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In this paper, we introduce the class of diagonally dominant with respect to a given LMI region D ⊂ C matrices. They are shown to possess the analogues of well-known properties of (classical) diagonally dominant matrices, e.g. their spectra are localized inside the region D . Moreover, we show that in some cases, diagonal D -dominance implies ( D , D ) -stability (i.e. the preservation of matrix spectra localization under multiplication by a positive diagonal matrix).

In this paper, we present a direct parallel block W Z algorithm, named DPBWZA, for the solution of block tridiagonal toeplitz–block–toeplitz (TBT) linear system A x = f . The algorithm is based on the proposed block W Z factorization of the coefficient matrix A . Existence of the block W Z factorization for block tridiagonal TBT block diagonally dominant matrix is proved. Error analysis of the parallel algorithm DPBWZA is presented and numerical stability of the algorithm is established. Numerical experiments are conducted to demonstrate the efficiency, stability and accuracy of the Direct Parallel Block W Z Algorithm on GPU platform. Forward and backward errors are computed and DPBWZA is found to be highly accurate, numerically stable and as efficient as the subroutine csrlsvlu of GPU accelerated cuSolverSP library.

In this paper, we generalize and analyze the concepts of diagonal dominance and irreducibility in the framework of column and row representative matrices of a set. Our analysis includes the definition of particular sets of M- and H-matrices. We also analyze the form that the matrices of the introduced irreducible sets must have and the implications of the obtained results on the solution of vertical and horizontal linear complementarity problems. In this context, we prove that the projected Jacobi and the projected Gauss-Seidel methods for horizontal linear complementarity problems converge when the matrices of the problem satisfy one of the introduced generalizations of strict diagonal dominance or of irreducible diagonal dominance. This extends current convergence results.

Analytical solutions to heat conduction problems are not straight forward and it becomes more difficult for the difficult – to – machine materials like Ti6Al4V. Computational methods for analysis of such kind of problems using specific numerical techniques helps to solve engineering heat conduction problems with accuracy and greater speed. As compared to one dimensional heat conduction problem, two dimensional and three-dimensional cases offer great difficulty in analytically solving such kind of problems. The present work covers the 2D rectangular heat conduction problem being solved utilizing the finite-difference scheme. The results from analytical solutions were utilized as the guideline for comparison with the computational scheme. The results of the present scheme were validated for the accuracy of the output results. The diagonal dominance of the solution matrix helped us to use Gauss-Seidel iterative scheme for the present study. The computational work has been carried out using in house code developed in Python release 3.7.0.

This paper presents a new fast power series solution method to solve the Hierarchal Method of Moment (MoM) matrix for a large complex, perfectly electric conducting (PEC) 3D structures. The proposed power series solution converges in just two (2) iterations which is faster than the conventional fast solver–based iterative solution. The method is purely algebraic in nature and, as such applicable to existing conventional methods. The method uses regular fast solver Hierarchal Matrix (H-Matrix) and can also be applied to Multilevel Fast Multipole Method Algorithm (MLFMA). In the proposed method, we use the scaling of the symmetric near-field matrix to develop a diagonally dominant overall matrix to enable a power series solution. Left and right block scaling coefficients are required for scaling near-field blocks to diagonal blocks using Schur’s complement method. However, only the right-hand scaling coefficients are computed for symmetric near-field matrix leading to saving of computation time and memory. Due to symmetric property, the left side-block scaling coefficients are just the transpose of the right-scaling blocks. Next, the near-field blocks are replaced by scaled near-field diagonal blocks. Now the scaled near-field blocks in combination with far-field and scaling coefficients are subjected to power series solution terminating after only two terms. As all the operations are performed on the near-field blocks, the complexity of scaling coefficient computation is retained as O(N)O⁢(N). The power series solution only involves the matrix-vector product of the far-field, scaling coefficients blocks, and inverse of scaled near-field blocks. Hence, the solution cost remains O(NlogN)O⁢(N⁢l⁢o⁢g⁢N). Several numerical results are presented to validate the efficiency and robustness of the proposed numerical method.

We introduce three new classes JGD ( n ) , LUDD ( n ) and DWDD ( n ) of n × n complex matrices with variants of the diagonal dominance property. Each of the three classes contains singular matrices. We investigate in depth the relations between the three classes and their relations with the classes of diagonally dominant and generalized diagonally dominant matrices. We also study the problem of providing necessary and sufficient conditions for the singularity of a matrix in each of the three classes JGD ( n ) , LUDD ( n ) and DWDD ( n ) . We present an application of the theory of the classes JGD ( n ) , LUDD ( n ) and DWDD ( n ) to the iterative Jacobi method for the solution of the linear system A x = b .

For an invertible H-matrix A, a classical result involving A − 1 and its comparison matrix was proved by Ostrowski. More recently, these were improved by Neumaier and Kolotilina. In this article, we have endeavoured to generalize these inequalities for the group inverse of A, in the presence of certain conditions, involving a specific extension of a notion of diagonal dominance. A result of Fan for invertible M-matrices is also extended for H-matrices.

Consider the classical problem of solving a general linear system of equations $$Ax=b$$ . It is well known that the (successively over relaxed) Gauss–Seidel scheme and many of its variants may not converge when A is neither diagonally dominant nor symmetric positive definite. Can we have a linearly convergent G–S type algorithm that works for anyA? In this paper we answer this question affirmatively by proposing a doubly stochastic G–S algorithm that is provably linearly convergent (in the mean square error sense) for any feasible linear system of equations. The key in the algorithm design is to introduce a nonuniform double stochastic scheme for picking the equation and the variable in each update step as well as a stepsize rule. These techniques also generalize to certain iterative alternating projection algorithms for solving the linear feasibility problem $$A x\le b$$ with an arbitrary A, as well as high-dimensional minimization problems for training over-parameterized models in machine learning. Our results demonstrate that a carefully designed randomization scheme can make an otherwise divergent G–S algorithm converge.

In this paper, we study diagonal dominance of the stiffness matrix resulted from the piecewise linear finite element discretisation of the integral fractional Laplacian under global homogeneous Dirichlet boundary condition in one spatial dimension. We first derive the exact form of this matrix in the frequency space which is extendable to multi-dimensional rectangular elements. Then we give the complete answer when the stiffness matrix can be strictly diagonally dominant. As one application, we apply this notion to the construction of maximum principle preserving schemes for the fractional-in-space Allen–Cahn equation, and provide ample numerical results to verify our findings.

Nonzero-sum stochastic differential games with impulse controls offer a realistic and far-reaching modelling framework for applications within finance, energy markets, and other areas, but the difficulty in solving such problems has hindered their proliferation. Semi-analytical approaches make strong assumptions pertaining to very particular cases. To the author's best knowledge, the only numerical method in the literature is the heuristic one we put forward to solve an underlying system of quasi-variational inequalities. Focusing on symmetric games, this paper presents a simpler, more precise and efficient fixed-point policy-iteration-type algorithm which removes the strong dependence on the initial guess and the relaxation scheme of the previous method. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. A novel provably convergent single-player impulse control solver is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise very challenging problems, and even some which go beyond the scope of the currently available theory., 33 pages, 9 figures, 1 table

© 2018 Elsevier Inc. An upper bound for the infinity norm for the inverse of Dashnic–Zusmanovich type matrices is given. It is proved that the upper bound is sharper than the well-known Varah's bound for strictly diagonally dominant matrices. By introducing a new subclass of P-matrices: Dashnic–Zusmanovich type B-matrices (DZ-type-B-matrices), and using the proposed infinity norm bound, an error bound is given for the linear complementarity problems of DZ-type-B-matrices. We also give a new pseudospectra localization to measure the distance to instability.

Motivated by the growing successful use of fractional differential equations in the modeling of different important phenomena, in this paper we derive tools for practical analysis of the robust asymptotic stability of a (incommensurate) fractional order linear system. First, the concept of fractional pseudospectra is introduced. Second, driven by the simplicity and usefulness of spectral localizations in the analysis of various matrix properties, we introduce adequate localization techniques using the ideas that come from diagonally dominant matrices, in order to localize the fractional pseudospectra. In such way, many theoretical and practical applications of pseudospectra (robust stability, transient behavior, nonnormal dynamics, etc.) in fractional order differential systems can be linked to the specificity of the matrix entries, allowing one to understand certain phenomena in practice better. Third, we consider the fractional distance to instability in l ∞ , l 1 and l 2 norms, and determine efficient lower bounds. Finally, this novel approach is implemented on the realistic model of empirical food web to link the stability (that incorporates hereditary dynamics of living organisms) with the empirical data and their uncertainty limitations.

A new class of matrices, called Dashnic–Zusmanovich type (DZ-type) matrices, which is similar to, but different from Dashnic–Zusmanovich (DZ) matrices, is introduced and proved to be a subclass of nonsingular H-matrices. The relations between strictly diagonally dominant matrices, doubly strictly diagonally dominant matrices, DZ matrices and DZ-type matrices are discussed. Finally, by the non-singularity of DZ-type matrices, a new eigenvalue localization set for matrices is given.

We obtain new bounds for the entries of the inverse of a diagonally-dominant tridiagonal matrix which improve the best previous ones, due to H.-B. Li et al. We apply our bounds to the tridiagonal matrices arising in the second-order finite-difference discretization of certain boundary-value problems of parabolic type, establishing asymptotically optimal bounds. Fil: Dratman, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina

We derive a finite volume method for two-sided fractional diffusion equations with RiemannLiouville derivatives in one spatial dimension. The method applies to non-uniform meshes, with arbitrary nodal spacing. The discretisation utilises the integral definition of the fractional derivatives, and we show that it leads to a diagonally dominant matrix representation, and a provably stable numerical scheme. Being a finite volume method, the numerical scheme is fully conservative, and the ability to locally refine the mesh can produce solutions with more accuracy for the same number of nodes compared to a uniform mesh, as we demonstrate numerically.

In this article we present a new characterization of inverse M-matrices, inverse row diagonally dominant M-matrices and inverse row and column diagonally dominant M-matrices, based on the positivity of certain inner products., Comment: 9 pages

The problem of finding a k x k submatrix of maximum volume of a matrix A is of interest in a variety of applications. For example, it yields a quasi-best low-rank approximation constructed from the rows and columns of A. We show that such a submatrix can always be chosen to be a principal submatrix not only for symmetric semidefinite matrices but also for diagonally dominant matrices. Then we analyze the low-rank approximation error returned by a greedy method for volume maximization, cross approximation with complete pivoting. Our bound for general matrices extends an existing result for symmetric semidefinite matrices and yields new error estimates for diagonally dominant matrices. In particular, for doubly diagonally dominant matrices the error is shown to remain within a modest factor of the best approximation error. We also illustrate how the application of our results to cross approximation for functions leads to new and better convergence results. (C) 2020 Elsevier Inc. All rights reserved.

In this paper, we study the positive stability of P-matrices. We prove that a matrix A is positive stable if A is a P 2 -matrix and there is at least one nested sequence of principal submatrices of A each of which is also a P 2 -matrix. This result generalizes the result by Carlson which shows the positive stability of sign-symmetric P-matrices and the result by Tang, Simsek, Ozdaglar and Acemoglu which shows the positive stability of strictly row (column) square diagonally dominant for every order of minors P-matrices.

The class of weakly chained diagonally dominant B-matrices, a subclass of P-matrices is introduced. Error bounds for the linear complementarity problem are presented when the involved matrix is a weakly chained diagonally dominant B-matrix. Numerical examples are given to show the sharpness of the proposed bounds.

To reduce the computational complexity of the well-established recursive Kullback–Leibler (RKL) method for real-time applications, a recursive estimation method of the unknown transition probabilities (TPs) for the jump Markov linear system (JMLS) is developed in this study. The authors first explore an underlying idea that the RKL estimate of a diagonally dominant TP matrix (TPM) can be constructed by the estimate of each row vector of the TPM under the minimum K–L divergence criterion using observations at specific time steps. A modified derivation of the numerical solution to the RKL estimate that can avoid redundant likelihood computations is then exploited to estimate the specific row vector of the TPM per time step. The developed TP estimation method is computationally more efficient than either the RKL method or the maximum likelihood method, in particular for the JMLS defined over a high-dimensional state space or a multi-dimensional model space. The effectiveness of the developed TP estimation method is verified through a numerical example.

We give upper and lower bounds on the determinant of a small perturbation of the identity matrix. The lower bounds are best possible, and in most cases they are stronger than well-known bounds due to Ostrowski and other authors. The upper bounds are best possible if a skew-Hadamard matrix of the same order exists.

In this paper, an implicit scheme is presented for a meshless compressible Euler solver based on the Least Square Kinetic Upwind Method (LSKUM). The Jameson and Yoon's split flux Jacobians formulation is very popular in finite volume methodology, which leads to a scalar diagonal dominant matrix for an efficient implicit procedure (Jameson & Yoon, 1987). However, this approach leads to a block diagonal matrix when applied to the LSKUM meshless method. The above split flux Jacobian formulation, along with a matrix-free approach, has been adopted to obtain a diagonally dominant, robust and cheap implicit time integration scheme. The efficacy of the scheme is demonstrated by computing 2D flow past a NACA 0012 airfoil under subsonic, transonic and supersonic flow conditions. The results obtained are compared with available experiments and other reliable computational fluid dynamics (CFD) results. The present implicit formulation shows good convergence acceleration over the RK4 explicit procedure. Further, the accuracy and robustness of the scheme in 3D is demonstrated by computing the flow past an ONERA M6 wing and a clipped delta wing with aileron deflection. The computed results show good agreement with wind tunnel experiments and other CFD computations.

We generalize the bounds on the inverses of diagonally dominant matrices obtained in [16] from scalar to block tridiagonal matrices. Our derivations are based on a generalization of the classical condition of block diagonal dominance of matrices given by Feingold and Varga in [11]. Based on this generalization, which was recently presented in [3], we also derive a variant of the Gershgorin Circle Theorem for general block matrices which can provide tighter spectral inclusion regions than those obtained by Feingold and Varga., 18 pages, 5 figures

A high-order compact finite difference method is presented for solving the three-dimensional (3D) time-fractional convection–diffusion equation (of order α ∈ ( 1 , 2 ) ). The original equation is first transformed to a fractional diffusion-wave equation, then using fourth-order Pade approximation for spatial derivatives and the center difference method for time derivative respectively, a fully discrete implicit compact scheme is obtained. Furthermore, based on different splitting terms, three unconditionally stable ADI compact schemes with optimal convergence order are developed respectively. The resulting schemes in each ADI solution step corresponding to a strictly diagonally dominant matrix equation can be solved using the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments show that these schemes can significantly improve the time accuracy.

In this paper, an implicit method of decomposition of cubic Hermite splines using a new type of multiwavelets with supercompact supports is investigated. A splitting algorithm of wavelet-transforms of solving two three-diagonal systems of linear equations with strict diagonal dominance in parallel is justified. Results of some numerical experiments are presented.

In this paper we investigate block SSOR multisplittings. When the coefficient matrix is a block H-matrix or a (generalized) block strictly diagonally dominant matrix, the convergence of the parallel block SSOR multisplitting method for solving nonsingular linear systems is proved. Two numerical examples are given to illustrate the theoretical results.

It is known that the Schur complements of doubly diagonally dominant matrices are doubly diagonally dominant. In this paper, we obtain an estimate for the doubly diagonally dominant degree on the Schur complement of strictly doubly diagonally dominant matrices. Then, as an application we obtain that the eigenvalues of the Schur complements are located in the Brauer Ovals of Cassini of the original matrices under certain conditions. As another application, we obtain an upper bound for the infinity norm on the inverse on the Schur complement of strictly doubly diagonally dominant matrices. Further, based on the derived results, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and can compute out the results faster.

Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of disconnected complete components. This result rigorously demonstrates that, except in special cases, positive definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions.

In the first part of the paper, we deal with Euclidean Jordan algebraic generalizations of some results of Brualdi on inclusion regions for the eigenvalues of complex matrices using directed graphs. As a consequence, the theorems of Brauer–Ostrowski and Brauer on the location of eigenvalues are extended to the setting of Euclidean Jordan algebras. In the second part, motivated by the work of Li and Tsatsomeros on the class of doubly diagonally dominant matrices with complex entries and its subclasses, we present some inter-relations between the H -property, generalized strict diagonal dominance, invertibility, and strict double diagonal dominance in Euclidean Jordan algebras. In addition, we show that in a Euclidean Jordan algebra, the Schur complements of a strictly doubly diagonally dominant element inherit this property.

In this article, we develop an exponential high order compact alternating direction implicit (EHOC ADI) method for solving three dimensional (3D) unsteady convection–diffusion equations. The method, which requires only a regular seven-point 3D stencil similar to that in the standard second-order methods, is second order accurate in time and fourth-order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit (ADI) solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas–Gunn ADI method and the Karaa's high-order compact ADI method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

It is shown that a 2 × 2 complex matrix A is diagonally equivalent to a matrix with two distinct eigenvalues iff A is not strictly triangular. It is established in this paper that every 3 × 3 nonsingular matrix is diagonally equivalent to a matrix with 3 distinct eigenvalues. More precisely, a 3 × 3 matrix A is not diagonally equivalent to any matrix with 3 distinct eigenvalues iff det A = 0 and each principal minor of A of order 2 is zero. It is conjectured that for all n ⩾ 2 , an n × n complex matrix is not diagonally equivalent to any matrix with n distinct eigenvalues iff det A = 0 and every principal minor of A of order n - 1 is zero.

In this paper, we analyze the relation between some classes of matrices with variants of the diagonal dominance property. We establish a sufficient condition for a generalized doubly diagonally dominant matrix to be invertible. Sufficient conditions for a matrix to be strictly generalized diagonally dominant are also presented. We provide a sufficient condition for the invertibility of a cyclically diagonally dominant matrix. These sufficient conditions do not assume the irreducibility of the matrix.

We firstly consider the block dominant degree for I-(II-)block strictly diagonally dominant matrix and their Schur complements, showing that the block dominant degree for the Schur complement of an I-(II-)block strictly diagonally dominant matrix is greater than that of the original grand block matrix. Then, as application, we present some disc theorems and some bounds for the eigenvalues of the Schur complement by the elements of the original matrix. Further, by means of matrix partition and the Schur complement of block matrix, based on the derived disc theorems, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and the numerical example illustrates that the iteration can compute out the results faster.

An accurate moving particle method for incompressible flow calculations is presented in this article. The major distinctive feature in this proposition is the insertion of a pressure mesh within the particle cloud to handle the continuity constraint. It is motivated by the thought that pressure should be a field variable rather than a material one moving with fluid flow. Both the diffusion and convection operators are executed on the particle locations, while the projection operator to retain a convergence-free velocity field is manipulated on the inserted pressure mesh. It will yield a diagonally dominant and constant-coefficient matrix equation to update the pressure field. Besides the advantages gained to enforce the continuity constraint, the auxiliary mesh can significantly enhance the particle searching efficiency in the particle smoothing process. Numerical verifications on some benchmark problems indicate that the present proposition will provide accurate results for incompressible flow calculations.

Improved formulations of the discretized pressure equation and boundary treatments in co-located, equal-order, control-volume finite-element methods for the prediction of incompressible fluid flow are presented in the context of steady, planar two-dimensional problems. Three-node triangles and polygonal-cross-section control volumes, created by joining element centroids to midpoints of the sides, are used. The proposed improvements maintain the strength of coefficients in the discretized pressure equations, and help in keeping them diagonally dominant. Thus they facilitate convergence of iterative solution methods and the results do not display physically untenable features. Solutions of two test problems are presented and discussed.

In diffusion transport, the flux at a point is typically modeled in terms of the local gradient of a potential. When heterogeneities are present, this local model can break down and it may be more appropriate to model the diffusion flux as a weighted sum of gradients present throughout the domain. Here a discrete nonlocal flux model—consistent with control-volume implementations—is developed. This scheme is referred to as the control-volume weighted flux scheme (CVWFS). The key component is the modeling of the diffusion flux at a given control-volume face in terms of a weighted sum of gradients at that face and at faces up- and downstream. Criteria for choosing the weights are proposed. This results in numerical solution schemes in which the coefficient matrix is diagonally dominant, has positive off-diagonal elements, and zero row sums. For a particular power-law weighting scheme it is shown how the CVWFS is related to the definition of the Caputo fractional derivative and the one-shift Grunwald approxim...

We present a structured perturbation theory for the LDU factorization of (row) diagonally dominant matrices and we use this theory to prove that a recent algorithm of Ye (Math Comp 77(264):2195–2230, 2008) computes the L, D and U factors of these matrices with relative errors less than 14n 3 u, where u is the unit roundoff and n × n is the size of the matrix. The relative errors for D are componentwise and for L and U are normwise with respect the “max norm” $${\|A\|_M = \max_{ij} |a_{ij}|}$$. These error bounds guarantee that for any diagonally dominant matrix A we can compute accurately its singular value decomposition and the solution of the linear system Ax = b for most vectors b, independently of the magnitude of the traditional condition number of A and in O(n 3) flops.

The inverse mean first passage time problem is given a positive matrix M ∈ R n , n , then when does there exist an n-state discrete-time homogeneous ergodic Markov chain C , whose mean first passage matrix is M? The inverse M-matrix problem is given a nonnegative matrix A, then when is A an inverse of an M-matrix. The main thrust of this paper is to show that the existence of a solution to one of the problems can be characterized by the existence of a solution to the other. In so doing we extend earlier results of Tetali and Fiedler.

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.

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.

This paper develops a novel linear system based approach for computing ‖ A - 1 ‖ ∞ , the Skeel condition number of an M -matrix A , and the positive diagonal matrices D guaranteeing that AD be a strictly diagonally dominant matrix. Theoretic analysis and simulation results justify the validity of the proposed approach. Moreover, the proposed linear system model can be implemented by application-specific integrated circuits, and then it has asynchronous parallel processing ability and can achieve high computing performance.

Let A be a weakly chained diagonally dominant ( wcdd ) M -matrix, an upper bound for ‖ A - 1 ‖ ∞ is presented and further applied to establish an lower bound for the smallest eigenvalue of A . Effectiveness of the new upper bound is shown numerically.

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.

Let A be a real strictly diagonally dominant M-matrix. We give a sharp upper bound for A - 1 ∞ . Furthermore, the lower bound of the smallest eigenvalue q ( A ) is established.

The paper presents new two-sided bounds for the infinity norm of the inverse for the so-called PM -matrices, which form a subclass of the class of nonsingular M -matrices and contain the class of strictly diagonally dominant matrices. These bounds are shown to be monotone with respect to the underlying partitioning of the index set, and the equality cases are analyzed. Also an upper bound for the infinity norm of the inverse of a PH -matrix (whose comparison matrix is a PM -matrix) is derived. The known Ostrowski, Ahlberg–Nilson–Varah, and Moraca bounds are shown to be special cases of the upper bound obtained.

Discrete maximum principles are derived for finite element discretizations of nonlinear elliptic systyems with cooperative and weakly diagonally dominant coupling. The results are achieved via an algebraic discrete maximum principle organized in a Hilbert space framework and, in the case of simplicial elements, are obtained under weakened acute type conditions for the finite element (FE) meshes.

Although upwind discretization of convection will lead to a diagonally dominant coefficient matrix, on arbitrary grids the latter is not necessarily positive real, i.e. its symmetric part need not be positive definite ('negative diffusion'). Especially on contracting-expanding grids this property can be lost. The paper discusses a conservative (finite-volume) upwind variant for which the latter property is guaranteed to hold, irrespective of grid (ir)regularity. Further, empirically it is found that often its global discretization error is smaller than that of the 'traditional' (finite-difference) upwind method. Finally, it is shown that in many situations its extremal eigenvalues at the outer side of the spectrum move towards the imaginary axis, thus enhancing the stability of explicit time-integration methods. (C) 2007 IMACS. Published by Elsevier B.V. All rights reserved.

In the first part, we obtain two easily calculable lower bounds for ‖ A - 1 ‖ , where ‖ · ‖ is an arbitrary matrix norm, in the case when A is an M-matrix, using first row sums and then column sums. Using those results, we obtain the characterization of M-matrices whose inverses are stochastic matrices. With different approach, we give another easily calculable lower bounds for ‖ A - 1 ‖ ∞ and ‖ A - 1 ‖ 1 in the case when A is an M-matrix. In the second part, using the results from the first part, we obtain our main result, an easily calculable upper bound for ‖ A - 1 ‖ 1 in the case when A is an SDD matrix, thus improving the known bound. All mentioned norm bounds can be used for bounding the smallest singular value of a matrix.

We prove that if A=[Aij]∈RN,N is a block symmetric matrix and y is a solution of a nearby linear system (A+E)y=b, then there exists F=FT such that y solves a nearby symmetric system (A+F)y=b, if A is symmetric positive definite or the matricial norm μ(A)=(‖Aij‖2) is diagonally dominant. Our blockwise analysis extends existing normwise and componentwise results on preserving symmetric perturbations (cf. [J.R. Bunch, J.W. Demmel, Ch. F. Van Loan, The strong stability of algorithms for solving symmetric linear systems, SIAM J.Matrix Anal. Appl. 10 (4) (1989) 494–499; D. Herceg, N. Krejić, On the strong componentwise stability and H-matrices, Demonstratio Mathematica 30 (2) (1997) 373–378; A. Smoktunowicz, A note on the strong componentwise stability of algorithms for solving symmetric linear systems, Demonstratio Mathematica 28 (2) (1995) 443–448]).