Computing

Es ist bekannt, daß das klassische Newton-Verfahren kubisch gegen eine einfache Nullstelle konvergiert, wenn die zweite Ableitung an der Nullstelle verschwindet. Wir zeigen zunächst, daß sich diese Eigenschaft nicht auf das Intervall-Newton-Verfahren überträgt. Verwendet man jedoch anstelle der intervallmäßigen Auswertung der Ableitung die Mittelwertform oder die zentrierte Form, so erhält man wieder kubische Konvergenz.

Nowadays, the Internet of Things (IoT) networks provide benefits to humans in numerous domains by empowering the projects of smart cities, healthcare, industrial enhancement and so forth. The IoT networks include nodes, which deliver the data to the destination. However, the network nodes’ connectivity is affected by the nodes’ removal caused due to the malicious attacks. The ideal plan is to construct a topology that maintains nodes’ connectivity after the attacks and subsequently increases the network robustness. Therefore, for constructing a robust scale-free network, two different mechanisms are adopted in this paper. First, a Multi-Population Genetic Algorithm (MPGA) is used to deal with premature convergence in GA. Then, an entropy based mechanism is used, which replaces the worst solution of high entropy population with the best solution of low entropy population to improve the network robustness. Second, two types of Edge Swap Mechanisms (ESMs) are proposed. The Efficiency based Edge Swap Mechanism (EESM) selects the pair of edges with high efficiency. While the second ESM named as EESM-Assortativity, transforms the network topology into an onion-like structure to achieve maximum connectivity between similar degree network nodes. Further, Hill Climbing (HC) and Simulated Annealing (SA) methods are used for optimizing the network robustness. The simulation results show that the proposed MPGA Entropy has 9% better network robustness as compared to MPGA. Moreover, both the proposed ESMs effectively increase the network robustness with an average of 15% better robustness as compared to HC and SA. Furthermore, they increase the graph density as well as network’s connectivity.

In this paper, we present the tangential block decomposition for block-tridiagonal matrices which is in many aspects similar to the frequency filtering method by Wittum [8] and also to the tangential frequency filtering decomposition by Wagner [6]–[7]. In opposite to the methods of Wittum and Wagner, for the class of model problems our approach does not use any test vectors for its implementation. Similar to many iterative methods, it needs only bounds for extremal eigenvalues. Theoretical properties of our scheme are similar to those for the ADI-method. The practical convergence of the presented method is illustrated by numerical examples.

For the matricesA mentioned in the headline we determine the limit points up to which there is possible a real factorization of the formA=QQ T . HereQ=(q ij ) is a circulant matrix, where from the elementsq ij andq ji withi≠j always one element is vanishing.

An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method are the coefficient matrix and zero energy modes, which are determined from nodal coordinates and knowledge of the differential equation. Efficiency of the resulting algorithm is demonstrated by computational results on real world problems from solid elasticity, plate bending, and shells.

The interval numberi (G) of a graphG withn vertices is the lowest integerm such thatG is the intersection graph of some family of setsI 1, ...,I n with everyI i being the union of at mostm real intervals. In this article, an idea is presented for the algorithmic determination ofi (G), ifG is triangle-free. An example for the application of these considerations is given.

An automatic quadrature algorithm especially designed for double integration of functions with some form of singular behaviour on the boundary of the integration region is described, and its FORTRAN code is presented. The algorithm is based on the use of the product trapezoidal rule, after a non-linear transformation of the integrand in both variables renders a new integrand function whose derivatives vanish on the (transformed) boundary. Numerical results demonstrate the ability of the algorithm to obtain high accuracies in dealing automatically with pathological singularities of non-specific types.

In the case of symmetric and positive definite plane elliptic boundary value problems, the condition numbers of the stiffness matrices arising from finite element discretizations grow only quadratically with the number of refinement levels, if one uses hierarchical bases of the finite element spaces instead of the usual nodal bases; see [9]. Here we show that results of the same type hold for nonsymmetric problems and we describe the interesting consequences for the solution of the discretized problems by Krylov-space methods.