scholarly journals Unconditional Applicability of Lehmer’s Measure to the Two-Term Machin-like Formula for π

2021 ◽  
Vol 23 ◽  
Author(s):  
Sanjar Abrarov ◽  
Rehan Siddiqui ◽  
Rajinder Jagpal ◽  
Brendan Quine
Keyword(s):  
Computational Efficiency ◽  
Iteration Method ◽  
Irrational Numbers ◽  
Tangent Function ◽  
Newton Raphson ◽  
The Given ◽  
Raphson Iteration

Lehmer defined a measure depending on numbers beta_i used in a Machin-like formula for pi. When the beta_i are integers, Lehmer's measure can be used to determine the computational efficiency of the given Machin-like formula for pi. However, because the computations are complicated, it is unclear if Lehmer's measure applies when one or more of the beta_i are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for pi as an example of the unconditional applicability of Lehmer's measure. This approach does not involve any irrational numbers and may allow calculating pi rapidly by the Newton-Raphson iteration method for the tangent function.

scholarly journals Calculation of Flight Data Fusion Coefficients Using Newton–Raphson Iterations

2020 ◽  
pp. 100-103
Author(s):  
Elkhan Nariman Sabziev
Keyword(s):  
Differential Equations ◽  
Data Fusion ◽  
Boundary Value Problem ◽  
Iteration Method ◽  
Measurement Errors ◽  
Continuous Model ◽  
System Of Differential Equations ◽  
Flight Data ◽  
Newton Raphson ◽  
Raphson Iteration

The problem of plotting the flight path of an aircraft based on flight data containing numerous measurement errors is investigated. A theoretical (continuous) model of the flight data fusion problem is proposed in the form of a boundary value problem for a system of differential equations with unknown coefficients. The application of the Newton–Raphson iteration method for calculating the sought-for coefficients is described.

scholarly journals Modeling and Parameter Extraction of PV Cell Using Single- and Two-Diode Model

2017 ◽  
Vol 2 (2) ◽  
pp. 06
Author(s):  
Bouchra Benabdelkrim ◽  
Ali Benatillah
Keyword(s):  
Iteration Method ◽  
Accurate Determination ◽  
Parameter Extraction ◽  
Solar Pv ◽  
Mathematical Methods ◽  
Pv System ◽  
Accurate Performance ◽  
Newton Raphson ◽  
Using Data ◽  
Raphson Iteration

Photovoltaic modules operate under a large range of conditions. This combined with the fact that manufacturers provide electrical parameters at specific conditions (STC). The present study proposes a comparison between single and double diode models of solar PV system and ensures the best suited model under specific environmental condition for accurate performance prediction. An important feature of these models is that its parameters can be determined using data commonly provided by module manufacturers on their published datasheets. Accurate determination of these parameters which arose from a diversification of models and methods dedicated to their estimations is still a challenge for researchers. In this paper the single and two diode models have been studied by mathematical methods based on simulated Newton-Raphson iteration method. Newton-Raphson iteration method is solved by MATLAB simulation. 

Variational Iteration Method for a Nonlinear Reaction-Diffusion Process

2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Shu-Qiang Wang ◽  
Ji-Huan He
Keyword(s):  
Exact Solution ◽  
Diffusion Process ◽  
Iteration Method ◽  
Unknown Parameter ◽  
Reaction Diffusion ◽  
Variational Iteration Method ◽  
Rigorous Derivation ◽  
Variational Iteration ◽  
Nonlinear Reaction ◽  
The Given

An extremely simple and elementary, but rigorous derivation of temperature distribution of a reaction-diffusion process is given using the variational iteration method. In this method, a trial function (an initial solution) is chosen with some unknown parameter, which is identified after a few iterations according to the given boundary conditions. Comparison with the exact solution shows that the method is very effective and convenient.

scholarly journals Estimation of the Extreme Distribution Model of Economic Losses Due to Outbreaks Using the POT Method with Newton Raphson Iteration

2021 ◽  
Vol 2 (1) ◽  
pp. 37-45
Author(s):  
Riza Adrian Ibrahim ◽  
Sukono Sukono ◽  
Riaman Riaman
Keyword(s):  
Estimation Method ◽  
Random Variable ◽  
Value Theory ◽  
Extreme Value ◽  
Distribution Model ◽  
Economic Losses ◽  
Extreme Distribution ◽  
Newton Raphson ◽  
The Government ◽  
Raphson Iteration

Extreme distribution is the distribution of a random variable that focuses on determining the probability of small values in the tail areaof the distribution. This distribution is widely used in various fields, one of which is reinsurance. An outbreak catastrophe is non-natural disaster that can pose an extreme risk of economic loss to a country that is exposed to it. To anticipate this risk, the government of a country can insure it to a reinsurance company which is then linkedto bonds in the capital market so that new securities are issued, namely outbreakcatastrophe bonds. In pricing, knowledge of the extreme distribution of economic losses due to outbreak catastrophe is indispensable. Therefore, this study aims to determine the extreme distribution model of economic losses due to outbreak catastrophe whose models will be determined by the approaches and methods of Extreme Value Theory and Peaks Over Threshold, respectively. The threshold value parameter of the model will be estimated by Kurtosis Method, while the other parameters will be estimated with Maximum Likelihood Estimation Method based on Newton-Raphson Iteration. The result of the research obtained is the resulting model of extreme value distribution of economic losses due to outbreak catastrophe that can be used by reinsurance companies as a tool in determining the value of risk in the outbreak catastrophe bonds.

New methods for projecting a 3D object onto a free-form surface

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Jingyu Pei ◽  
Xiaoping Wang ◽  
Leen Zhang ◽  
Yu Zhou ◽  
Jinyuan Qian
Keyword(s):  
Approximate Solution ◽  
New Method ◽  
Free Form ◽  
Parallel Projection ◽  
Content Type ◽  
3D Object ◽  
New Methods ◽  
Free Form Surface ◽  
Newton Raphson ◽  
Raphson Iteration

Purpose This paper aims to provide a series of new methods for projecting a three-dimensional (3D) object onto a free-form surface. The projection algorithms presented can be divided into three types, namely, orthogonal, perspective and parallel projection. Design/methodology/approach For parametric surfaces, the computing strategy of the algorithm is to obtain an approximate solution by using a geometric algorithm, then improve the accuracy of the approximate solution using the Newton–Raphson iteration. For perspective projection and parallel projection on an implicit surface, the strategy replaces Newton–Raphson iteration by multi-segment tracing. The implementation takes two mesh objects as an example of calculating an image projected onto parametric and implicit surfaces. Moreover, a comparison is made for orthogonal projections with Hu’s and Liu’s methods. Findings The results show that the new method can solve the 3D objects projection problem in an effective manner. For orthogonal projection, the time taken by the new method is substantially less than that required for Hu’s method. The new method is also more accurate and faster than Liu’s approach, particularly when the 3D object has a large number of points. Originality/value The algorithms presented in this paper can be applied in many industrial applications such as computer aided design, computer graphics and computer vision.

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