120219316
On bounds for the convergence of the SSOR method for H-matrices
In [1] G. Alefeld and R S. Varga show that if A is an H-matrix, then where and JA are, respectively, the SSOR and the Jacobi iteration matrices associated with A. We investigate the possibility for[M0001] when Ais an H-matrix and show that unless does not hold. Comparison is made with the sharpness of a similar bound on the ω's for which the SOR iteration matrix associated with A converges.
18512153
Algorithms for Computing Bases for the Perron Eigenspace with PrescribedNonnegativity and Combinatorial Properties
Let $P$ be an $n \times n$ nonnegative matrix. In this paper the authors introduce a method called the SCANBAS algorithm for computing a union of (Jordan) chains $\cal{C}$ corresponding to the Perron eigenvalue of $P$, such that $\cal{C}$ consists of nonnegative vectors only and such that at each height, $\cal{C}$ contains the maximal number of nonnegative vectors of that height possible in a height basis for the Perron eigenspace of $P$. It is further shown that $\cal{C}$ can be extended to a height basis for the Perron eigenspace of $P$. The chains are extracted from transform components of $P$ that are, in turn, polynomials in $P$. When the Perron eigenspace has a Jordan basis consisting of nonnegative vectors only, this algorithm computes such a basis. The paper concludes with various examples computed by the algorithm using MATLAB. The work here continues and deepens work on computing nonnegative bases for the Perron eigenspace from polynomials in the matrix already begun by Hartwig, Neumann, and Rose and by Neumann and Schneider.
121745003
The Kahan S.O.R. convergence bound for nonsingular and irreducible M-matrices
Abstract Let A be a nonsingular M -matrix, and let π be a block partitioning of A such that the diagonal blocks are square. Denote by J A π and A π ω the block Jacobi and the block S.O.R. iteration matrices arising from and associated with the partitioning π of A , respectively. In [5] Kahan showed that ρ ( A π ω J A π )]. Under the assumption that J A π is irreducible we examine the question of when ρ( A π ω' )=1 for certain recurring M -matrices.
120901930
Nonnegative jordan bases
In this paper we are concerned with the problem of when the generalized eigenspace of an n × n nonegative matrix A corresponding to its spectrai radius ρ has a nonnegative Jordan basis. Richman and Schneider have shown this to hold when, and only when, the Weyr characteristics of A corresponding to ρ is equal to the vector of levels in the singular graph of A Here we develop sufficient conditions for this equality to hold based upon the linear independence of a certain set of vectors which are generated according to a process of elimination of totally independent chains from the singular graph. We show that this condition comes close to being a further characterization for the existence of a nonnegative Jordan basis for this eigenspace and actually conjecture that this is the case. Finally, we show that this eigenspace has a Rothblum basis which is a Jordan basis if and only if the singular graph is a disjoint union of chains.
123146090
Corrections and additions to: "principal components of minus m-matrices"
122147649
Cone reachability for linear differential systems
Given a real nxn matrix A and a simplicial cone which is positively invariant under the linear differential system , we consider the problem of characterizing the closure of the set XA(K) of all initial points x(0) ∈ Rn which give rise to solutions x(t) that enter (and remain in) K. An explicit characterization of the closure of XA(K) is given first in case A is diagon-able with a real spectrum. Subsequently a characterization for the closure of XA(K) is given in case there exist k ∈ int K such that −Ak ∈ K, but in the absence of any other explicit assumptions on A.
120497198
A maximality theorem concerning extreme points
120206552
Beschränktheitsaussagen für sublineare und bisublineare Funktionale
118524160
Automatic Continuity of Linear Operators
The present survey is essentially based on some recent automatic continuity results which have been obtained by Ernst Albrecht and the author in the joint papers [1;2;3]. In contrast to the excellent account of automatic continuity theory in Sinclair's lecture notes [29] and in Dales' recent survey article [9], the emphasis of the present exposition lies on the automatic continuity problem for linear operators acting between spaces of functions and distributions. The continuity of such operators will be derived from certain conditions which naturally arise in the theory of linear systems, namely from time-invariance and causality on the one side and from dissipativity on the other. The automatic continuity of all time-invariant and causal linear operators turns out to be just one achievement of a far-reaching new theory on automatic continuity which also applies to the problem of continuity for algebrahomomorphisms, derivations, intertwining operators, and local operators. Some central features of this theory will be sketched; in particular, some extensions and variants of the classical principle of uniform boundedness will be discussed.