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 In so doing, the initial results of Varga in 1960 on regular splittings of matrices, and the subsequent unpublished results of Wonicki in 1973 on regular splittings of matrices, will be seen to be special cases of these new comparison theorems.Dedicated to Fritz Bauer on the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     

Comparison theorems for weak splittings of bounded operators

Summary Comparison theorems for weak splittings of bounded operators are presented. These theorems extend the classical comparison theorem for regular splittings of matrices by Varga, the less known result by Wonicki, and the recent results for regular and weak regular splittings of matrices by Neumann and Plemmons, Elsner, and Lanzkron, Rose and Szyld. The hypotheses of the theorems presented here are weaker and the theorems hold for general Banach spaces and rather general cones. Hypotheses are given which provide strict inequalities for the comparisons. It is also shown that the comparison theorem by Alefeld and Volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators.This work was supported by the National Science Foundation grants DMS-8807338 and INT-8918502     

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     

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     

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     

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     

A class of norms of iterative methods for solving systems of linear equations

A new class of norms which generalize norms previously investigated by Young [9, 14], Sheldon [4, 5], Golub [1], Golub and Varga [2], Varga [6], Wachspress [7], Young and Kincaid [12], Young [14], and Kincaid [3] is introduced. Expressions for these norms applied to the matrices associated with various iterative methods are developed.Work on this paper was sponsored by NSF Grant GP-8442 and Army Grant DA-ARO(D)-31-124-G1050 at The University of Texas at Austin.     

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.     

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.     

Precise domains of convergence for the block SSOR method associated withp-cyclic matrices

We apply Rouché's theorem to the functional equation relating the eigenvalues of theblock symmetric successive overrelaxation (SSOR) matrix with those of the block Jacobi iteration matrix found by Varga, Niethammer, and Cai, in order to obtain precise domains of convergence for the block SSOR iteration method associated withp-cyclic matricesA, p3. The intersection of these domains, taken over all suchp's, is shown to coincide with the exact domain of convergence of thepoint SSOR iteration method associated withH-matricesA. The latter domain was essentially discovered by Neumaier and Varga, but was recently sharpened by Hadjidimos and Neumann.Research supported in part by NSF Grant DMS 870064.     

Regular splittings and monotone iteration functions

Summary In this paper we introduce the set of so-called monotone iteration functions (MI-functions) belonging to a given function. We prove necessary and sufficient conditions in order that a given MI-function is (in a precisely defined sense) at least as fast as a second one.Regular splittings of a function which were initially introduced for linear functions by R.S. Varga in 1960 are generating MI-functions in a natural manner.For linear functions every MI-function is generated by a regular splitting. For nonlinear functions, however, this is generally not the case.     

On the sharpness of some upper bounds for the spectral radii of S.O.R. iteration matrices

Summary Sharpness is shown for three upper bounds for the spectral radii of point S.O.R. iteration matrices resulting from the splitting (i) of a nonsingularH-matrixA into the usualD–L–U, and (ii) of an hermitian positive definite matrixA intoD–L–U, whereD is hermitian positive definite andL=1/2(A–D+S) withS some skew-hermitian matrix. The first upper bound (which is related to the splitting in (i)) is due to Kahan [6], Apostolatos and Kulisch [1] and Kulisch [7], while the remaining upper bounds (which are related to the splitting in (ii)) are due to Varga [11]. The considerations regarding the first bound yield an answer to a question which, in essence, was recently posed by Professor Ridgway Scott: What is the largest interval in , 0, for which the point S.O.R. iterative method is convergent for all strictly diagonally dominant matrices of arbitrary order? The answer is, precisely, the interval (0, 1].Research supported in part by the Air Force Office of Scientific Research, and the Department of Energy     

Further study of alternating iterations for rectangular matrices

Theory of matrix splittings is a useful tool in the analysis of iterative methods for solving systems of linear equations. When two splittings are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. This helps to arrive at the conclusion that which splitting should one choose so that one can reach the desired solution of accuracy or the exact solution in a faster way. In the case of many splittings are provided, the comparison of the spectral radii is time-consuming. Such a situation can be overcome by introducing another iteration scheme which converges to the same solution of interest in a much faster way. In this direction, the theory of alternating iterations for real rectangular matrices is recently proposed. In this note, some more results to the theory of alternating iterations are added. A comparison result of two different alternating iteration schemes is then presented which will help us to choose the iteration scheme that will guarantee the faster convergence of the alternating iteration scheme. In addition to these results, a comparison result for proper weak regular splittings is also obtained.     

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     

Regular incomplete factorizations of real positive definite matrices

We study the classes of matrices which admit a regular incomplete factorization with respect to any graph set. We extend the construction to the class of real positive definite matrices. The convergence of the basic iteration associated with the splittings we have is discussed.     

A stable Richardson iteration method for complex linear systems

Summary The Richardson iteration method is conceptually simple, as well as easy to program and parallelize. This makes the method attractive for the solution of large linear systems of algebraic equations with matrices with complex eigenvalues. We change the ordering of the relaxation parameters of a Richardson iteration method proposed by Eiermann, Niethammer and Varga for the solution of such problems. The new method obtained is shown to be stable and to have better convergence properties.Research supported by the National Science Foundation under Grant DMS-8704196     

Iterative processes based on block <Emphasis Type="Italic">H</Emphasis>-splittings

Block H-splittings of block square matrices (which, in general, have complex entries) are examined. It is shown that block H-matrices are the only ones that admit this type of splittings. Iterative processes corresponding to these splittings are proved to be convergent. The concept of a simple splitting of a block matrix is introduced, and the convergence of iterative processes related to simple splittings of block H-matrices is investigated. Multisplitting and nonstationary iterative processes based on block H-splittings are considered. Sufficient conditions for their convergence are derived, and some estimates for the asymptotic convergence rate are given.     

Accelerated PMHSS iteration methods for complex symmetric linear systems

In this paper, we introduce and analyze an accelerated preconditioning modification of the Hermitian and skew-Hermitian splitting (APMHSS) iteration method for solving a broad class of complex symmetric linear systems. This accelerated PMHSS algorithm involves two iteration parameters α,β and two preconditioned matrices whose special choices can recover the known PMHSS (preconditioned modification of the Hermitian and skew-Hermitian splitting) iteration method which includes the MHSS method, as well as yield new ones. The convergence theory of this class of APMHSS iteration methods is established under suitable conditions. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. Theoretical analyses show that the upper bound σ₁(α,β) of the asymptotic convergence rate of the APMHSS method is smaller than that of the PMHSS iteration method. This implies that the APMHSS method may converge faster than the PMHSS method. Numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method.     

Exact convergence and divergence domains for the symmetric successive overrelaxation iterative (SSOR) method applied to H-matrices

In this paper, exact convergence and divergence domains for the SSOR iterative method, as applied to the class of H-matrices, are obtained. The theory of regular splittings and the recent results of Varga, Niethammer, and Cai are used as tools in establishing these convergence and divergence domains.     

M-矩阵线性互补问题模系多分裂迭代方法的收敛性

首先证明了M-矩阵的H-相容分裂都是正则分裂,反之不成立.这表明对于M-矩阵而言,其正则分裂包含H-相容分裂.然后针对系数矩阵为M-矩阵的线性互补问题,建立了两个收敛定理:一是模系多分裂迭代方法关于正则分裂的收敛定理;二是模系二级多分裂迭代方法关于外迭代为正则分裂和内迭代为弱正则分裂的收敛定理.