Comparisons of regular splittings of matrices

Summary In this article, new comparison theorems for regular splittings of matrices are derived. In so doing, the initial results of Varga in 1960 on regular splittings of matrices, and the subsequent unpublished results of Wonicki in 1973 on regular splittings of matrices, will be seen to be special cases of these new comparison theorems.Dedicated to Fritz Bauer on the occasion of his 60th birthdayResearch supported in part by the Air Force Office of Scientific Research, and by the Department of Energy     

A note on comparison theorems for nonnegative matrices

Summary In a recent paper, [4], Csordas and Varga have unified and extended earlier theorems, of Varga in [10] and Wonicki in [11], on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings. The main purpose of this note is to show a connection between the Csordas-Varga paper and a paper by Beauwens, [1], in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular. Monotonic norms already used in [1] play an important role in our work here.Research supported in part by NSF grant number DMS-8400879     

Comparison theorems for splittings of matrices

Summary. This paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices. On the basis of nonnegative matrix theory, comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived. When the coefficient matrix A is Hermitian positive (semi)definite, comparison theorems about two P-regular splittings and two parallel multisplitting methods are proved. Received April 4, 1998 / Revised version received October 18, 1999 / Published online November 15, 2001     

Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods

Summary. Given a nonsingular matrix , and a matrix of the same order, under certain very mild conditions, there is a unique splitting , such that . Moreover, all properties of the splitting are derived directly from the iteration matrix . These results do not hold when the matrix is singular. In this case, given a matrix and a splitting such that , there are infinitely many other splittings corresponding to the same matrices and , and different splittings can have different properties. For instance, when is nonnegative, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results hold in the symmetric positive semidefinite case. Given a singular matrix , not for all iteration matrices there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are examined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different cases where the matrix is monotone, singular, and positive (semi)definite are studied. Received September 5, 1995 / Revised version received April 3, 1996     

Some comparison theorems for nonnegative splittings of matrices

Summary In a recent paper the author has proposed some theorems on the comparison of the asymptotic rates of convergence of two nonnegative splittings. They extended the corresponding result of Miller and Neumann and implied the earlier theorems of Varga, Beauwens, Csordas and Varga. An open question by Miller and Neumann, which additional and appropriate conditions should be imposed to obtain strict inequality, was also answered. This article continues to investigate the comparison theorems for nonnegative splittings. The new results extend and imply the known theorems by the author, Miller and Neumann.The Project Supported by the Natural Science Foundation of Jiangsu Province Education Commission     

Alpha well-posedness,alpha stability,and the matrix theorems of H.-O. Kreiss

Summary For systems of partial differential equations with constant coefficients and for the corresponding difference equations the concepts of well-posedness and stability are introduced. These concepts are more general than strong well-posedness and stability on the one hand, and more restrictive than weak well-posedness (Petrovskii condition) and weak stability (von Neumann condition) on the other. Characterizations of these properties are established which partly extend the matrix theorems of H.-O. Kreiss. Also a Lax type theorem is valid in this setting.This author was supported by the Deutsche Forschungsgemeinschaft (Grant Go 261/4)     

Convergence of nested classical iterative methods for linear systems

Summary Classical iterative methods for the solution of algebraic linear systems of equations proceed by solving at each step a simpler system of equations. When this system is itself solved by an (inner) iterative method, the global method is called a two-stage iterative method. If this process is repeated, then the resulting method is called a nested iterative method. We study the convergence of such methods and present conditions on the splittings corresponding to the iterative methods to guarantee convergence forany number of inner iterations. We also show that under the conditions presented, the spectral radii of the global iteration matrices decrease when the number of inner iterations increases. The proof uses a new comparison theorem for weak regular splittings. We extend our results to larger classes of iterative methods, which include iterative block Gauss-Seidel. We develop a theory for the concatenation of such iterative methods. This concatenation appears when different numbers of inner interations are performed at each outer step. We also analyze block methods, where different numbers of inner iterations are performed for different diagonal blocks.Dedicated to Richard S. Varga on the occasion of his sixtieth birthdayP.J. Lanzkron was supported by Exxon Foundation Educational grant 12663 and the UNISYS Corporation; D.J. Rose was supported by AT&T Bell Laboratories, the Microelectronic Center of North Carolina and the Office of Naval Research under contract number N00014-85-K-0487; D.B. Szyld was supported by the National Science Foundation grant DMS-8807338.     

On the inverses of general tridiagonal matrices

In this work, the sign distribution for all inverse elements of general tridiagonal H-matrices is presented. In addition, some computable upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices are obtained. Based on the sign distribution, these bounds greatly improve some well-known results due to Ostrowski (1952) 23, Shivakumar and Ji (1996) 26, Nabben (1999) [21] and [22] and recently given by Peluso and Politi (2001) 24, Peluso and Popolizio (2008) 25 and so forth. It is also stated that the inverse of a general tridiagonal matrix may be described by 2n-2 parameters ( and ) instead of 2n+2 ones as given by El-Mikkawy (2004) 3, El-Mikkawy and Karawia (2006) 4 and Huang and McColl (1997) 10. According to these results, a new symbolic algorithm for finding the inverse of a tridiagonal matrix without imposing any restrictive conditions is presented, which improves some recent results. Finally, several applications to the preconditioning technology, the numerical solution of differential equations and the birth-death processes together with numerical tests are given.     

Ergodic theorems for noncommutative dynamical systems

Recently, E.C. Lance extended the pointwise ergodic theorem to actions of the group of integers on von Neumann algebras. Our purpose is to extend other pointwise ergodic theorems to von Neumann algebra context: the Dunford-Schwartz-Zygmund pointwise ergodic theorem, the pointwise ergodic theorem for connected amenable locally compact groups, the Wiener's local ergodic theorem for ₊ ^d and for general Lie groups.     

Convergence theorems for block splitting iterative methods for linear systems

In this paper, we will present the block splitting iterative methods with general weighting matrices for solving linear systems of algebraic equations Ax=b$\mathit{Ax}=b$ when the coefficient matrix A is symmetric positive definite of block form, and establish the convergence theories with respect to the general weighting matrices but special splittings. Finally, a numerical example shows the advantage of this method.     

On a class of matrices which arise in the numerical solution of Euler equations

Summary We study block matricesA=[A_ij], where every blockA _ij ^k,k is Hermitian andA _ii is positive definite. We call such a matrix a generalized H-matrix if its block comparison matrix is a generalized M-matrix. These matrices arise in the numerical solution of Euler equations in fluid flow computations and in the study of invariant tori of dynamical systems. We discuss properties of these matrices and we give some equivalent conditions for a matrix to be a generalized H-matrix.Research supported by the Graduiertenkolleg mathematik der Universität Bielefeld     

Convergent nonnegative matrices and iterative methods for consistent linear systems

Summary In this paper we study linear stationary iterative methods with nonnegative iteration matrices for solving singular and consistent systems of linear equationsAx=b. The iteration matrices for the schemes are obtained via regular and weak regular splittings of the coefficients matrixA. In certain cases when only some necessary, but not sufficient, conditions for the convergence of the iterations schemes exist, we consider a transformation on the iteration matrices and obtain new iterative schemes which ensure convergence to a solution toAx=b. This transformation is parameter-dependent, and in the case where all the eigenvalues of the iteration matrix are real, we show how to choose this parameter so that the asymptotic convergence rate of the new schemes is optimal. Finally, some applications to the problem of computing the stationary distribution vector for a finite homogeneous ergodic Markov chain are discussed.Research sponsored in part by US Army Research Office     

A note on properties of splittings of singular symmetric positive semidefinite matrices

Summary. Recently, Benzi and Szyld have published an important paper [1] concerning the existence and uniqueness of splittings for singular matrices. However, the assertion in Theorem 3.9 on the inheriting property of P-regular splitting for singular symmetric positive semidefinite matrices seems to be incorrect. As a complement of paper [1], in this short note we point out that if a matrix T is resulted from a P-regular splitting of a symmetric positive semidefinite matrix A, then splittings induced by T are not all P-regular. Received January 7, 1999 / Published online December 19, 2000     

Comparisons of weak regular splittings and multisplitting methods

Summary Comparison results for weak regular splittings of monotone matrices are derived. As an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings. This yields a very simple proof of results of Neumann-Plemmons on upper bounds, and establishes lower bounds, which has in special cases been conjectured by these authors.Dedicated to the memory of Peter Henrici     

On the analysis of the unsymmetric successive overrelaxation method when applied top-cyclic matrices

Summary A Determinantal Invariance, associated with consistently ordered weakly cyclic matrices, is given. The DI is then used to obtain a new functional equation which relates the eigenvalues of a particular block Jacobi iteration matrix to the eigenvalues of its associated Unsymmetric Successive Overrelaxation (USSOR) iteration matrix. This functional equation as well as the theory of nonnegative matrices and regular splittings are used to obtain convergence and divergence regions of the USSOR method.     

Semiconvergence of nonnegative splittings for singular matrices

Summary. In this paper, we discuss semiconvergence of the matrix splitting methods for solving singular linear systems. The concepts that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting is quasi-regular or quasi-nonnegative. The equivalent conditions for the semiconvergence are proved. Comparison theorem on convergence factors for two different quasi-nonnegative splittings is presented. As an application, the semiconvergence of the power method for solving the Markov chain is derived. The monotone convergence of the quasi-nonnegative splittings is proved. That is, for some initial guess, the iterative sequence generated by the iterative method introduced by a quasi-nonnegative splitting converges towards a solution of the system from below or from above. Received August 19, 1997 / Revised version received August 20, 1998 / Published online January 27, 2000     

Comparison theorems for a subclass of proper splittings of matrices

In this article, a convergence theorem and several comparison theorems are presented for a subclass of proper splittings of matrices introduced recently.     

Splittings of symmetric matrices and a question of Ortega

A complete answer is given to a problem posed in 1988 by Ortega concerning convergent splittings of symmetric matrices.     

Convergence of block iterative methods for linear systems with generalized H-matrices

The paper studies the convergence of some block iterative methods for the solution of linear systems when the coefficient matrices are generalized H$H$-matrices. A truth is found that the class of conjugate generalized H$H$-matrices is a subclass of the class of generalized H$H$-matrices and the convergence results of R. Nabben [R. Nabben, On a class of matrices which arises in the numerical solution of Euler equations, Numer. Math. 63 (1992) 411–431] are then extended to the class of generalized H$H$-matrices. Furthermore, the convergence of the block AOR iterative method for linear systems with generalized H$H$-matrices is established and some properties of special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper.