122751787
Varianten zum Konvergenzsatz von Simons und Anwendungen in der Choquettheorie
122589104
Bemerkungen zum von Neumannschen Minimaxtheorem
120206552
Beschränktheitsaussagen für sublineare und bisublineare Funktionale
122531296
On the Schur complement and the LU-factorization of a matrix
Further to a paper by Cottle [12] we discuss how the Lu-factorization of a matrix could be used to study its Schur complements and vice-versa. In addition, it is argued that the Schur complement provides a compact tool for studying problems rekated to the incomplete Choleski factorization.
120227102
Preconditionings and splittings for rectangular systems
SummaryRecently Eiermann, Marek, and Niethammer have shown how to applysemiiterative methods to a fixed point systemx=Tx+c which isinconsistent or in whichthe powers of the fixed point operator T have no limit, to obtain iterative methods which converge to some approximate solution to the fixed point system. In view of their results we consider here stipulations on apreconditioning QAx=Qb of the systemAx=b and, separately, on asplitting A=M−N which lead to fixed point systems such that, with the aid of a semiiterative method, the iterative scheme converges to a weighted Moore-Penrose solution to the systemAx=b. We show in several ways that to obtain a meaningful limit point from a semiiterative method requires less restrictions on the splittings or the reconditionings than those which have been required in the classical Picard iterative method (see, e.g., the works of Berman and Plemmons, Berman and Neumann, and Tanabe).We pay special attention to the case when the weighted Moore-Penrose solution which is sought is the minimal norm least squares solution toAx=b.
121748192
A note on generalizations of strict diagonal dominance for real matrices
Abstract We investigate classes of real square matrices possessing some weakened from of strict diagonal dominance of a real matrix whose diagonal entries are all positive. The intersection of each one of these classes with the set of all real matrices, with nonpositive off-diagonal elements, coincides with the set of all nonsingular M - matrices.
121142956
A conjecture concerning the Hadamard product of inverses of M-matrices
Abstract We conjecture that for an n × n matrix A which is an inverse of an M -matrix, the Hadamard product A ∘ A is also an inverse of an M -matrix. We have checked this conjecture without failure on many many examples. But here we show that for quite a few well known classes of inverses of M -matrices, the conjecture is true. It is known that the more general conjecture, that when A and B are n × n inverses of M -matrices, then A ∘ B is also an inverse of an M -matrix, is false. However, here too we are able to display some classes of inverses of M -matrices which are closed under taking Hadamard products.
120140045
Models of parallel chaotic iteration methods
Abstract We consider two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations Ax = b . In the first model each processor can carry out an arbitrary number of local iterations before the next global approximation to the solution is formed. In the second model any processor can update the global approximation which resides in the central processor at any time. This model is a generalization of a sequential iterative scheme due to Ostrowski called the free steering group Jacobi iterative scheme and a chaotic relaxation point iterative scheme due to Chazan and Miranker. We show that when A is a monotone matrix and all the splittings are weak regular, both models lead to convergent schemes.