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.     

On the global convergence of the Eberlein method for real matrices

Summary The global convergence proof of the column-and row-cyclic Eberlein diagonalization process for real matrices is given. The convergence to a fixed matrix in Murnaghan form is obtained with the well-known exception of complex-conjugate pairs of eigenvalues whose real parts are more than double.     

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     

A modified bisection algorithm for the determination of the eigenvalues of a symmetric tridiagonal matrix

Summary A modification to the well known bisection algorithm [1] when used to determine the eigenvalues of a real symmetric matrix is presented. In the new strategy the terms in the Sturm sequence are computed only as long as relevant information on the required eigenvalues is obtained. The resulting algorithm usingincomplete Sturm sequences can be shown to minimise the computational work required especially when only a few eigenvalues are required.The technique is also applicable to other computational methods which use the bisection process.     

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     

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     

On a new class of elementary matrices

Summary A new class of elementary matrices, a block-generalisation of plane rotations, is presented and the application in constructing quadratically convergent block diagonalisation algorithms of Jacobi type is discussed.     

On a class of Jacobi-like procedures for diagonalising arbitrary real matrices

Summary A new class of elementary matrices is presented which are convenient in Jacobi-like diagonalisation methods for arbitrary real matrices. It is shown that the presented transformations possess the normreducing property and that they produce an ultimate quadratic convergence even in the case of complex eigenvalues. Finally, a quadratically convergent Jacobi-like algorithm for real matrices with complex eigenvalues is presented.     

On the convergence of the conjugate gradient method for singular capacitance matrix equations from the Neumann problem of the Poisson equation

Summary It is shown analytically in this work that the conjugate gradient method is an efficient means of solving the singular capacitance matrix equations arising from the Neumann problem of the Poisson equation. The total operation count of the algorithm does not exceed constant timesN ²logN(N=1/h) for any bounded domain with sufficiently smooth boundary.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation under Grant No. MCS75-17385. Also partially supported by the Energy Research and Development Administration     

Strong underrelaxation in Kaczmarz's method for inconsistent systems

Summary We investigate the behavior of Kaczmarz's method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.     

The rate of convergence of an approximate matrix factorization semi-direct method

Summary The definition of acceleration parameters for the convergence of a sparseLU factorization semi-direct method is shown to be based on lower and upper bounds of the extreme eigevalues of the iteration matrix. Optimum values of these parameters are established when the eigenvalues of the iteration matrix are either real or complex. Estimates for the computational work required to reduce theL ₂ norm of the error by a specified factor are also given.     

Rank-one modification of the symmetric eigenproblem

Summary An algorithm is presented for computing the eigensystem of the rank-one modification of a symmetric matrix with known eigensystem. The explicit computation of the updated eigenvectors and the treatment of multiple eigenvalues are discussed. The sensitivity of the computed eigenvectors to errors in the updated eigenvalues is shown by a perturbation analysis.Support for this research was provided by NSF grants MCS 75-06510 and MCS 76-03139Support for this research was provided by the Applied Mathematics Division, Argonne National Laboratory, Argonne, IL 60439, USA     

An alternative givens ordering

Summary A new Givens ordering is shown, empirically and by an approximate theoretical analysis, to take appreciably fewer stages than the standard scheme. Sharper error bounds than Gentleman's ensue, and the scheme is better suited to parallel computation. Other schemes, less efficient but more easily analysed, are discussed. The effect of a possible limit in practice on the number of simultaneous computations is considered.     

AB-Algorithm and its modifications for the spectral problems of linear pencils of matrices

Summary This paper describes and algorithm and its modifications for solving spectral problems for linear pencils of matrices both regular as well as singular.     

Generalizations of the projection method with applications to SOR theory for hermitian positive semidefinite linear systems

Summary Ann×n complex matrixB is calledparacontracting if B₂1 and 0x[N(I-B)]Bx₂<x₂. We show that a productB=B _k B _k–1 ...B ₁ ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofB in terms of the subdominant singular values of theB _i 's and in terms of the angles between certain subspaces. Our results here extend earlier results due to Halperin and due to Smith, Solomon and Wagner. We also determine necessary and sufficient conditions forn numbers in the interval [0, 1] to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product. In the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the SOR iteration matrix associated with ann×n hermitian positive semidefinite matrixA none of whose diagonal entries vanish.The work of this author was supported in part by NSF Research Grant No. MCS-8400879     

Eigenvector matrices of symmetric tridiagonals

Summary A simple test is given for determining whether a given matrix is the eigenvector matrix of an (unknown) unreduced symmetric tridiagonal matrix. A list of known necessary conditions is also provided. A lower bound on the separation between eigenvalues of tridiagonals follows from our Theorem 3.Dedicated to Professor F.L. Bauer on the occasion of his 60th birthdayThe first author gratefully acknowledges support from ONR Contract N00014-76-C-0013     

On the rate of convergence of the preconditioned conjugate gradient method

Summary We derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum. We present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function.     

Comparison of splittings used with the conjugate gradient algorithm

We consider several possible criteria for comparing splittings used with the conjugate gradient algorithm. We present sharp upper bounds for the error at each step of the algorithm and compare several widely used splittings with respect to their effect on these sharp upper bounds. We then consider a more stringent comparison test, and present necessary and sufficient conditions for the error at each step with one splitting to be smaller than that with another, for all pairs of corresponding initial guesses.     

Adaptive procedure for estimating parameters for the nonsymmetric Tchebychev iteration

Summary An iteration based upon the Tchebychev polynomials in the complex plane can be used to solve large sparse nonsymmetric linear systems whose eigenvalues lie in the right half plane. The iteration depends upon two parameters which can be chosen from knowledge of the convex hull of the spectrum of the linear operator. This paper deals with a procedure based upon the power method for dynamically estimating the convex hull of the spectrum. The stability of the procedure is discussed in terms of the field of values of the operator. Results show the adaptive procedure to be an effective method of determining parameters. The Tchebychev iteration compares favorably with several competing iterative methods.This work was supported in part by the National Science Foundation under grants NSF GJ-36393 and DCR 74-23679 (NSF)     

LU decompositions of generalized diagonally dominant matrices

Summary Using the simple vehicle ofM-matrices, the existence and stability ofLU decompositions of matricesA which can be scaled to diagonally dominant (possibly singular) matrices are investigated. Bounds on the growth factor for Gaussian elimination onA are derived. Motivation for this study is provided in part by applications to solving homogeneous systems of linear equationsAx=0, arising in Markov queuing networks, input-output models in economics and compartmental systems, whereA or –A is an irreducible, singularM-matrix.This paper extends earlier work by Funderlic and Plemmons and by Varga and Cai.Research sponsored by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide CorporationResearch supported in part by the National Science Foundation under Grant No. MCS 8102114Research supported in part by the U.S. Army Research Office under contract no. DAAG 29-81-k-0132