Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {T n (f )}, under suitable assumptions on the associated generating function f . In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {T −1 n (g)T n (f )}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f Numer Algor in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators.

The need for smart e-learning environments is resulting in new challenges for researchers and practitioners to develop intelligent systems that can be used to automate the Higher Education (HE) activities in an intelligent way. Some common examples of such activities are “analyzing, finding, and ranking the right resource to teach a course,” “analyzing and finding the people with common research interests to start joint research projects,” and “using data analytics and machine reasoning techniques for conducting the exams with different levels of complexities.” Ontological reasoning and smart data analytics can play an important role in analyzing and automating these HE activities and processes. In this paper, we present a framework named as Higher Education Activities and Processes Automation Framework (HEAPAF). The HEAPAF framework can be used to identify, extract, process, and produce the semantically enriched data in machine understandable format from different educational resources. We also present the Higher Education Ontology (HEO) that we designed and developed to accommodate the HE data and then to perform analysis and reasoning on it. As a proof of concept, we present a case study on the topic, “analyzing, finding, and ranking the right resources to teach a course,” which can dramatically improve the learning patterns of students in the growing smart educational environment. Finally, we provide the evaluation of our framework as evidence of its competency and consistency in improving academic analytics for educational activities and processes by using machine reasoning.

A frozen Jacobian iterative method is proposed for solving systems of nonlinear equations. In particular, we are interested in solving the systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). In a single instance of the proposed iterative method DEDF, we evaluate two Jacobians, one inversion of the Jacobian and four function evaluations. The direct inversion of the Jacobian is computationally expensive, so, for a moderate size, LU factorization is a good direct method to solve the linear system. We employed the LU factorization of the Jacobian to avoid the direct inversion. The convergence order of the proposed iterative method is at least eight, and it is nine for some particular classes of problems. The discretization of IVPs and BVPs is employed by using Jacobi-Gauss-Lobatto collocation (J-GL-C) method. A comparison of J-GL-C methods is presented in order to choose best collocation method. The validity, accuracy and the efficiency of our DEDF are shown by solving eleven IVPs and BVPs problems. 379 ordinary differential equations. They used implicit Runge-Kutta method to solve the system of ordinary differential equations and obtained highly accurate numerical solutions, for four different kinds of nonlinear 1+1 Schrödinger equations. In another article Bhrawy et al. [5] solved the nonlinear reaction-diffusion equations by using J-GL-C method. In the solution of the complex generalized Zakharov system of equation [4], the J-GL-C method is also the method of discretization. The further application of pseudospectral collocation techniques for solving nonlinear IVPs and BVPs can be found in [6,8,9,17] and references therein. The J-GL-C method is a parametric pseudospectral collocation method. By selecting different values of the parameters, we can get different pseudospectral collocation methods. The Legendre, Chebyshev, and Gegenbauer pseudospectral collocation methods are special cases of J-GL-C method [16].The Jacobi polynomials are the eigenfunctions of a singular Strum-Liouville problem [16] (1 − x 2 )σ (x) + (θ − φ + (θ + φ + 2)x)σ (x) + n(n + θ + φ + 1)σ(x) = 0 .The following recurrence relation produces the Jacobi polynomials J