Numerical differentiation for two-dimensional scattered data on arbitrary domain base on Hermite extension with an implicit iteration process

<abstract><p>In this paper, we develop a method for numerical differentiation of two-dimensional scattered input data on arbitrary domain. A Hermite extension approach is used to realize the approximation and a modified implicit iteration method is proposed to stabilize the approximation process. For functions with various smooth conditions, the numerical solution process of the method is uniform. The error estimates are obtained and numerical results show that the new method is effective. The advantage of the method is that it can solve the problem in any domain.</p></abstract>

Approximate Factorization Method Using Alternating Cell Direction Implicit Method: Comparison of Convergence Characteristics Using Basic Model Equations

In this paper, a fast implicit iteration scheme called the alternating cell directions implicit (ACDI) method is combined with the approximate factorization scheme. The use of fast implicit iteration methods with unstructured grids is hardly. The proposed method allows fast implicit formulations to be used in unstructured meshes, revealing the advantages of fast implicit schemes in unstructured meshes. Fast implicit schemes used in structured meshes have evolved considerably and are much more accurate and robust, and are faster than explicit schemes. It is a crucial novel development that such developed schemes can be applied to unstructured schemes. In steady incompressible potential flow, the convergence character of the scheme is compared with the Runge-Kutta order 4 (RK4), Laasonen, point Gauss–Seidel iteration, old version ACDI, and line Gauss–Seidel iteration methods. The scheme behaves like an approximation of the fully implicit method (Laasonen) up to an optimum pseudo-time-step size. This is a highly anticipated result because the approximate factorization method is an approach to a fully implicit formulation. The results of the numerical study are compared with other fast implicit methods (e.g., the point and line Gauss–Seidel methods), the RK4 method, which is an explicit scheme, and the Laasonen method, which is a fully implicit scheme. The study increased the accuracy of the ACDI method. Thus, the new ACDI method is faster in unstructured grids than other methods and can be used for any mesh construction.

Fixed point results of an implicit iterative scheme for fractal generations

<abstract><p>In this paper, we derive the escape criteria for general complex polynomial $ f(x) = \sum_{i = 0}^{p}a_{i}x^{i} $ with $ p\geq2 $, where $ a_{i} \in \mathbb{C} $ for $ i = 0, 1, 2, \dots, p $ to generate the fractals. Moreover, we study the orbit of an implicit iteration (i.e., Jungck-Ishikawa iteration with $ s $-convexity) and develop algorithms for Mandelbrot set and Multi-corn or Multi-edge set. Moreover, we draw some complex graphs and observe how the graph of Mandelbrot set and Multi-corn or Multi-edge set vary with the variation of $ a_{i} $'s.</p></abstract>

Low-Rank Alternating Direction Implicit Iteration in pyMOR

The low-rank alternating direction implicit (LR-ADI) iteration is an effective method for solving large-scale Lyapunov equations. In the software library pyMOR, solutions to Lyapunov equations play an important role when reducing a model using the balanced truncation method. In this article we introduce the LR-ADI iteration as well as pyMOR, while focusing on its features which are relevant for integrating the iteration into the library. We compare the run time of the iteration's pure pyMOR implementation with those achieved by external libraries available within the pyMOR framework.

Strong Convergence Theorems for Variational Inequalities and Common Fixed-Point Problems Using Relaxed Mann Implicit Iteration Methods

Mann-like iteration methods are significant to deal with convex feasibility problems in Banach spaces. We focus on a relaxed Mann implicit iteration method to solve a general system of accretive variational inequalities with an asymptotically nonexpansive mapping in the intermediate sense and a countable family of uniformly Lipschitzian pseudocontractive mappings. More convergence theorems are proved under some suitable weak condition in both 2-uniformly smooth and uniformly convex Banach spaces.

On New Quicker N-Tupled Xed Point Implicit Iterations for Contractive-Like Mappings and Comparison of their Rate of Convergence in Hyperbolic Metric Spaces

In this paper, we study the convergence of new implicit iterations dealing with n-tupled fixed point results for nonlinear contractive-like mappings on W-hyperbolic metric spaces. Herein, we demonstrate that our newly implicit iteration schemes have faster rate of convergence than implicit S-iteration process, implicit Ishikawa and Mann type iteration processes. Furthermore, a numerical simulation to improve our theoretical results is obtained.