On square roots of M-operators

AbstractUsing a cone order in a real Banach space, the concept of an M-operator is discussed, and the existence and uniqueness of the square roots of M-operators are studied. In this way most of the results of the paper by Alefeld and Schneider are generalized to an infinite dimensional case. For some finite dimensional situations our results also seem to be new

Collatz-Wielandt numbers in general partially ordered spaces

AbstractIt has been shown by Förster and Nagy that the convergence of the Collatz-Wielandt numbers to bound the spectral radius of a K-nonnegative linear operator T in a partially ordered Bannach space essentially depends upon the starting vector of the iteration process. In this paper necessary and sufficient conditions for convergence are presented under rather general hypotheses; e.g., the emptiness of the interior of the order cone K is admitted. We also present min sup and max inf characterizations of the spectral radius in the spirit of earlier work

Aggregation/disaggregation as a theoretical tool

The main aim of this contribution is to establish convergence of some iterative procedures that play an important role in the PageRank computation. The problems we are interested in are considered in two recent papers by Ipsen and Selee, and Lee, Golub andZenios. Both these papers present new ideas to solve the celebrated problem of the PageRank. Our aim is to show that the results and some generalizations of them can be proven via an application of the iterative aggregation/disaggregation methods. One of the results may be of particular interest. It concerns a proof that the two-stage algorithm proposed by Lee, Golub and Zenios does compute the PageRank. This problem has been raised in the literature. We answer this question in positive by showing appropriate necessary and sufficient conditions. In addition a short proof of the celebrated Google lemma is presented

Convergence of iterative aggregation/disaggregation methods based on splittings with cyclic iteration matrices

Iterative aggregation/disaggregation methods (IAD) belong to competitive tools for computation the characteristics of Markov chains as shown in some publications devoted to testing and comparing various methods designed to this purpose. According to Dayar T., Stewart W.J., ``Comparison of partitioning techniques for two-level iterative solvers on large, sparse Markov chains,\u27\u27 SIAM J. Sci. Comput., Vol.21, No. 5, 1691-1705 (2000), the IAD methods are effective in particular when applied to large ill posed problems. One of the purposes of this paper is to contribute to a possible explanation of this fact. The novelty may consist of the fact that the IAD algorithms do converge independently of whether the iteration matrix of the corresponding process is primitive or not. Some numerical tests are presented and possible applications mentioned; e.g. computing the PageRank

Ljusternik acceleration and the extrapolated S.O.R. method

summary:A general semi-iterative acceleration technique is described for improving the convergence of stationary iterative methods. By applying this technique to the successive over relaxation (S.O.R.) iterations with a particular nonoptimal relaxation factor an acceleration of the rate of convergence is obtained which is superior to the optimal S.O.R

Monotony of solutions of some difference and differential equations

AbstractIn this contribution motivated by some analysis of the first author concerning bounds of topological entropy it is shown that a well known sufficient condition for a difference and differential equation with constant real coefficients to possess strictly monotone solution appears to be also necessary. Transparent proofs of adequate generalizations to Banach space analogs are presented