A Family of higher-order convergent iterative methods for computing the Moore-Penrose inverse

In this paper, we extend the iterative method for computing the inner inverse of a matrix proposed in Li and Li [W.G. Li, Z. Li, A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix, Applied Mathematics and Computation 215 (2010) 3433-3442] to compute the Moore-Penrose inverse of a matrix, and show that the generated sequence converges to the Moore-Penrose inverse of a matrix in a higher order. The performance of the method is tested on some randomly generated matrices.     

A note on higher-order surface radiation conditions

One way of improving the surface radiation-condition approachmight be to use a condition higher than the second order. Here,this possibility is examined. For this purpose, surface radiationconditions higher than the second order are derived by a methodwhich is, to a certain extent, similar to the method introducedby Jones (1988, IMA J. Appl. Math. 41, 21–30). It is shownthat the first- and the second-order conditions are identicallyequal to the corresponding conditions given by Jones. Then,the third- and the fourth-order conditions, together with conditionsobtained by the mode-annihilation method are tested againstthe second order, and exact results are given for an impedancesphere and for a penetrable sphere in a variety of circumstances.It has been observed that introduction of these higher-orderradiation conditions from moderate to high frequencies improvesthe approximation considerably in comparison with result sobtained by the use of a second-order radiation condition, especiallyin cases in which creeping waves are less pervasive.     

A higher-order Markov model for the Newsboy's problem

Markov models are commonly used in modelling many practical systems such as telecommunication systems, manufacturing systems and inventory systems. However, higher-order Markov models are not commonly used in practice because of their huge number of states and parameters that lead to computational difficulties. In this paper, we propose a higher-order Markov model whose number of states and parameters are linear with respect to the order of the model. We also develop efficient estimation methods for the model parameters. We then apply the model and method to solve the generalised Newsboy's problem. Numerical examples with applications to production planning are given to illustrate the power of our proposed model.     

A framework for robustness to ambiguity of higher-order beliefs

Standard type spaces induce belief structures defined by precise beliefs. This paper proposes and analyzes simple procedures for constructing perturbations of such belief structures in which beliefs have a degree of ambiguity. Specifically, we construct ambiguous type spaces whose induced (ambiguous) belief hierarchies approximate the standard, precise, belief hierarchies corresponding to the initial type space. Based on a metric that captures the resulting approximation, two alternative procedures to construct such perturbations are introduced, and are shown to yield a simple and intuitive characterization of convergence to the initial unperturbed environment. As a special case, one of these procedures is shown to characterize the set of all finite perturbations. The introduced perturbations and their convergence properties provide conceptual foundations for the analysis of robustness to ambiguity of various solutions concepts, and for various decision rules under ambiguity.     

A Steffensen-like method and its higher-order variants

For solving nonlinear equations, we suggest a second-order parametric Steffensen-like method, which is derivative free and only uses two evaluations of the function in one step. We also suggest a variant of the Steffensen-like method which is still derivative free and uses four evaluations of the function to achieve cubic convergence. Moreover, a fast Steffensen-like method with super quadratic convergence and a fast variant of the Steffensen-like method with super cubic convergence are proposed by using a parameter estimation. The error equations and asymptotic convergence constants are obtained for the discussed methods. The numerical results and the basins of attraction support the proposed methods.     

A recursive method for computing interpolants

In this paper, we describe a recursive method for computing interpolants defined in a space spanned by a finite number of continuous functions in R^d${\mathbb{R}}^d$. We apply this method to construct several interpolants such as spline interpolants, tensor product interpolants and multivariate polynomial interpolants. We also give a simple algorithm for solving a multivariate polynomial interpolation problem and constructing the minimal interpolation space for a given finite set of interpolation points.     

A higher-order finite element scheme for incompressible lubrication calculations

A higher-order finite element scheme is formulated for incompressible lubrication calculations, based on the energy functional of the lubricating system (derived from the variational technique) and the hierarchical approximation concept. The current formulation ensures the pressure continuity across inter-element mating boundaries. Since this is a hierarchical formulation, it facilitates convergence studies of results. Numerical examples are provided to demonstrate the accuracy of the proposed method, the simplicity of modeling, applications, and the convergence characteristics of numerical solutions.     

A higher-order Godunov method for modeling finite deformation in elastic-plastic solids

In this paper we develop a first-order system of conservation laws for finite deformation in solids, describe its characteristic structure, and use this analysis to develop a second-order numerical method for problems involving finite deformation and plasticity. The equations of mass, momentum, and energy conservation in Lagrangian and Eulerian frames of reference are combined with kinetic equations of state for the stress and with caloric equations of state for the internal energy, as well as with auxiliary equations representing equality of mixed partial derivatives of the deformation gradient. Particular attention is paid to the influence of a curl constraint on the deformation gradient, so that the characteristic speeds transform properly between the two frames of reference. Next, we consider models in rate-form for isotropic elastic-plastic materials with work-hardening, and examine the circumstances under which these models lead to hyperbolic systems for the equations of motion. In spite of the fact that these models violate thermodynamic principles in such a way that the acoustic tensor becomes nonsymmetric, we still find that the characteristic speeds are always real for elastic behavior, and essentially always real for plastic response. These results allow us to construct a second-order Godunov method for the computation of three-dimensional displacement in a one-dimensional material viewed in the Lagrangian frame of reference. We also describe a technique for the approximate solution of Riemann problems in order to determine numerical fluxes in this algorithm. Finally, we present numerical examples of the results of the algorithm.     

A note on the controllability of higher-order linear systems

In this paper, a new condition for the controllability of higher-order linear dynamical systems is obtained. The suggested test contains rank conditions of suitably defined matrices and is based on the notion of compound matrices and the Binet-Cauchy formula.     

A general view on computing communities

We define a community structure of a graph as a partition of the vertices into at least two sets with the property that each vertex has connections to relatively many vertices in its own set compared to any other set in the partition and refer to the sets in such a partition as communities  . We show that it is NP-hard to compute a community containing a given set of vertices. On the other hand, we show how to compute a community structure in polynomial time for any connected graph containing at least four vertices except the star graph _Sn$S_n$. Finally, we generalize our results and formally show that counterintuitive aspects are unavoidable for any definition of a community structure with a polynomial time algorithm for computing communities containing specific vertices.     

Solutions for higher-order dynamic equations on time scales

Let T be a time scale. We study the existence of positive solutions for the nonlinear four-point singular boundary value problem with higher-order p-Laplacian dynamic equations on time scales. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with p-Laplacian operator are obtained.     

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

A simplicial homotopy algorithm for computing zero points on polytopes

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A new algorithm for computing modified z-transforms

A new method of computing modified z-tranforms is introduced,which is easy to implement and use. The new algorithm is explainedin detail, and a compact listing is included. Six examples aregiven to demonstrate different uses of the algorithm. Assumptionsused to develop the basic algorithm are explained, and proceduresto overcome the restrictions of these assumptions are explainedand demonstrated. How to solve multirate digital systems problemsusing the introduced algorithm is demonstrated with an example.Time-saving and memory-saving modifications of the algorithmfor multirate problems are also explained.     

A mesh transformation method for computing microstructures

Summary. A numerical method is established to solve the problem of minimizing a nonquasiconvex potential energy. Convergence of the method is proved both in the case on its own and in the case when it is combined with a weak boundary condition. Numerical examples are given to show that the method, especially when applied together with a continuation method and some other numerical techniques, is not only successful and efficient in solving problems with laminated microstructures but also capable of computing more complicated microstructures. Received March 17, 2000 / Published online April 5, 2001     

A new rule for computing Clebsch-Gordan series

A new combinatorial rule for computing the Clebsch-Gordan series of a tensor product of irreducible SU(n)-representations is introduced. This rule provides an efficient algorithm for computing features of the Clebsch-Gordan multiplicities, such as the “characteristic nullspaces” of tensor operators, that are not easily computed by other methods. Furthermore, unlike the Littlewood-Richardson rule, this new rule generalizes, in principle, to other semisimple Lie groups. This generalization and its consequences for tensor operators are discussed in some detail. As an application, it is shown that as n increases, it is possible to find characteristic nullspaces for SU(n)-tensor operators, which have arbitrarily large multiplicity.