Secant Method Research Articles

Numerical Solution of Nonlinear Equations in Maple

The main purpose of this paper is to obtain the real roots of an expression using the Numerical method, bisection method, Newton's method and secant method. Root analysis is calculated using specific, precise starting points and numerical methods and is represented by Maple. In this research, we used Maple software to analyze the roots of nonlinear equations by special methods, and by showing geometric diagrams, we examined the relevant examples. In this process, the Newton-Raphson method, the algorithm for root access, is fully illustrated by Maple. Also, the secant method and the bisection method were demonstrated by Maple by solving examples and drawing graphs related to each method.

AN ITERATIVE, BRACKETING & DERIVATIVE-FREE METHOD FOR NUMERICAL SOLUTION OF NON-LINEAR EQUATIONS USING STIRLING INTERPOLATION TECHNIQUE

In this article, an iterative, bracketing and derivative-free method have been proposed with the second-order of convergence for the solution of non-linear equations. The proposed method derives from the Stirling interpolation technique, Stirling interpolation technique is the process of using points with known values or sample points to estimate values at unknown points or polynomials. All types of problems (taken from literature) have been tested by the proposed method and compared with existing methods (regula falsi method, secant method and newton raphson method) and it’s noted that the proposed method is more rapidly converges as compared to all other existing methods. All problems were solved by using MATLAB Version: 8.3.0.532 (R2014a) on my personal computer with specification Intel(R) Core (TM) i3-4010U CPU @ 1.70GHz with RAM 4.00GB and Operating System: Microsoft Windows 10 Enterprise Version 10.0, 64-Bit Server, x64-based processor.

NUMERICAL ITERATIVE METHOD OF OPEN METHODS WITH CONVERGE CUBICALLY FOR ESTIMATING NONLINEAR APPLICATION EQUATIONS

Finding the single root of nonlinear equations is a classical problem that arises in a practical application in Engineering, Physics, Chemistry, Biosciences, etc. For this purpose, this study traces the development of a novel numerical iterative method of an open method for solving nonlinear algebraic and transcendental application equations. The proposed numerical technique has been founded from Secant Method and Newton Raphson Method, and the proposed method is compared with the Modified Newton Method and Variant Newton Method. From the results, it is pragmatic that the developed numerical iterative method is improving iteration number and accuracy with the assessment of the existing cubic method for estimating a single root nonlinear application equation.

Singularity problems from source functions as source nodes located near boundaries; numerical methods and removal techniques

Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain S. The source function qln|PQ*¯| is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by qln|PQ*¯| as the source node Q* is located near the boundary Γ(=∂S). So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q* located near Γ. In this paper, an additional FS as, d0ln|PQ0¯|, is added to the original source nodes in the traditional MFS, and the point charge d0(=q) and the source node Q0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q* is located inside S but near Γ, both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function qln|PQ*¯| as the source node Q* is located near Γ.

Performance One-step secant Training Method for Forecasting Cases

The training function used in the ANN method, especially backpropagation, can produce different forecasting accuracy, depending on the method parameters given and the data to be predicted. This paper aims to analyze the ability and performance of one of the training functions in the backpropagation algorithm, namely One-step secant, which can later be used or used as a reference in the case of data forecasting. This method is able to update the values of bias and weights according to the one-step secant method. The analysis process uses a dataset of Foreign Exchange Reserves (US $ Million) in Indonesia 2011-2020. Based on this dataset, the dataset will be divided into two parts. The training data uses the 2011-2014 and 2015 dataset as the training data target. Meanwhile, the test data used 2016-2019 and 2020 as the target test data. The analysis process uses 5 experimental architectures, namely 4-5-1, 4-7-1, 4-9-1, 4-11-1 and 4-13-1. The results of the research based on the analysis obtained the best network architecture 4-11-1 with an MSE Training value of 0.00000012, MSE testing/performance of 0.00115144 (the smallest compared to other architectures) and Epoch 343 Iterations.

Predictive Energy Management of Hybrid Electric Vehicles via Multi-Layer Control

This paper presents predictive energy management of hybrid electric vehicles (HEVs) via computationally efficient multi-layer control. First, we formulate an optimization problem by considering driveability and a penalty for using service brakes in the objective function to optimize gear, engine on/off, engine clutch state, and power-split decisions subject to constraints on the battery state of charge (SOC) and charge sustenance. Then, we split it into two control layers, including a supervisory control in a higher layer and a local power-split control in a lower layer. In the supervisory layer, a gear and powertrain mode manager (PM) is designed, and optimal gear, engine on/off and clutch states are obtained by using a combination of dynamic programming (DP) and Pontryagin's minimum principle (PMP). Moreover, a real-time iteration Secant method is proposed to calculate optimal battery costate such that the constraint on charge sustenance is satisfied. In the local controller layer, a linear quadratic tracking method (LQT) is used to optimally split power between the engine and the electric machine and keep battery SOC within its bounds.

Algoritma Fungsi Perlatihan pada Machine Learning berbasis ANN untuk Peramalan Fenomena Bencana

Research has been carried out with several training functions using standard backpropagation methods, One-Step Secant (OSS), and Bayesian regulation. The purpose of this study was to (i) analyze the Performance accuracy (Performance) of the standard backpropagation method and (ii) optimize the training function with the One-Step Secant (OSS) and Bayesian regulation methods to obtain comparison results of the three methods in the search for the best results implementation of disaster phenomenon forecasting data. The research method is based on quantitative methods with times-series data on disaster phenomena in Indonesia over the last ten years (2011-2020) which were analyzed using two network architecture models, namely 4-8-1 and 4-10-1. The results showed that the 4-8-1 architectural model with the Bayesian regulation training function method was able to optimize quite well through accelerating training time and resulted in a low MSE measurement, although not the lowest with an epoch value of 197 iterations and a Performance of 0.0148480766. The lowest epoch value is generated by the OSS method, but it Performs poorly. The best Performance is produced by the standard backpropagation method with the traingd training function, but the training process for achieving convergence is also too long. In general, it can be concluded that the 4-8-1 architectural model with Bayesian regulation can be used to predict (predict) the phenomenon of natural disasters in Indonesia because the training time to achieve convergence is not too long and Performs exceptionally well.

A NEW SECOND ORDER DERIVATIVE FREE METHOD FOR NUMERICAL SOLUTION OF NON-LINEAR ALGEBRAIC AND TRANSCENDENTAL EQUATIONS USING INTERPOLATION TECHNIQUE

Its most important task in numerical analysis to find roots of nonlinear equations, several methods already exist in literature to find roots but in this paper, we introduce a unique idea by using the interpolation technique. The proposed method derived from the newton backward interpolation technique and the convergence of the proposed method is quadratic, all types of problems (taken from literature) have been solved by this method and compared their results with another existing method (bisection method (BM), regula falsi method (RFM), secant method (SM) and newton raphson method (NRM)) it’s observed that the proposed method have fast convergence. MATLAB/C++ software is used to solve problems by different methods.

Secant Method Superiority Over Newton’s Method in Solving Scalar Nonlinear Equations

This study is aimed to pinpoint the cases or situations that makes the performance of secant method superior than the performance of New- ton’s method in solving certain selected nonlinear equations. Despite the convergence order of Newton’s method that is higher, that does not necessarily mean it performs faster than secant method for all nonlinear problems. In addition, both methods are almost identical in their approach and secant method has advantages that can make its performances superior or equivalent to Newton’s method for some nonlinear equations. Four numerical experiments are given to clarify the situations in which the performance of secant method are proven to be superior over Newton’s method.

Fully Coupled Turbojet Engine Computational Fluid Dynamics Simulations and Cycle Analyses Along the Equilibrium Running Line

Abstract This work presents fully coupled computational fluid dynamics (CFD) simulations and thermodynamic cycle analyses of a small-scale turbojet engine at several conditions along the equilibrium running line. The CFD simulations use a single mesh for the entire engine, from the intake to the exhaust, allowing information to travel in all directions. The CFD simulations are performed along the equilibrium running line by using the iterative Secant method to compute the fuel flow rate required to match the compressor and turbine power. The freestream pressure and temperature and shaft angular speed are the only inputs needed for the CFD simulations. To evaluate the consistency of the CFD results with thermodynamic cycle results, outputs from the CFD simulations are prescribed as inputs to the cycle model. This approach enables on-design and off-design cycle calculations to be performed without requiring turbomachinery performance maps. In contrast, traditional off-design cycle analyses require either scaling, calculating, or measuring compressor and turbine maps with boundary condition assumptions. In addition, the CFD simulations and the cycle analyses are compared with measurements of the turbojet engine. The CFD simulations, thermodynamic cycle analyses, and measurements agree in terms of total temperature and pressure at the diffuser–combustor interface, air and fuel mass flow rate, equivalence ratio, and thrust. The developed methods to perform CFD simulations from the intake to the exhaust of the turbojet engine are expected to be useful for guiding the design and development of future small-scale gas turbine engines.

Quantifying input error in hydrologic modeling using the Bayesian error analysis with reordering (BEAR) approach

Input error in hydrologic models, mainly arising from observed precipitation, can impair model calibration. Precise and reliable identification of input error is important for improved model parameter estimation. However, limited information about the nature of the error and the memory of the hydrologic system make it challenging to disentangle input error from the total residual error. Based on a Sequential Monte Carlo sampling algorithm, Bayesian error analysis with reordering (BEAR) method is developed to quantify the unknown input error series by sampling errors from an assumed error distribution and reordering them with inferred error ranks via the secant method, rather than estimating their values directly. The results of a synthetic case and a real case using the hydrologic models GR4J and HYMOD show several benefits of this new approach: 1) the reordering strategy via the secant method can significantly improve the efficiency and accuracy of input error quantification, and consequently promote the parameter estimation; 2) an autoregressive model can address the persistence of hydrologic residual errors in calibration; 3) the lag-time between the forcing data and the corresponding response can be explicitly acknowledged when the system exhibits a delayed response. This developed BEAR algorithm can be extended to other environmental modeling studies with correlated or/and delayed responses although its ability is limited by the impacts of model structural error and the output observational error.

Evaluating the influence of humidity on asphalt mixture performance by the flow number test

Moisture damage greatly influences the performance of asphalt pavements. However, existing studies evaluating the moisture sensitivity of asphalt mixtures either ignore the influence of water vapor or use inappropriate testing methods. This study examined the moisture sensitivity of asphalt mixtures by using the flow number (FN) test. Four levels of the humidity condition (2%, 50%, 80% and 100% RH) were first designed to condition specimens, after which the FN test was employed to assess the moisture susceptibility of mixtures. The results show that it took 50 days for specimens to reach humidity equilibrium at Humidity Level 1, while approximately 4300 h (180 days) were required for specimens to achieve the respective Humidity Levels 2, 3, and 4. The evaluation indicators—in other words, FN, the onset of the secondary stage N4, their corresponding permanent strains (PS2 and PS3), and the slope of secondary stage S2—were then determined by applying a two-step secant method. It was observed that N4 and FN exhibit a gradually decreasing tendency, while PS2 and PS3 tend to increase with the rise in humidity level, but this phenomenon seems indiscernible in some groups. A newly modified exponential function model was proposed to characterize the interaction between humidity levels and S2. It was also demonstrated that the slope of the secondary stage is positively correlated with humidity conditions, which indicates that more deformation will occur with a faster deformation rate during higher humidity levels. The combined use of FN and S2 is a recommended way to assess the trend of permanent deformation of asphalt mixtures under different humidity conditions.

Effective Controls of Fixed Capacitor-Thyristor Controlled Reactors for Power Quality Improvement in Shipboard Microgrids

Addressing power quality issues in shipboard microgrids (SMs), which are mainly attributable to the increased installation of power converters, has received much attention recently. To this end, static VAR compensators, such as thyristor-switched capacitors and fixed capacitors-thyristor controlled reactors (FCs-TCRs), can be effective solutions. Controlling these compensators, however, is not a trivial task as it involves sophisticated operations, especially estimating the firing angle, which should be carried out based on some nonlinear equations. This article aims to propose the application of some simple yet numerically efficient algorithms based on bisection, Newton-Raphson, false position, and secant methods for estimating the firing angle of the FC-TCR. The effectiveness and robustness of these algorithms are demonstrated via modeling of the FC-TCR with the electrical power system of a practical hybrid ferry under MATLAB/Simulink environment, where the results proved that the enhanced power quality issues respect the IEC standards 61000-4-7/30. Furthermore, an experimental setup consists of a digital signal processor and a programmable source is used to demonstrate that these techniques can be effectively applied in real-time applications.

Influence of cable tension history on the monitoring of cable tension using magnetoelastic inductance method

Cable tension monitoring is vital for the health monitoring of cable-stayed bridges. During the service of bridges, cable tension fluctuates rather than monotonously changes. However, existing research works pay little attention to the influence of tension history. In this article, the influence of the tension history on the monitoring of cable tension was studied. To guide the experiment, the magnetization theory of ferromagnetic materials and the electromagnetic induction principle were combined to analyze the theory of the magnetoelastic inductance method. The magnetoelastic inductance method characterized cable tension by sensor inductance. Based on the theoretical analysis, tension monitoring experiments were carried out to figure out the influences of design tension and tension variation. Experimental results showed the design tension and the tension variation influenced the relationship between the inductance and the tension. To monitor the fluctuating tension, a secant method was proposed. When the tension changed less than 30% of the design tension, the tension can be ascertained by the secant method. The experimental results demonstrated that the influence of the tension history should be considered when the design tension was different or the tension variation was large. Besides, the influence of the tension history analyzed in this article is suitable for other tension monitoring methods based on the magnetic properties of ferromagnetic materials.

Determination of PV Model Parameters Using Bisection and Secant Methods

The mathematical modeling of solar cell device is used to demonstrate the equivalent circuit of single-diode solar cell model operating under environmental conditions. In this work, bisection (BM) and secant (SM) Methods currently exists to demonstrate the non-linear electrical behavior of this device. The proposed method is tested in order to solve non-linear equation of PV cell with various values of load resistance are characterized. Comparisons between these numerical examples have been compared. The main idea of the present work is to calculate the PV parameters numerically using different methods with the comparison between them. The results showed that secant method is the best because it has least iteration to obtain the optimal values of the PV parameters.

Electromechanical coupling characteristics of double-layer piezoelectric quasicrystal actuators

Electromechanical coupling characteristics of double-layer piezoelectric quasicrystal micro/nano-actuator are investigated by considering the effects of nanoscale interactions and nonlocal scale. The analytical models in the phonon, phason, and electric fields are derived via state vector formulation and propagator matrix method based on quasicrystal nonlocal elasticity theory. Governing equations of the models with geometrical nonlinearity are solved using the secant method. All nonlinear effects of the electrostatic force, van der Waals force, Casimir force, and nonlocal parameters on the mechanical behaviors of quasicrystal actuators are studied. Our results show that the increment of van der Waals force and Casimir force enlarges phonon stress and displacement. When the initial gap is larger than 3.72 nm, the effect of van der Waals force is more significant than that of Casimir force. The phonon-phason coupling elastic coefficient and the nonlocal scale parameter respectively have positive and negative effects on the static responses. Besides, the influence of the Casimir force on mechanical behavior depends on the volume ratio and scale of the quasicrystal nano-actuator.

Flexible Power Point Tracking for Solar Photovoltaic Systems Using Secant Method

Grid-connected photovoltaic (PV) systems impose challenges like voltage fluctuations, low system inertia, and power quality issues. The need to tackle these challenges led to the introduction of flexible power point tracking (FPPT), where the PV power output is controlled by an energy management system, rather than solely operating the PV systems on the maximum power point. The requirement of fast transient response implies that algorithms such as the one proposed in this article are desirable. The proposed algorithm uses the secant method to achieve significantly improved results in comparison to the existing methods. The method also simplifies the prediction of variations during changes in the environment and power reference, hence, results in reduced oscillation around the set-point and faster convergence. Experimental validation is presented in this article for supporting the claims. The results in terms of accuracy, convergence rate, steady-state oscillations, and cumulative error are bench-marked against one of the most recent FPPT methods.

Real-Time Distributed Economic Dispatch Adapted to General Convex Cost Functions: A Secant Approximation-Based Method

The increasing penetration of distributed energy resources (DERs) in power systems has aroused interest in distributed economic dispatch (DED). While quadratic cost functions are usually adopted, those of DERs are not necessarily quadratic in practice. Existing distributed algorithms perform unsatisfying convergence rates under nonquadratic cost functions. Therefore, this article proposes a secant approximation-based method (SAM) for general convex cost functions to achieve efficient convergence. The marginal cost functions are represented by linear functions in particular price intervals. With a dynamic ratio consensus (DRC) approach, the DERs with linear marginal cost functions can rapidly reach price consensus. The approximated prices can, in turn, determine new price intervals, which are narrower and more accurate.Case studies based on real-world data verify the improved convergence rate under general convex cost functions and exhibit growing performance in larger-scale cases. The proposed algorithm can be applied to DED in systems with high penetration of DERs and massive agents with general convex cost functions.

Achieving an extended convergence analysis for the secant method under a restricted Hölder continuity condition

Achieving an extended convergence analysis for the secant method under a restricted Hölder continuity condition

Homotopy Analysis-Based Hybrid Genetic Algorithm and Secant Method to Solve IVP and Higher-Order BVP

The analytical solution of the Initial Value Problem (IVP) and the Boundary Value Problem (BVP) is an essential issue in numerous engineering applications. However, it cannot usually be realized by simple methods. Accordingly, utilizing semi-analytical solution methods such as the homotopy analysis method (HAM) is an alternative. However, it depends on calculating a dedicated convergence parameter. Different algorithms were proposed to realize this parameter. Thus, converting the residual function into an optimization problem represented a promising solution. The resulted optimization problem can be solved either using traditional analytical methods or meta-heuristic algorithms. In this paper, a new algorithm called Homotopy Analysis Method-based hybrid Genetic algorithm and Secant method (HAMGS) is proposed for solving such optimization problems. The proposed hybridization is based on utilizing the Secant method in the crossover process of the genetic algorithm to improve and accelerate the convergence. Moreover, this hybridization will protect the secant method from divergence in searching for new individuals. As a result, the convergence control parameter can be detected algebraically without drawing the h-curves. This facilitates identifying the minimum number of HAM terms to reach the solution, which reduces the computational burden remarkably. The presumed improvement of pairing the genetic algorithm and Secant method for improving the HAM solver is verified via four IVPs and three higher-order BVPs. The results corroborate the competence of the proposed algorithm and its ability to solve such problems efficiently.