scholarly journals Semi-Local Analysis and Real Life Applications of Higher-Order Iterative Schemes for Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 92
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Ioannis K. Argyros ◽  
Sanjeev Kumar
Keyword(s):  
Nonlinear Systems ◽  
Applied Mathematics ◽  
Real Life ◽  
Higher Order ◽  
Local Analysis ◽  
First Order ◽  
The Family ◽  
Lipschitz Constants ◽  
Theoretical Results ◽  
Real World Problems

Our aim is to improve the applicability of the family suggested by Bhalla et al. (Computational and Applied Mathematics, 2018) for the approximation of solutions of nonlinear systems. Semi-local convergence relies on conditions with first order derivatives and Lipschitz constants in contrast to other works requiring higher order derivatives not appearing in these schemes. Hence, the usage of these schemes is improved. Moreover, a variety of real world problems, namely, Bratu’s 1D, Bratu’s 2D and Fisher’s problems, are applied in order to inspect the utilization of the family and to test the theoretical results by adopting variable precision arithmetics in Mathematica 10. On account of these examples, it is concluded that the family is more efficient and shows better performance as compared to the existing one.

scholarly journals Stability and Controllability of Nonlinear Systems

2020 ◽  
pp. 51-60
Author(s):  
S. T. Cotterell ◽  
I. Davies ◽  
L. C. Abraham
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Equilibrium Solution ◽  
Indirect Method ◽  
Real Life ◽  
Linearization Method ◽  
Controlled System ◽  
The Stability ◽  
Theoretical Results

This paper is aimed at establishing new stability and controllability results for nonlinear systems. The approach is to use the Lyapunov indirect method to obtain the stability of the equilibrium solution of the uncontrolled nonlinear system by applying the Jacobi’s linearization method and the controllability of the controlled system obtained by the rank criterion for properness. Example is given with a real-life application to illustrate the effectiveness of the theoretical results.

scholarly journals Consensus Tracking for Multiagent Systems with Nonlinear Dynamics

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Runsha Dong
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Systems ◽  
Numerical Simulations ◽  
Multiagent Systems ◽  
Second Order ◽  
Control Laws ◽  
First Order ◽  
Consensus Tracking ◽  
Network Topologies ◽  
Theoretical Results

This paper concerns the problem of consensus tracking for multiagent systems with a dynamical leader. In particular, it proposes the corresponding explicit control laws for multiple first-order nonlinear systems, second-order nonlinear systems, and quite general nonlinear systems based on the leader-follower and the tree shaped network topologies. Several numerical simulations are given to verify the theoretical results.

scholarly journals Extension of First Order Predictive Functional Controllers to Handle Higher Order Internal Models

2008 ◽  
Vol 18 (2) ◽  
pp. 229-239 ◽  
Author(s):  
Mohamed Khadir ◽  
John Ringwood
Keyword(s):  
Higher Order ◽  
Internal Models ◽  
Input Output ◽  
Model Decomposition ◽  
First Order ◽  
The Family ◽  
Armax Model ◽  
Cascaded Model ◽  
Process Plants ◽  
Functional Control

Extension of First Order Predictive Functional Controllers to Handle Higher Order Internal ModelsPredictive Functional Control (PFC), belonging to the family of predictive control techniques, has been demonstrated as a powerful algorithm for controlling process plants. The input/output PFC formulation has been a particularly attractive paradigm for industrial processes, with a combination of simplicity and effectiveness. Though its use of a lag plus delay ARX/ARMAX model is justified in many applications, there exists a range of process types which may present difficulties, leading to chattering and/or instability. In this paper, instability of first order PFC is addressed, and solutions to handle higher order and difficult systems are proposed. The input/output PFC formulation is extended to cover the cases of internal models with zero and/or higher order pole dynamics in an ARX/ARMAX form, via a parallel and cascaded model decomposition. Finally, a generic form of PFC, based on elementary outputs, is proposed to handle a wider range of higher order oscillatory and non-minimum phase systems. The range of solutions presented are supported by appropriate examples.

Higher Order Two-Point Efficient Family of Halley Type Methods for Simple Roots

2016 ◽  
Vol 13 (04) ◽  
pp. 1641016 ◽  
Author(s):  
Ramandeep Behl ◽  
S. S. Motsa
Keyword(s):  
Nonlinear Equations ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Higher Order ◽  
The Other ◽  
Sixth Order ◽  
Order Derivative ◽  
Initial Value ◽  
First Order ◽  
The Family

In this paper, we proposed a new highly efficient two-point sixth-order family of Halley type methods that do not require any second-order derivative evaluation for obtaining simple roots of nonlinear equations, numerically. In terms of computational cost, each member of the family requires two-function and two first-order derivative evaluations per iteration. On the account of the results obtained, it is found that our proposed methods are efficient and show better performance than existing sixth-order methods available in the literature. Further, it is also noted that larger basins of attraction belong to our methods as compared to the existing ones. On the other hand, the existing methods are slower and have darker basins while some of them are too sensitive upon the choice of the initial value.

scholarly journals Learning higher-order logic programs

2019 ◽  
Vol 109 (7) ◽  
pp. 1289-1322 ◽  
Author(s):  
Andrew Cropper ◽  
Rolf Morel ◽  
Stephen Muggleton
Keyword(s):  
Inductive Logic Programming ◽  
Inductive Logic ◽  
Higher Order ◽  
Order Logic ◽  
Sample Complexity ◽  
Logic Programs ◽  
Higher Order Logic ◽  
First Order ◽  
Hypothesis Space ◽  
Theoretical Results

AbstractA key feature of inductive logic programming is its ability to learn first-order programs, which are intrinsically more expressive than propositional programs. In this paper, we introduce techniques to learn higher-order programs. Specifically, we extend meta-interpretive learning (MIL) to support learning higher-order programs by allowing for higher-order definitions to be used as background knowledge. Our theoretical results show that learning higher-order programs, rather than first-order programs, can reduce the textual complexity required to express programs, which in turn reduces the size of the hypothesis space and sample complexity. We implement our idea in two new MIL systems: the Prolog system $$\text {Metagol}_{ho}$$ Metagol ho and the ASP system $$\text {HEXMIL}_{ho}$$ HEXMIL ho . Both systems support learning higher-order programs and higher-order predicate invention, such as inventing functions for and conditions for . We conduct experiments on four domains (robot strategies, chess playing, list transformations, and string decryption) that compare learning first-order and higher-order programs. Our experimental results support our theoretical claims and show that, compared to learning first-order programs, learning higher-order programs can significantly improve predictive accuracies and reduce learning times.

scholarly journals Convergence and Stability of a Parametric Class of Iterative Schemes for Solving Nonlinear Systems

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 86
Author(s):  
Alicia Cordero ◽  
Eva G. Villalba ◽  
Juan R. Torregrosa ◽  
Paula Triguero-Navarro
Keyword(s):  
Nonlinear Systems ◽  
Dynamical Behavior ◽  
Order Convergence ◽  
Hammerstein Integral Equation ◽  
Iterative Schemes ◽  
The Third ◽  
The Family ◽  
The Stability ◽  
Theoretical Results ◽  
Parametric Class

A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations is presented. This study gives us important information about the stability and reliability of the members of the family. The numerical results obtained by applying different elements of the family for solving the Hammerstein integral equation and the Fisher’s equation confirm the theoretical results.

scholarly journals Ball Convergence for Combined Three-Step Methods Under Generalized Conditions in Banach Space

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1002
Author(s):  
R. A. Alharbey ◽  
Ioannis K. Argyros ◽  
Ramandeep Behl
Keyword(s):  
Applied Sciences ◽  
Banach Space ◽  
Iterative Method ◽  
Applied Mathematics ◽  
Higher Order ◽  
Order Derivative ◽  
Numerical Examples ◽  
First Order ◽  
Seventh Order ◽  
Ball Convergence

Problems from numerous disciplines such as applied sciences, scientific computing, applied mathematics, engineering to mention some can be converted to solving an equation. That is why, we suggest higher-order iterative method to solve equations with Banach space valued operators. Researchers used the suppositions involving seventh-order derivative by Chen, S.P. and Qian, Y.H. But, here, we only use suppositions on the first-order derivative and Lipschitz constrains. In addition, we do not only enlarge the applicability region of them but also suggest computable radii. Finally, we consider a good mixture of numerical examples in order to demonstrate the applicability of our results in cases not covered before.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Challenging the Multidimensional Conception of Perceived Person-Environment Fit

2020 ◽  
pp. 1-9
Author(s):  
Julian M. Etzel ◽  
Gabriel Nagy
Keyword(s):  
Higher Education ◽  
University Students ◽  
Educational Outcomes ◽  
Factor Model ◽  
Higher Order ◽  
Substantial Proportion ◽  
Unique Variance ◽  
First Order ◽  
Person Environment Fit ◽  
Order Factor

Abstract. In the current study, we examined the viability of a multidimensional conception of perceived person-environment (P-E) fit in higher education. We introduce an optimized 12-item measure that distinguishes between four content dimensions of perceived P-E fit: interest-contents (I-C) fit, needs-supplies (N-S) fit, demands-abilities (D-A) fit, and values-culture (V-C) fit. The central aim of our study was to examine whether the relationships between different P-E fit dimensions and educational outcomes can be accounted for by a higher-order factor that captures the shared features of the four fit dimensions. Relying on a large sample of university students in Germany, we found that students distinguish between the proposed fit dimensions. The respective first-order factors shared a substantial proportion of variance and conformed to a higher-order factor model. Using a newly developed factor extension procedure, we found that the relationships between the first-order factors and most outcomes were not fully accounted for by the higher-order factor. Rather, with the exception of V-C fit, all specific P-E fit factors that represent the first-order factors’ unique variance showed reliable and theoretically plausible relationships with different outcomes. These findings support the viability of a multidimensional conceptualization of P-E fit and the validity of our adapted instrument.

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