Vol 10, No 1 (2017)

In this paper, a strongly NP-hard problem of finding a family of disjoint subsets with given cardinalities in a finite set of points from a Euclidean space is considered. Minimization of the sum over all required subsets of the sum of the squared distances from the elements of these subsets to their geometric centers is used as the search criterion. It is proved that if the coordinates of the input points are integer and the space dimension and the number of required subsets are fixed (i.e., bounded by some constants), the problem is a pseudopolynomial time solvable one.

This paper deals with the numerical simulations of a system of diffusion-reaction equations in the context of a porous medium. We start by giving a microscopic model and then the upscaled version (i.e., homogenized or continuum model) of it from the previous works of the author. Since with the help of homogenization we obtain the macroscopic description of amodel that is microscopically heterogeneous, via these numerical simulations, we show that this macroscopic description approximates the microscopicmodel, which contains the heterogeneities and oscillating terms at the pore scale such as diffusion coefficients.

This paper is concerned with the semilocal convergence of a continuation method between two third-order iterative methods, namely, the Halley’s and the convex acceleration of Newton’s method, also known as the Super-Halley’s method. This convergence analysis is discussed using the recurrence relations approach. This approach simplifies the analysis and leads to improved results. The convergence analysis is established under the assumption that the second Frëchet derivative satisfies Lipschitz continuity condition. An existence-uniqueness theorem is given. Also, a closed form of error bound is derived in terms of a real parameter α ∈ [0, 1]. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences [15], we observed that our analysis gives better results. Further, we have observed that for particular values of the α, our analysis reduces to those for the Halley’s method (α = 0) and the convex acceleration of Newton’s method (α = 1), respectively, with improved results.

The main objective and inspiration in the construction of two- and three-point with memory method is to attain the utmost computational efficiency, without any additional function evaluations. At this juncture, we have modified the existing fourth and eighth order without memory method with optimal order of convergence by means of different approximations of self-accelerating parameters. The parameters have been calculated by Hermite interpolating polynomial, which accelerates the order of convergence of the without memory methods. In particular, the R-order convergence of the proposed two- and three-step with memory methods is increased from four to five and eight to ten. One more advantage of these methods is that the condition f′(x) ≠ 0, in the neighborhood of the required root, imposed on Newton’s method, can be removed. Numerical comparison is also stated to confirm the theoretical results.