Gaussian elimination is stable for the inverse of a diagonally dominant matrix

Let be a row diagonally dominant matrix, i.e.,

where with We show that no pivoting is necessary when Gaussian elimination is applied to Moreover, the growth factor for does not exceed The same results are true with row diagonal dominance being replaced by column diagonal dominance.

     

On the parallel GSAOR method for block diagonally dominant matrices

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.     

Invex Functions and Generalized Convexity in Multiobjective Programming

Martin (Ref. 1) studied the optimality conditions of invex functions for scalar programming problems. In this work, we generalize his results making them applicable to vectorial optimization problems. We prove that the equivalence between minima and stationary points or Kuhn–Tucker points (depending on the case) remains true if we optimize several objective functions instead of one objective function. To this end, we define accurately stationary points and Kuhn–Tucker optimality conditions for multiobjective programming problems. We see that the Martin results cannot be improved in mathematical programming, because the new types of generalized convexity that have appeared over the last few years do not yield any new optimality conditions for mathematical programming problems.     

Uzawa block relaxation domain decomposition method for a two-body frictionless contact problem

We propose a Uzawa block relaxation domain decomposition method for a two-body frictionless contact problem. We introduce auxiliary variables to separate subdomains representing linear elastic bodies. Applying a Uzawa block relaxation algorithm to the corresponding augmented Lagrangian functional yields a domain decomposition algorithm in which we have to solve two uncoupled linear elasticity subproblems in each iteration while the auxiliary variables are computed explicitly using Kuhn–Tucker optimality conditions.     

Error bounds for analytic systems and their applications

Using a 1958 result of Lojasiewicz, we establish an error bound for analytic systems consisting of equalities and inequalities defined by real analytic functions. In particular, we show that over any bounded region, the distance from any vectorx in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated atx. For quadratic systems satisfying certain nonnegativity assumptions, we show that this exponent is equal to 1/2. We apply the error bounds to the Karush—Kuhn—Tucker system of a variational inequality, the affine variational inequality, the linear and nonlinear complementarity problem, and the 0–1 integer feasibility problem, and obtain new error bound results for these problems. The latter results extend previous work for polynomial systems and explain why a certain square-root term is needed in an error bound for the (monotone) linear complementarity problem.The research of this author is based on work supported by the Natural Sciences and Engineering Research Council of Canada under grant OPG0090391.The research of this author is based on work supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739 and by the Office of Naval Research under grant 4116687-01.     

Sharp thresholds for the phase transition between primitive recursive and Ackermannian Ramsey numbers

We compute the sharp thresholds on g at which g-large and g-regressive Ramsey numbers cease to be primitive recursive and become Ackermannian.We also identify the threshold below which g-regressive colorings have usual Ramsey numbers, that is, admit homogeneous, rather than just min-homogeneous sets.     

The convergence of Jacobi–Davidson iterations for Hermitian eigenproblems

Rayleigh quotient iteration is an iterative method with some attractive convergence properties for finding (interior) eigenvalues of large sparse Hermitian matrices. However, the method requires the accurate (and, hence, often expensive) solution of a linear system in every iteration step. Unfortunately, replacing the exact solution with a cheaper approximation may destroy the convergence. The (Jacobi‐) Davidson correction equation can be seen as a solution for this problem. In this paper we deduce quantitative results to support this viewpoint and we relate it to other methods. This should make some of the experimental observations in practice more quantitative in the Hermitian case. Asymptotic convergence bounds are given for fixed preconditioners and for the special case if the correction equation is solved with some fixed relative residual precision. A dynamic tolerance is proposed and some numerical illustration is presented. Copyright © 2002 John Wiley & Sons, Ltd.     

Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems

Necessary and sufficient conditions are obtained for the existence of symmetric positive solutions to the boundary value problem

     

An upper and lower solution approach for singular boundary value problems with sign changing non‐linearities

We obtain via Schauder's fixed point theorem new results for singular second‐order boundary value problems where our non‐linear term f(t,y,z) is allowed to change sign. In particular, our problem may be singular at y=0, t=0 and/or t=1. Copyright © 2002 John Wiley & Sons, Ltd.     

An Analog of the Poincarée Separation Theorem for Normal Matrices and the Gauss–Lucas Theorem

We establish an analog of the Cauchy–Poincarée separation theorem for normal matrices in terms of majorization. A solution to the inverse spectral problem (Borg type result) is also presented. Using this result, we generalize and extend the Gauss–Lucas theorem about the location of roots of a complex polynomial and of its derivative. The generalization is applied to prove old conjectures due to de Bruijn–Springer and Schoenberg.     

The Construction of an Evolution System in the Hyperbolic Case and Applications

We study the Cauchy problem for abstract linear and quasi–linear non–autonomous evolution equations of hyperbolic type using semigroup theory. Under weak differentiability assumptions on the time regularity of the coefficients we prove well–posedness and regularity of a solution. The abstract results are illustrated by their application to a series of equations of mathematical physics.     

On conditions for and the order of approximation of functions by operators of classS 2m

Following P. P. Korovkin, we study conditions for the convergence of operators of classesS _2m to continuous functions and the asymptotics of approximation by such operators to differentiable functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 654–661, May, 2000.     

On Lower Bounds for Lp Norms in the Central Limit Theorem for Independent and m-Dependent Random Variables

We derive lower bounds for L_p norms , in the central limit theorem for independent and m–dependent random variables with finite fifth order absolute moments and for independent and m–dependent identically distributed random variables with fourth order moments.