Optimal iterative processes for root-finding

The article presents a methodological approach to studying iterative processes in the school course of geometry, by the example of constructing a Koch snowflake fractal curve and calculating a few characteristics of it. The interactive creative environment 1C:MathKit is chosen to visualize the method discussed. By performing repetitive constructions and algebraic calculations using ICT tools, students acquire a steady skill of work with geometric objects of various levels of complexity, comprehend the possibilities of mathematical interpretation of iterative processes in practice, and learn how to understand the dialectical unity between finite and infinite parameters of flat geometric figures. When students are getting familiar with such contradictory concepts and categories, that replenishes their experience of worldview comprehension of the subject areas they study through the concept of “big ideas”. The latter allows them to take a fresh look at the processes in the world around. The article is a matter of interest to schoolteachers of computer science and mathematics, as well as university scholars who teach the course “Concepts of modern natural sciences”.

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Iterative Processes for Resilient Transboundary Water Management: Collaboratively Governing the Okavango for Adaptation

SSRN Electronic Journal ◽

10.2139/ssrn.2039023 ◽

2012 ◽

Author(s):

Olivia Odom Green ◽

Barbara A. Cosens ◽

Ahjond S. Garmestani

Keyword(s):

Water Management ◽

Transboundary Water ◽

Iterative Processes ◽

Transboundary Water Management

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DoA Estimation Using Neural Tangent Kernel under Electromagnetic Mutual Coupling

Electronics ◽

10.3390/electronics10091057 ◽

2021 ◽

Vol 10 (9) ◽

pp. 1057

Author(s):

Qifeng Wang ◽

Xiaolin Hu ◽

Xiaobao Deng ◽

Nicholas E. Buris

Keyword(s):

Mutual Coupling ◽

Deep Neural Network ◽

Doa Estimation ◽

Antenna Element ◽

Root Finding ◽

Multiple Signal ◽

Estimation Problems ◽

Limiting Case ◽

Signal Doa Estimation ◽

Polynomial Root Finding

Antenna element mutual coupling degrades the performance of Direction of Arrival (DoA) estimation significantly. In this paper, a novel machine learning-based method via Neural Tangent Kernel (NTK) is employed to address the DoA estimation problem under the effect of electromagnetic mutual coupling. NTK originates from Deep Neural Network (DNN) considerations, based on the limiting case of an infinite number of neurons in each layer, which ultimately leads to very efficient estimators. With the help of the Polynomial Root Finding (PRF) technique, an advanced method, NTK-PRF, is proposed. The method adapts well to multiple-signal scenarios when sources are far apart. Numerical simulations are carried out to demonstrate that this NTK-PRF approach can handle, accurately and very efficiently, multiple-signal DoA estimation problems with realistic mutual coupling.

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A Comparative Study among New Hybrid Root Finding Algorithms and Traditional Methods

Mathematics ◽

10.3390/math9111306 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1306

Author(s):

Elsayed Badr ◽

Sultan Almotairi ◽

Abdallah El Ghamry

Keyword(s):

Comparative Study ◽

Numerical Results ◽

Traditional Methods ◽

Running Time ◽

Root Finding ◽

Average Running Time ◽

False Position ◽

Newton Raphson ◽

Number Of Iterations

In this paper, we propose a novel blended algorithm that has the advantages of the trisection method and the false position method. Numerical results indicate that the proposed algorithm outperforms the secant, the trisection, the Newton–Raphson, the bisection and the regula falsi methods, as well as the hybrid of the last two methods proposed by Sabharwal, with regard to the number of iterations and the average running time.

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On the stability of asynchronous iterative processes

Mathematical Systems Theory ◽

10.1007/bf01692062 ◽

1987 ◽

Vol 20 (1) ◽

pp. 137-153 ◽

Cited By ~ 26

Author(s):

John N. Tsitsiklis

Keyword(s):

Iterative Processes ◽

The Stability

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An implicit multistep numerical method for real-time simulation of stiff systems

SIMULATION ◽

10.1177/00375497211021653 ◽

2021 ◽

pp. 003754972110216

Author(s):

Zhang Lei ◽

Li Jie ◽

Wang Menglu ◽

Liu Mengya

Keyword(s):

Differential Equations ◽

Numerical Method ◽

Real Time ◽

Physical System ◽

Physical Models ◽

Stiff Systems ◽

Real Time Simulation ◽

Stiff Ordinary Differential Equations ◽

Root Finding ◽

Time Simulation

Simulating a physical system in real-time is widely used in equipment design, test, and validation. Though an implicit multistep numerical method excels at solving physical models that are usually composed of stiff ordinary differential equations, it is not suitable for real-time simulation because of state discontinuity and massive iterations for root finding. Thus, a method based on the backward differential formula is presented. It divides the main fixed step of real-time simulation into limited minor steps according to computing cost and accuracy demand. By analyzing and testing its capability, this method shows advantage and efficiency in real-time simulation, especially when the system contains stiff equations. A simulation application will have more flexibility while using this method.

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