scholarly journals A Parallel Fourth Order Rosenbrock Method: Construction, Analysis and Numerical Comparison

2014 ◽  
Vol 1 (1) ◽  
pp. 45-68 ◽  
Author(s):  
R. Ponalagusamy ◽  
K. Ponnammal
Keyword(s):  
Fourth Order ◽  
Numerical Comparison ◽  
Rosenbrock Method ◽  
Method Construction

Continuous Galerkin Petrov Time Discretization Scheme for the Solutions of the Chen System

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
S. Hussain ◽  
Z. Salleh
Keyword(s):  
Fourth Order ◽  
Three Dimensional ◽  
Time Discretization ◽  
Dimensional System ◽  
Numerical Comparison ◽  
Chen System ◽  
Time Step ◽  
Quadratic Nonlinearities ◽  
Continuous Galerkin ◽  
Three Dimensional System

In this paper, the continuous Galerkin Petrov time discretization (cGP) scheme is applied to the Chen system, which is a three-dimensional system of ordinary differential equations (ODEs) with quadratic nonlinearities. In particular, we implement and analyze numerically the higher order cGP(2)-method which is found to be of fourth order at the discrete time points. A numerical comparison with classical fourth-order Runge–Kutta (RK4) is given for the presented problem. We look at the accuracy of the cGP(2) as the Chen system changes from a nonchaotic system to a chaotic one. It is shown that the cGP(2) method gains accurate results at larger time step sizes for both cases.

scholarly journals Modified King’s Family for Multiple Zeros of Scalar Nonlinear Functions

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 827
Author(s):  
Ramandeep Behl ◽  
Munish Kansal ◽  
Mehdi Salimi
Keyword(s):  
Fourth Order ◽  
Numerical Experiments ◽  
Scientific Literature ◽  
Numerical Comparison ◽  
Nonlinear Functions ◽  
Multiple Zeros ◽  
Cpu Time ◽  
Order Scheme ◽  
Function Strategy ◽  
Exhaustive Study

There is no doubt that there is plethora of optimal fourth-order iterative approaches available to estimate the simple zeros of nonlinear functions. We can extend these method/methods for multiple zeros but the main issue is to preserve the same convergence order. Therefore, numerous optimal and non-optimal modifications have been introduced in the literature to preserve the order of convergence. Such count of methods that can estimate the multiple zeros are limited in the scientific literature. With this point, a new optimal fourth-order scheme is presented for multiple zeros with known multiplicity. The proposed scheme is based on the weight function strategy involving functions in ratio. Moreover, the scheme is optimal as it satisfies the hypothesis of Kung–Traub conjecture. An exhaustive study of the convergence is shown to determine the fourth order of the methods under certain conditions. To demonstrate the validity and appropriateness for the proposed family, several numerical experiments have been performed. The numerical comparison highlights the effectiveness of scheme in terms of accuracy, stability, and CPU time.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

Exploring the Boundaries of the Number–Temporal Order Association

2015 ◽  
Vol 62 (3) ◽  
pp. 198-205 ◽  
Author(s):  
Dana Ganor-Stern
Keyword(s):  
Embodied Cognition ◽  
Mental Representation ◽  
Temporal Order ◽  
Past Research ◽  
Numerical Comparison ◽  
Negative Number ◽  
Comparison Task ◽  
Negative Numbers ◽  
The Past

Past research has shown that numbers are associated with order in time such that performance in a numerical comparison task is enhanced when number pairs appear in ascending order, when the larger number follows the smaller one. This was found in the past for the integers 1–9 ( Ben-Meir, Ganor-Stern, & Tzelgov, 2013 ; Müller & Schwarz, 2008 ). In the present study we explored whether the advantage for processing numbers in ascending order exists also for fractions and negative numbers. The results demonstrate this advantage for fraction pairs and for integer-fraction pairs. However, the opposite advantage for descending order was found for negative numbers and for positive-negative number pairs. These findings are interpreted in the context of embodied cognition approaches and current theories on the mental representation of fractions and negative numbers.

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