A convergence criterion for Secant method with approximate zeros

1999 ◽  
Vol 6 (3) ◽  
pp. 575-588
Author(s):  
S. Kim
Keyword(s):  
Secant Method ◽  
Convergence Criterion

Determination of Base Flipping Free Energy Landscapes from Nonequilibrium Stratification

2019 ◽  
Author(s):  
Xiaohui Wang ◽  
Zhaoxi Sun
Keyword(s):  
Free Energy ◽  
Energy Landscape ◽  
Sampling Technique ◽  
Umbrella Sampling ◽  
Energy Simulation ◽  
Nonlinear Dependence ◽  
Free Energy Landscape ◽  
Convergence Criterion ◽  
Base Flipping ◽  
Convergence Behavior

<p>Correct calculation of the variation of free energy upon base flipping is crucial in understanding the dynamics of DNA systems. The free energy landscape along the flipping pathway gives the thermodynamic stability and the flexibility of base-paired states. Although numerous free energy simulations are performed in the base flipping cases, no theoretically rigorous nonequilibrium techniques are devised and employed to investigate the thermodynamics of base flipping. In the current work, we report a general nonequilibrium stratification scheme for efficient calculation of the free energy landscape of base flipping in DNA duplex. We carefully monitor the convergence behavior of the equilibrium sampling based free energy simulation and the nonequilibrium stratification and determine the empirical length of time blocks required for converged sampling. Comparison between the performances of equilibrium umbrella sampling and nonequilibrium stratification is given. The results show that nonequilibrium free energy simulation is able to give similar accuracy and efficiency compared with the equilibrium enhanced sampling technique in the base flipping cases. We further test a convergence criterion we previously proposed and it comes out that the convergence behavior determined by this criterion agrees with those given by the time-invariant behavior of PMF and the nonlinear dependence of standard deviation on the sample size. </p>

scholarly journals Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 169
Author(s):  
Avram Sidi
Keyword(s):  
Nonlinear Equations ◽  
Local Convergence ◽  
Real Root ◽  
Numerical Procedure ◽  
Secant Method ◽  
Convergence Theory ◽  
Simple Complex ◽  
Real Roots ◽  
Complex Roots ◽  
Appl Math

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

scholarly journals A NOVEL IMAGE SEGMENTATION APPROACH BASED ON NEUTROSOPHIC SET AND IMPROVED FUZZY C-MEANS ALGORITHM

2011 ◽  
Vol 07 (01) ◽  
pp. 155-171 ◽  
Author(s):  
H. D. CHENG ◽  
YANHUI GUO ◽  
YINGTAO ZHANG
Keyword(s):  
Image Segmentation ◽  
Point Of View ◽  
Neutrosophic Set ◽  
Convergence Criterion ◽  
Complex Objects ◽  
Fuzzy C Means ◽  
Segmentation Methods ◽  
Segmentation Algorithms ◽  
Segmentation Approach ◽  
Fuzzy C Means Algorithm

Image segmentation is an important component in image processing, pattern recognition and computer vision. Many segmentation algorithms have been proposed. However, segmentation methods for both noisy and noise-free images have not been studied in much detail. Neutrosophic set (NS), a part of neutrosophy theory, studies the origin, nature, and scope of neutralities, as well as their interaction with different ideational spectra. However, neutrosophic set needs to be specified and clarified from a technical point of view for a given application or field to demonstrate its usefulness. In this paper, we apply neutrosophic set and define some operations. Neutrosphic set is integrated with an improved fuzzy c-means method and employed for image segmentation. A new operation, α-mean operation, is proposed to reduce the set indeterminacy. An improved fuzzy c-means (IFCM) is proposed based on neutrosophic set. The computation of membership and the convergence criterion of clustering are redefined accordingly. We have conducted experiments on a variety of images. The experimental results demonstrate that the proposed approach can segment images accurately and effectively. Especially, it can segment the clean images and the images having different gray levels and complex objects, which is the most difficult task for image segmentation.

scholarly journals CFD Simulation of the Dispersion of Exhaust Gases in a Traffic Loaded Street of Astana, Kazakhstan

2016 ◽  
Vol 9 (2) ◽  
pp. 158-166
Author(s):  
Ardak Akhatova ◽  
Assylan Kassymov ◽  
Meruyert Kazmaganbetova ◽  
Luis Ramon Rojas-Solórzano
Keyword(s):  
Cfd Simulation ◽  
Maximum Level ◽  
Weather Conditions ◽  
Control Measures ◽  
Convergence Criterion ◽  
Emission Rates ◽  
Wall Functions ◽  
Mesh Density ◽  
Volume Method ◽  
Future Work

The aim of this paper is to consider one of the most traffic-loaded regions of Astana city (Kazakhstan) and to determine the concentration of carbon-monoxide (CO) in the air during the peak hours. CFD analysis based on the SolidWorks-EFD platform was used to simulate the dispersion of contaminants given the estimated emission rates and weather conditions at the crossroad of Bogenbay Batyr and Zhenis Avenues in Astana. Turbulence prediction was based on k-ε model with wall functions. The governing equations were discretized using the finite volume method and a 2nd order spatial scheme. The mesh verification was based on 1% convergence criterion for a 50% of mesh density increment; air pressure near the wall of a selected building was chosen as the parameter to control the convergence. Numerical results are presented for prevailing conditions during all 4 seasons of the year, demonstrating that the highest levels of CO are recorded in summer and reach the values up to 11.2 ppm which are still lower than the maximum level admitted for humans. Nevertheless, obtained results show that Astana is gradually becoming a city that is likely to reach the critical levels of pollutants in the nearest future if control measures are not taken with enough anticipation. As for a future work, it is proposed to perform in-situ validation of specific scenarios to check and support the results obtained with CFD and to develop then specific policies for tackling the problem before it becomes evident.

Design and Analysis of a High Pressure Turbine Using Computational Methods for Small Gas Turbine Application

2013 ◽  
Author(s):  
Kishor Kumar ◽  
R. Prathapanayaka ◽  
S. V. Ramana Murthy ◽  
S. Kishore Kumar ◽  
T. M. Ajay Krishna
Keyword(s):  
High Pressure ◽  
Gas Turbine ◽  
Computational Methods ◽  
Gas Turbine Engine ◽  
Flow Coefficient ◽  
Design Parameters ◽  
Convergence Criterion ◽  
Turbine Engine ◽  
Design Point ◽  
Ansys Cfx

This paper describes the aerodynamic design and analysis of a high-pressure, single-stage axial flow turbine suitable for small gas turbine engine application using computational methods. The specifications of turbine were based on the need of a typical high-pressure compressor and geometric restrictions of small gas turbine engine. Baseline design parameters such as flow coefficient, stage loading coefficient are close to 0.23 and 1.22 respectively with maximum flow expansion in the NGV rows. In the preliminary design mode, the meanline approach is used to generate the turbine flow path and the design point performance is achieved by considering three blade sections at hub, mean and tip using the AMDC+KO+MK+BSM loss models to meet the design constraints. An average exit swirl angle of less than 5 degrees is achieved leading to minimum losses in the stage. Also, NGV and rotor blade numbers were chosen based on the optimum blade solidity. Blade profile is redesigned using the results from blade-to-blade analysis and through-flow analysis based on an enhanced Dawes BTOB3D flow solver. Using PbCFD (Pushbutton CFD) and commercially available CFD software ANSYS-CFX, aero-thermodynamic parameters like pressure ratios, aerodynamic power, and efficiencies are computed and these results are compared with one another. The boundary conditions, convergence criterion, and turbulence model used in CFD computations are set uniform for comparison with 8 per cent turbulence intensity. Grid independence study is performed at design point to optimize the grid density for off-design performance predictions. ANSYS-CFX and PbCFD have predicted higher efficiency of 3.4% and 1.2% respectively with respect to targeted efficiency of 89 per cent.

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