Newton-Raphson Algorithm Research Articles

Minorize–maximize algorithm for the generalized odds rate model for clustered current status data

AbstractCurrent status data are widely used in epidemiology and public health, where the only observable information is the random inspection time and the event status at inspection. This article presents a unified methodology to analyze such complex data subject to clustering. Given the random clustering effect, the time to event is assumed to follow a semiparametric generalized odds rate (GOR) model. The nonparametric component of the GOR model is approximated via penalized splines, with a set of knot points that increase with the sample size. The within‐subject correlation is accounted for by a random (frailty) effect. For estimation, a novel MM algorithm is developed that allows the separation of the parametric and nonparametric components of the model. This separation makes the problem conducive to applying the Newton–Raphson algorithm that quickly returns the roots. The work is accompanied by a complexity analysis of the algorithm, a rigorous asymptotic proof, and the related semiparametric efficiency of the proposed methodology. The finite sample performance of the proposed method is assessed via simulation studies. Furthermore, the proposed methodology is illustrated via real data analysis on periodontal disease studies accompanied by diagnostic checks to identify influential observations.

Comparison of Some Methods for Estimating the Survival Function and Failure Rate for the Exponentiated Expanded Power Function Distribution


 We have presented the distribution of the exponentiated expanded power function (EEPF) with four parameters, where this distribution was created by the exponentiated expanded method created by the scientist Gupta to expand the exponential distribution by adding a new shape parameter to the cumulative function of the distribution, resulting in a new distribution, and this method is characterized by obtaining a distribution that belongs for the exponential family. We also obtained a function of survival rate and failure rate for this distribution, where some mathematical properties were derived, then we used the method of maximum likelihood (ML) and method least squares developed (LSD) to estimate the parameters and because of the nonlinear relationship between the parameters, numerical algorithms were used to find the estimates of the two methods. They are Newton-Raphson (NR) and Nelder mead (NM) algorithms to improve the estimators, and a Monte Carlo simulation experiment was conducted to evaluate the performance of the two algorithms' estimates, and the average integrated error criterion (IMSE) was used to compare the survival function estimates and the failure rate. The results showed the efficiency of the maximum likelihood method estimates and least squares developed using the two algorithms (NR, NM) where their results were close, and this shows the new distribution efficiency (EEPF) for modeling survival data.

Recycling Newton–Krylov algorithm for efficient solution of large scale power systems

Power flow calculations are crucial for the study of power systems, as they can be used to calculate bus voltage magnitudes and phase angles, as well as active and reactive power flows on lines. In this paper, a new approach, the Recycling Newton–Krylov (ReNK) algorithm, is proposed to solve the linear systems of equations in Newton–Raphson iterations. The proposed method uses the Generalized Conjugate Residuals with inner orthogonalization and deflated restarting (GCRO-DR) method within the Newton–Raphson algorithm and reuses the Krylov subspace information generated in previous Newton runs. We evaluate the performance of the proposed method over the traditional direct solver (LU) and iterative solvers (Generalized Minimal Residual Method (GMRES), the Biconjugate Gradient Stabilized Method (Bi-CGSTAB) and Quasi-Minimal Residual Method (QMR)) as the inner linear solver of the Newton–Raphson method. We use different test systems with a number of busses ranging from 300 to 70000 and compare the number of iterations of the inner linear solver (for iterative solvers) and the CPU times (for both direct and iterative solvers). We also test the performance of the ReNK algorithm for contingency analysis and for different load conditions to simulate optimization problems and observe possible performance gains.

A consistent algorithm for finite-strain visco-hyperelasticity and visco-plasticity of amorphous polymers

Shared rheology elements of soft tissues and additive-manufactured amorphous thermoplastics benefit from a common visco-hyperelastic and visco-plastic constitutive framework. For finite-strain and general visco/elastic/plastic constitutive laws, allowing full anisotropy, we use the Kröner–Lee decomposition of the deformation gradient combined with Mandel stress-based yield function. Relatively weak conditions are necessary for specific laws to be incorporated in this framework. For example, stress-like variables present in visco-elasticity are removed from the list of constitutive unknowns. Constitutive iteration is performed for the elastic Cauchy–Green tensor Ce and the plastic multiplier increment Δλ. The source is here the right Cauchy–Green tensor provided by any discretization. For the integration of the flow law we adopt a scaled/squared series approximation of the matrix exponential. The exact Jacobian of the second Piola–Kirchhoff stress is determined with respect to this source, consistent with the integrator. The resulting system is produced by symbolic source-code generation for each yield function and hyperelastic strain-energy density function with each of the viscous terms added. The constitutive system is solved by a damped Newton–Raphson algorithm with the corresponding stress error being extensively assessed with stress error maps. A cellular beam structure is analyzed and results compared with experiments from our group. Some of the symbolic calculations and closed-form solutions are beyond what is manually achievable and therefore the symbolic sources for this work are made available in a source code repository.

Optimization of Object Injection Current in the Development of Electrical Impedance Tomography for Bone Fracture Detection

Electrical Impedance Tomography (EIT) nearly has all the parameters of the ideal of a modality tomography. There is one parameter that makes the EIT modality not yet ideal. It is the low spatial resolution of the image. Unfortunately, it is limited in the range of 5 percent to 10 percent of the object diameter. The causal factor is the limited number of measurement data obtained. The focus of research on EIT at now is the development of functional applications. The novel application developed in this research is to detect bone fractures, mainly due to sports injuries. The hardware characteristics are certainly different from other applications. Optimization of the injection current to match the most resistive characteristics of bone compared to all human body tissues has been carried out. The image of bone is reconstructed using the adjacent method of data collection and the Newton-Raphson iterative algorithm with Tikhonov regularization. The optimization results are seen from the high quality and contrast of the reconstructed image for various injection current values. The current injection frequency is selected at 10 kHz. In the results of this research, the injection current that can produce the highest quality and contrast is 0.3 mA.

Two-step incremental procedure associated with the normal flow technique applied to trusses

To achieve the nonlinear structural behavior, there is a need to trace of their equilibrium path in the space of load-displacement. Truss systems are commonly implemented in several structural systems, including high-span bridges and bracing of the supporting structure of tall buildings. Our study adapts a two-step method with cubic convergence into an incremental-iterative procedure to analyze the geometric nonlinear behavior of trusses. The solution method is combined with the Linear Arc-Length path-following technique. To find the approximate root of nonlinear equation system in the two-step method, two formulas are used. Structures are discretized using the Positional Finite Element Method and all truss bars are assumed to remain linear elastic. The correction of the nodal coordinates subincrement vector is performed using the Normal Flow technique. A computational algorithm was implemented using the free program Scilab. Our numerical results show that, when compared to the standard and modified Newton-Raphson algorithms, the new algorithm decreases the number of iterations and the computing time in the nonlinear analysis of trusses. Equilibrium paths with force and/or displacement limit points are obtained with good precision.

Impact of Openings on the In-Plane Strength of Confined and Unconfined Masonry Walls: A Sustainable Numerical Study

While openings are an essential requirement in buildings as a source of access, fresh air and sunlight, these openings cause a reduction in the lateral stiffness and torsional resistance of masonry wall units. A detailed numerical investigation was carried out to explore the impact of the opening percentage on the in-plane stiffness and lateral strength of unconfined and confined masonry wall panels prepared using calcium silicate bricks, for sustainable masonry structures. A commercially available FEM package (ANSYS) was used to carry out comparative analysis of ten wall panels, five of each type (confined and unconfined masonry walls) with concentrically located openings of varying sizes (0% to 16.5%). A simplified micro-modeling technique following the Newton Raphson Algorithm was adopted. Results revealed that the confined masonry approach unveiled a more reliable anti-seismic response along with improved in-plane strength in the case of confined masonry walls. The failure type shifted from pure flexural to more of a blend of shear and flexure after the opening percentage increased to 10.09% in unconfined masonry walls, which was not the case where confinement was provided. Based on the outcomes, it is strongly recommended to adopt confined masonry in highly seismic-prone areas to avoid catastrophic damage caused by earthquakes.

Computation of Maximum-Likelihood Parameters of the Generalized Logistic Distribution by Three-Step Newton–Raphson Algorithm

Computation of Maximum-Likelihood Parameters of the Generalized Logistic Distribution by Three-Step Newton–Raphson Algorithm

An inverse kinematics method for robots after geometric parameters compensation

It is difficult to apply the inverse kinematics (IK) of the nominal model of robots directly after the error compensation of geometric parameters. This paper proposed an instruction pose-based compensation method to overcome this problem. Its principle is very simple: the pose error between the calibrated model and the nominal model is compensated to the instruction pose, and then the new instruction pose is solved by the IK algorithm of the nominal model, and the above steps are repeated until the termination condition. According to the principle, we obtained the final iterative expression of the pose error and determined its convergence conditions and convergence. Experiments on the calibrated spherical-wrist and non-spherical-wrist robots show that the theoretical analysis is correct. It is concluded that the calibrated robot could be simplified into a structure with an analytic solution. Then, the IK of the original robot can be obtained by nesting the analytic solution into the proposed method. The proposed method can converge the pose error norm to the order of 10−10 in only a finite number of iterations and takes less time than the Newton-Raphson algorithm.

A composite voltage stability index for integrated energy systems based on L-index and the minimum eigenvalue of reduced Jacobian matrix

The closer interdependency of subsystems in integrated energy systems (IESs) and the continuous increase of electricity/gas/heat loads have brought great challenges to the stable operation of IESs. In response to the static voltage stability problem of IESs, this paper proposes a composite voltage stability index (CVSI) based on the minimum eigenvalue of a reduced Jacobian matrix and the known L-index. First, a unified multi-energy flow model for integrated electrical, natural gas and heating system is established, which comprises the steady-state modeling of each energy subsystem and coupling components in IESs. Then, an integrated multi-energy flow calculation method based on the Newton–Raphson algorithm is developed to solve the unified steady-state model of IESs. On the basis, the unified Jacobian matrix after multi-energy flow convergence can be obtained. Moreover, a reduced Jacobian matrix considering the impacts of multi-energy state variables on the reactive power of electrical power system (EPS) is constructed. Afterward, the CVSI is proposed, which combines the minimum eigenvalue of the constructed reduced Jacobian matrix and the L-index derived from bus voltage calculation formula. The proposed CVSI takes both the voltage stability margin of whole system and the voltage stability of electrical buses into consideration. Finally, numerical simulations are exploited to verify the effectiveness and applicability of the proposed voltage stability index.

DeepPhysics: A physics aware deep learning framework for real‐time simulation

AbstractReal‐time simulation of elastic structures is essential in many applications, from computer‐guided surgical interventions to interactive design in mechanical engineering. The finite element method is often used as the numerical method of reference for solving the partial differential equations associated with these problems. Deep learning methods have recently shown that they could represent an alternative strategy to solve physics‐based problems. In this article, we propose a solution to simulate hyper‐elastic materials using a data‐driven approach, where a neural network is trained to learn the nonlinear relationship between boundary conditions and the resulting displacement field. We also introduce a method to guarantee the validity of the solution. In total, we present three contributions: an optimized data set generation algorithm based on modal analysis, a physics‐informed loss function, and a hybrid Newton–Raphson algorithm. The method is applied to two benchmarks: a cantilever beam and a propeller. The results show that our network architecture trained with a limited amount of data can predict the displacement field in less than a millisecond. The predictions on various geometries, topologies, mesh resolutions, and boundary conditions are accurate to a few micrometers for nonlinear deformations of several centimeters of amplitude.

Part 1: 조류해석에 있어서 무효전력 제어 모선의 동작 알고리즘에 관한 연구

Recently, distributed energy resources (DERs), including not only an active load that controls active power and reactive power but also power electronics-based inverters capable of supplying reactive power, have been introduced into the power generation, transmission, and distribution grids. For example, properly controlled reactive power can reduce voltage variations and losses. Therefore, the objective of this study is to present an algorithm to control reactive power in power-flow studies. For this purpose, this study introduces the Newton-Raphson power-flow analysis algorithm, which is able to examine the effect of reactive power control and management on voltage variations, and combines the proposed algorithm into the Newton-Raphson power-flow algorithm. This study also verifies the proposed algorithm in various test systems, including IEEE 30-bus test system. As a case study, the proposed algorithm can be used for the power-flow analysis studies of reactive power control and management.

Statistical Analysis of the Lifetime Distribution with Bathtub-Shaped Hazard Function under Lagged-Effect Step-Stress Model

In survival analysis, applying stress is often used to accelerate an experiment. Stress can be discontinuous, and the step-stress model is applied widely due to its flexibility. However, in reality, when new stress is applied, it often does not take effect immediately, but there will be a lagged effect. Under the lagged-effect step-stress model, the statistical inference of the Chen distribution is discussed. The Chen distribution is an important life distribution as its risk function is bathtub-shaped with certain parameters. In this paper, the maximum likelihood estimators are presented and the Newton–Raphson algorithm is used. According to the form of risk function under this model, the explicit expressions of least squares estimators are obtained. The calculation methods of asymptotic confidence intervals and coverage probabilities are proposed by using the observed Fisher matrix. Finally, to evaluate the performance of the above estimation methods, a Monte Carlo simulation study is provided.

An iterative approach for the exact solution of the pairing Hamiltonian

A new iterative algorithm is established for the exact solution of the standard pairing problem, based on the Richardson-Gaudin method using the polynomial approach. It provides efficient and robust solutions for both spherical and deformed systems at a large scale. The key to its success is that the initial guess for the solutions of such a large set of the non-linear equations is provided in a physically meaningful and controllable manner. Moreover, one reduces the large-dimensional problem to a one-dimensional Monte Carlo sampling procedure, which improves the algorithm's efficiency and avoids the non-solutions and numerical instabilities that persist in most existing approaches. We calculated the ground state and low-lying excited states of equally spaced systems at different pairing strengths G. We then applied the model to study the quantum phase transitional Sm isotopes and the actinide nuclei Pu isotopes, where an excellent agreement with experimental data is obtained. Program summaryProgram Title: IterV1.mCPC Library link to program files:https://doi.org/10.17632/rjnbhgk2p6.1Licensing provisions: GPLv3Programming language: MathematicaNature of problem: The program calculates exact pairing energies based on a new iterative algorithm. The key is the procedure of determining the initial guesses for the large-set non-linear equations involved in a controllable and physically motivated manner. It provides an efficient and robust solver for both spherical and deformed systems in super large model spaces.Solution method: The new iterative algorithm approach starts with simple systems with k nucleon pairs and n=k levels, which can be solved iteratively by including one pair and one level at each step using the Newton-Raphson algorithm with a Monte Carlo sampling procedure. Then it takes the solutions of those systems as initial values and obtain the converged results for the full space by gradually adding the remaining levels. In this way, one reduces the k-dimensional Monte Carlo sampling procedure to a one-dimensional sampling, which improves the efficiency of the algorithm and avoids the non-solutions and numerical instabilities.

A Projection Transformation Method for Eliminating Crosstalk in Networked Thermopile Arrays

The networked thermopile array in row-column fashion suffered from the crosstalk caused by parasitic parallel paths. To measure accurate two-dimensional thermal information, it is of great importance to eliminate the crosstalk. In this work, based on the response voltage non-linearity of thermopile pixels as well as the response voltage non-uniformity of array, a projection transformation method that suits for small-scale networked thermopile arrays is proposed. It transforms the crosstalk elimination problem into an overdetermined non-linear least squares problem, and is solved by the Newton-Raphson iterative algorithm. Its reconstruction error is analyzed carefully by Monte Carlo simulation, and its ability to implement non-uniformity and defective pixel correction is also discussed. Its validity is verified experimentally using a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3 \times 3$ </tex-math></inline-formula> networked thermopile array. The experimental result shows that the original IR image can be reconstructed from measured IR images by this method, and the reconstruction error is significantly decreased from 17.03% to 2.10%.

Two fast and accurate routines for solving the elliptic Kepler equation for all values of the eccentricity and mean anomaly

Context. The repetitive solution of Kepler’s equation (KE) is the slowest step for several highly demanding computational tasks in astrophysics. Moreover, a recent work demonstrated that the current solvers face an accuracy limit that becomes particularly stringent for high eccentricity orbits. Aims. Here we describe two routines, ENRKE and ENP5KE, for solving KE with both high speed and optimal accuracy, circumventing the abovementioned limit by avoiding the use of derivatives for the critical values of the eccentricity e and mean anomaly M, namely e &gt; 0.99 and M close to the periapsis within 0.0045 rad. Methods. The ENRKE routine enhances the Newton-Raphson algorithm with a conditional switch to the bisection algorithm in the critical region, an efficient stopping condition, a rational first guess, and one fourth-order iteration. The ENP5KE routine uses a class of infinite series solutions of KE to build an optimized piecewise quintic polynomial, also enhanced with a conditional switch for close bracketing and bisection in the critical region. High-performance Cython routines are provided that implement these methods, with the option of utilizing parallel execution. Results. These routines outperform other solvers for KE both in accuracy and speed. They solve KE for every e ∈ [0, 1 − ϵ], where ϵ is the machine epsilon, and for every M, at the best accuracy that can be obtained in a given M interval. In particular, since the ENP5KE routine does not involve any transcendental function evaluation in its generation phase, besides a minimum amount in the critical region, it outperforms any other KE solver, including the ENRKE, when the solution E(M) is required for a large number N of values of M. Conclusions. The ENRKE routine can be recommended as a general purpose solver for KE, and the ENP5KE can be the best choice in the large N regime.

Estimation of parameters of the logistic exponential distribution under progressive type-I hybrid censored sample

ABSTRACT The present paper addresses the problem of estimation of model parameters of the logistic exponential distribution based on progressive type-I hybrid censored sample. The maximum likelihood estimates are obtained and computed numerically using Newton–Raphson algorithm. Further, the Bayes estimates are derived under the squared error, LINEX and the generalized entropy loss functions. Two types (independent and bivariate) of prior distributions are considered for the purpose of Bayesian estimation. It is seen that the Bayes estimates are not of explicit forms. Thus, Lindley’s approximation technique is employed to get approximate Bayes estimates. Interval estimates of the parameters based on the normal approximation of the maximum likelihood estimates and the log-transformed maximum likelihood estimates are constructed. The highest posterior density credible intervals are obtained by using the importance sampling method. Numerical simulation is performed to see the performance of the proposed estimation techniques. A real-life data set is considered and analysed for the purpose of illustrations.

Using (ECM) Algorithm to Estimate the Missing Values and Make Comparison between (MLE) and (GA) Algorithm for Estimating Parameters of Multivariate Skew Normal Distribution (MSN)

The estimation of statistical parameters for multivariate data leads to waste in the information if the missing values are neglected, which will subsequently lead to inaccurate estimates. Therefore, the incomplete data must be estimated using one of the statistical estimation methods to obtain accurate results and thus obtaining good estimates for the parameters.The aim of this paper is to estimate the missing values for the multivariate skew normal distribution function using the Expectation Conditional Maximization (ECM) algorithm. After estimating the missing values, the parameters are estimated using Maximum Likelihood Estimation (MLE) with the Newton-Raphson algorithm, as well as using the Genetic Algorithm (GA). Using simulation, the Mean Squared Error (MSE) was calculated to find out which method is the best for estimation by comparing the two methods using different sample sizes (400, 600, and 800). The (GA) that is based on the (ECM) algorithm to estimate the missing values proved to be better and more efficient than the (MLE) method in terms of the results.

Load Flow Study of Togo and Benin Transmission Power Network by the Newton-Raphson Method

Within the framework of the Economic Community of West African States (ECOWAS) countries power networks interconnection project carried by the West African Power Pool (WAPP), operating guidelines have been defined by WAPP and ECOWAS Regional Electricity Regulatory Authority (ERERA) to ensure efficient operation of the interconnected network. Each operator is obliged to comply with these directives to guarantee the stability and reliability of energy supply to ECOWAS countries. The purpose of this paper is to carry out the study of the power flow on the existing Togo and Benin power transmission network and to propose solutions for compliance with the requirements relating to the voltage level required on the interconnection lines or at connection points with neighboring power network operators. The dynamic stability of generators will also be studied to assess the behavior of the network during major disturbances. CEB (Communauté Electrique du Bénin in french) is the power transmission network operator of the Republic of Togo and the Republic of Benin. After the inventory of CEB transmission and production infrastructure and the consumption report, the model of CEB power network is produced as well as the single-line diagram. The power flow calculation is performed with the Newton-Raphson algorithm and the node voltages were calculated using the Cyme software. From the voltage values obtained at the various nodes and depending on the behavior of each generator, solutions are proposed to improve the voltage plan of the power network in accordance with the recommendations of the Regulatory Authority.

A Heuristic Feature Selection in Logistic Regression Modeling with Newton Raphson and Gradient Descent Algorithm

Binary choices, such as success or failure, acceptance or rejection, high or low, heavy or light, and so on, can always be used to express decision-making. Based on the known predictor feature values, a classification model can be used to predict an unknown categorical value. The logistic regression model is a commonly used classification approach in a variety of scientific domains. The goal of this research is to create a logistic regression model with a heuristic approach for selecting input characteristics and to compare the Newton Raphson and gradient descent (GD) algorithms for estimating parameters. Among predictor traits, there were four that met the criterion for being both dependent on the target and independent of one another. Also, optional features In Malang, Indonesia, researchers used the Chi-square test to find four significant characteristics that increase the incidence of pregnant women developing preeclampsia: age (X1), parity (X2), history of hypertension (X3) and salty food consumption (X6). In the above work author proposed, the logistic regression model developed using the gradient descent approach had a lower risk of error than the logistic regression model generated using the Newton Raphson algorithm. The model with the gradient descent approach has a precision of 98.54 percent and an F1 score of 97.64 percent, while the model with the Newton Raphson algorithm has a precision of 86.34 percent and an F1 score of 72.55 percent.