matrix A with high accuracy. The method is performed by splitting the given matrix A = D - B where D is a strictly diagonally dominant matrix and it can be easily constructed based on matrix A and B = k ‖ A ‖ such that k can be selected as an arbitrary number greater than or equal to 1. The accuracy of presented algorithm is compatible with any numerical method. The theory of the method and its numerical results are also presented.

The theory of M-matrix and H-matrix plays an important role in many fields such as computational mathematics, mathematical physics, etc. Recently, some iterative algorithms for identifying H-matrices have been proposed, but these methods are all designed for implementing on a sequential computer, and may be ineffective for large scalar matrices. In this paper, based on the previous ideas, we propose a parallel iterative criterion implemented on the distributed-memory multiprocessor machines, which is convergent and costs less computational time than the earlier ones, we also present several numerical examples to show its effectiveness.

M-matrices and H-matrices play an important role in computational mathematics, mathematical physics, and theory of dynamical systems. Recently, some iterative methods have been proposed for identifying H-matrices. In this paper, we provide two new convergent methods, which need fewer iterations than the earlier ones, we also propose a corresponding non-parameter method for irreducible situation, and several numerical examples are given.

In this paper, we firstly establish a class of generalized AOR (GAOR) methods for solving a linear complementarity problem LCP(M, q), whose special case reduces to generalized SOR (GSOR) method. Then, some sufficient conditions for convergence of the GAOR and GSOR methods are presented, when the system matrix M is an H-matrix, M-matrix and a strictly or irreducible diagonally dominant matrix. Moreover, when M is an L-matrix, we discuss the monotone convergence of the new methods. Lastly, we report some computational results with the proposed methods.

A non-parameter iterative method for generalized diagonally dominant matrices (i.e. H-matrices) was proposed by Li et al. [L. Li, H. Niki, M. Sasanabc, A non-parameter criterion for generalized diagonally dominant matrices, Int. J. Comput. Math. 71 (1999) 267–275]. In this paper, we provide an improved algorithm by means of interleaved iteration, the new method is always convergent and needs fewer iterations than that of Li et al.; we also provide a corresponding algorithm for a general matrix, which decreases the wasteful computations when the given matrix is not an H-matrix. Several numerical examples for the effectiveness of the proposed algorithms are presented.

An algorithmic characterization of H-matrices was provided by Huang et al. [Comput. Math. Appl. 48 (2004) 1587-1601]. In this paper, we propose a new non-parameter method, which is always convergent in finite iterative steps for H-matrices and needs fewer number of iterations than that of Huang et al.; we also provide an improved algorithm for a general matrix, which decreases the wasteful computations when the given matrix is not an H-matrix. Several numerical examples for the effectiveness of the proposed algorithms are presented.

In this paper we extend and improve one algorithm to solve the problem of determining whether a matrix is generalized diagonally dominant or not. A new algorithm is proposed which is suited for reducible matrices and numerical evidence for the effectiveness of the proposed algorithm is presented. We also think about the properties of the positive diagonal matrix for some kind of reducible matrices.

In this paper, some practical sufficient conditions for convergence of AOR method for solving Ax = b with A is a doubly diagonally dominant matrix are presented, and results in [L.J. Cvekovic, D. Herceg, Convergence theory for AOR method, J. Comput. Math. 8 (2) (1990) 128–134] are improved. The results obtained are also generalized to H -matrices.

In this paper, we obtain bounds for the spectral radius of the matrix l ω , r , which is the iterative matrix of generalized accelerated overrelaxation (GAOR) iterative method. We consider a linear system Hy = f , where H is an n × n diagonally dominant real matrix, f is an n -real vector of known values and y is an n -real vector of unknown variables. Sufficient conditions for the convergence of GAOR method will be presented.

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. The numerical tests illustrate that the method works very well even for very ill-conditioned linear systems. y well even for very ill-conditioned linear systems.

Diagonal dominance of matrices and M-matrices (and H-matrices) play an important role in stability theory of dynamical systems. But it is difficult to determine the scaling matrix G=diag(g"1,...,g"n) (g"1,...,g"n>0) with AG being a strictly diagonally dominant matrix. In this paper, an convergent iterative algorithm for determine the scaling matrix for an irreducible M-matrix (H-matrix) A is presented.

The linear complementarity problem can be solved by modified AOR method given in [Appl. Math. Comput. 140 (2003) 53]. In the same paper the convergence was proved for the H-matrix case, using the estimation of spectral radius of corresponding matrix. In this paper we present the other possibility for obtaining convergence result. We use the estimation of maximum norm, and surprisingly, obtain convergence area which can be better. First, we consider SDD (strictly diagonally dominant) matrix case, and after that H-matrix case.

In this paper we present a new overlapped two-way parallel method for solving tridiagonal linear systems on a bulk-synchronous parallel (BSP) computer. We develop a theoretical study of the computational cost for this new method and we compare it with the experimental times measured on an IBM SP2 using switch hardware for the communications between processors. Using the cost model, we also obtain theoretical results on a CRAY T3E and we achieve a study on the optimum number of processors.

We extend and improve recently proposed algorithms to solve the problem of minimizing the distance from a given matrix to the cone of symmetric and diagonally dominant matrices with positive diagonal (SDD^+). We present a variety of criteria to select a subset of the supporting hyperplanes of the faces of SDD^+, and also of the polar cone (SDD^+)^0, to then apply Dykstra's alternating projection method. These selections reduce the number of projections and therefore reduce the required computational work. In all our new algorithms, the symmetry and the diagonal dominance of the obtained matrix are guaranteed. Preliminary numerical experiments indicate that some of the selection criteria produce a significant reduction in CPU time.

This note points out that the main results of [Appl. Math. Comput. 114 (2000) 255] is not true. We show that (1) Theorem 2.1 in [Appl. Math. Comput. 114 (2000) 255] is well known, (2) There are no nonsingular matrices satisfying the sufficient conditions for ensuring diagonally dominance given in Theorem 3.1, and (3) Theorem 4.1 for preconditioning p-cyclic matrices is not true. We also prove that p-cyclic matrices can be column diagonally preconditioned, with a special row permutation if required, to be row diagonally dominant under some assumptions.

In order to solve a linear complementarity problem (LCP(M,q)), when M is a 2-cyclic matrix, a class of modified AOR (MAOR) methods based on MAOR methods for solving linear system, whose special case reduces modified SOR (MSOR) method, is proposed. Some sufficient conditions for convergence of the MAOR and MSOR methods are given, when the system matrix M is an H-matrix, M-matrix and a strictly or irreducible diagonally dominant matrix. When M is an L-matrix, their monotone convergence are discussed.

This paper deals with two computationally efficient iterative methods for determining the response of nonclassically damped dynamic systems. Rigorous analytical convergence results related to these iterative methods are provided. Sufficient conditions under which these two iterative schemes are convergent are derived. Three different kinds of damping matrices, namely (i) strongly diagonally dominant, (ii) irreducible and weakly diagonally dominant, and (iii) symmetric and positive definite damping matrices, are considered. Asymptotic rates of convergence are discussed. Theoretical results are illustrated with numerical examples that show vastly improved rates of convergence when compared to earlier iterative schemes.

The recent application of the diagonal dominance in the development of the optimization algorithms in the wireless sensor networks design, has been done by Yuan and Yu (2006) [14] , extended in Yu et al. (2006) [9] , and surveyed in Machado and Tekinay [11] . In this paper, we will use the concept of generalized diagonal dominance, to improve the obtained results regarding the power control game, in three directions. We also discuss the applicability of such improvements.

For symmetric positive definite matrices with weak diagonal dominance, the algebraic sense of smooth errors is discussed and convergence of algebraic multigrid (AMG) methods is proved. It is found that the results of AMG for symmetric positive definite M-matrices with weak diagonal dominance also hold without the M-matrix assumption. The key result is the new formula x^[email protected]?ia"i"[email protected]?j [email protected]"[email protected]"j|a"i"j|+|a"j"i|x^2"[email protected][email protected]?j [email protected]"[email protected]"j|a"i"j|x"[email protected]"[email protected]"ia"i"j|a"i"j|x"j^2.