A PARALLEL COMPUTATION SCHEME FOR IMPLICIT RUNGE-KUTTA METHODS AND THE ITERATIVELY B-CONVERGENCE OF ITS NEWTON ITERATIVE PROCESS

In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and discuss the iteratively B-convergence of the Newton iterative process for solving the algebraic equations of the scheme, secondly we present a strategy providing initial values parallelly for the iterative process. Finally, some numerical results show that our parallel scheme is higher efficient as N is not so large.

A novel algorithm for evaluating cement azimuthal density based on perturbation theory in horizontal well

Cement density monitoring plays a vital role in evaluating the quality of cementing projects,which is of great significance to the development of oil and gas.However,the presence of inhomogeneous cement distribution and casing eccentricity in horizontal wells often complicates the accurate evaluation of cement azimuthal density.In this regard,this paper proposes an algorithm to calculate the cement azimuthal density in horizontal wells using a multi-detector gamma-ray detection system.The spatial dynamic response functions are simulated to obtain the influence of cement density on gamma-ray counts by the perturbation theory,and the contribution of cement density in six sectors to the gamma-ray recorded by different detectors is obtained by integrating the spatial dynamic response functions.Combined with the relationship between gamma-ray counts and cement density,a multi-parameter calculation equation system is established,and the regularized Newton iteration method is employed to invert casing eccentricity and cement azimuthal density.This approach ensures the stability of the inversion process while simultaneously achieving an accuracy of 0.05 g/cm^(3) for the cement azimuthal density.This accuracy level is ten times higher compared to density accuracy calculated using calibration equations.Overall,this algorithm enhances the accuracy of cement azimuthal density evaluation,provides valuable technical support for the monitoring of cement azimuthal density in the oil and gas industry.

ANCIENT CHINESE ALGORITHM: THE YING BUZU SHU (METHOD OF SURPLUS AND DEFICIENCY)VS NEWTON ITERATION METHOD

Air exploratory discussion of an ancient Chinese algorithm, the Ying Buzu Shu, in about 2nd century BC, known as the rule of double false position in the West is given. In addition to pointing out that the rule of double false position is actually a translation version of the ancient Chinese algorithm, a comparison with well-known Newton iteration method is also made. If derivative is introduced, the ancient Chinese algorithm reduces to the Newton method. A modification of the ancient Chinese algorithm is also proposed, and some of applications to nonlinear oscillators are illustrated.

The Continuous Analogy of Newton’s Method for Solving a System of Linear Algebraic Equations

We propose a continuous analogy of Newton’s method with inner iteration for solving a system of linear algebraic equations. Implementation of inner iterations is carried out in two ways. The former is to fix the number of inner iterations in advance. The latter is to use the inexact Newton method for solution of the linear system of equations that arises at each stage of outer iterations. We give some new choices of iteration parameter and of forcing term, that ensure the convergence of iterations. The performance and efficiency of the proposed iteration is illustrated by numerical examples that represent a wide range of typical systems.

Variational iteration method for solving the mechanism of the Equatorial Eastern Pacific El Nino-Southern Oscillation

A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO model are obtained and the asymptotic behaviour of solution for corresponding problem is considered.

A Family of Fifth-order Iterative Methods for Solving Nonlinear Equations

In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects.

A Newton type iterative method for heat-conduction inverse problems

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods are considered. The implicit iterative method is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.

The Initial Guess Estimation Newton Method for Power Flow in Distribution Systems

With the increasing integration of distributed generations U+0028 DGs U+0029, there is a demand for DGs to play a more important role on the voltage regulation. Meanwhile, the high penetration of DGs could raise a technical problem that the distribution system may operate with bi-directional power flow, leading to the inadequacy of the traditional power flow. Considering this new scenario in distribution system power flow, the convergence theorem is proposed, which contributes to develop a novel selection method of the initial guess closed to the convergent solution. Moreover, to ensure the fast rate of power flow convergence, the theorem of the maximum iterations estimation is also proposed. Based on the two proposed theorems, an Initial Guess Estimation Newton method is proposed, considering different operational status of DGs and initial guess sensitivity simultaneously. Based on the standard node systems, Tongliao grid, and 69 system of USA, three simulation examples are provided to illustrate the effectiveness of the proposed method. © 2017 Chinese Association of Automation.

Modified Efficient Families of Two and Three-Step Predictor-Corrector Iterative Methods for Solving Nonlinear Equations

In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.

A New Modification of Newton Method with Cubic Convergence

Newton’s method is used to find the roots of a system of equations f (x) = 0. It is one of the most important procedures in numerical analysis, and its applicability extends to differential equations and integral equations. Analysis of the method shows a quadratic convergence under certain assumptions. For several years, researchers have improved the method by proposing modified Newton methods with salutary efforts. A modification of the Newton’s method was proposed by McDougall and Wotherspoon [1] with an order of convergence of 1+ √2. On a new type of methods with cubic convergence was proposed by H. H. H. Homeier [2]. In this article, we present a new modification of Newton method based on secant method. Analysis of convergence shows that the new method is cubically convergent. Our method requires an evaluation of the function and one of its derivatives.

Newton-type methods and their modifications for inverse heat conduction problems

This paper studies to numerical solutions of an inverse heat conduction problem.The effect of algorithms based on the Newton-Tikhonov method and the Newton-implicit iterative method is investigated,and then several modifications are presented.Numerical examples show the modified algorithms always work and can greatly reduce the computational costs.

A SIGNIFICANT IMPROVEMENT ON NEWTON’S ITERATIVE METHOD

For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .

The Hidden Geometry of the Babylonian Square Root Method

We propose and demonstrate an original geometric argument for the ancient Babylonian square root method, which is analyzed and compared to the Newton-Raphson method. Based on simple geometry and algebraic analysis the former original iterated map is derived and reinterpreted. Time series, fixed points, stability analysis and convergence schemes are studied and compared for both methods, in the approach of discrete dynamical systems.

关于Newton-like-iterative方法新的收敛性定理(英文)

用迭代法求解Newton-like法中的方程,T.J . Ypma提出Newton-like-iterative方法。在其早期的文章中,不精确牛顿法理论用来研究Newton-like-iterative方法的收敛性。与以往方法不同,今提出用不精确Newton-like法做相关的收敛性分析,所得定理更加简单,同时具有仿射不变性。

ITERATIVE REGULARIZATION METHODS FOR NONLINEAR ILL-POSED OPERATOR EQUATIONS WITH M-ACCRETIVE MAPPINGS IN BANACH SPACES

In this paper, a modified Newton type iterative method is considered for ap- proximately solving ill-posed nonlinear operator equations involving m-accretive mappings in Banach space. Convergence rate of the method is obtained based on an a priori choice of the regularization parameter. Our analysis is not based on the sequential continuity of the normalized duality mapping.

A Fourth-order Covergence Newton-type Method

A fourth-order convergence method of solving roots for nonlinear equation, which is a variant of Newton＇s method given. Its convergence properties is proved. It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end, numerical tests are given and compared with other known Newton and Newton-type methods. The results show that the proposed method has some more advantages than others. It enriches the methods to find the roots of non-linear equations and it is important in both theory and application.

On Newton-Like Methods for Solving Nonlinear Equations

In this paper, we present a family of general New to n-like methods with a parametric function for finding a zero of a univariate fu nction, permitting f′(x)=0 in some points. The case of multiple roots is n ot treated. The methods are proved to be quadratically convergent provided the w eak condition. Thus the methods remove the severe condition f′(x)≠0. Based on the general form of the Newton-like methods, a family of new iterative meth ods with a variable parameter are developed.

Comparison of Iterative Feedback Tuning Search Techniques

Iterative feedback tuning is an attractive method for industry as it is a model free approach using experiments conducted on the plant to tune controller parameters. Classically Gauss-Newton iterative methods are used in IFT to update the controller parameters in the negative gradient direction of a specified design criterion function. Levenburg-Marquardt and Trust-Region strategies offer attractive advantages to Gauss-Newton in many applications,these alternative methods are given and results from simulation presented. A discussion on the differences between line search methods and Trust-Region methods is given showing the Trust-Region search direction is more flexible. Step size selection is often the limiting factor and it is found that with unknown step size values and initial controller parameters the Trust-Region is the best selection,where as if overshoot is a concern Levenburg-Marquardt is a good choice.Gauss-Newton method provides quick convergence and a fast response time however it shows more dependence on the step size.

Simpson’s Method for Solution of Nonlinear Equation

The programs offered for solving nonlinear equations, usually the old method, such as alpha, chordal movement, Newton, etc. have been used. Among these methods may Newton’s method of them all be better and higher integration. In this paper, we propose the integration method for finding the roots of nonlinear equation we use. In this way, Newton’s method uses integration methods to obtain. In previous work, [1] and [2] presented numerical integration methods such as integration, trapezoidal and rectangular integration method that are used. The new method proposed here, uses Simpson’s integration. With this method, the approximation error is reduced. The calculated results show that this hypothesis is confirmed.

Exact Bayesian and Fiducial Limits for the Mean of Lognormal Distribution

In this paper, the exact Bayesian limits, taking conjugate and noninformative prior distribution, and the exact fiducial limits for the mean of the lognormal distribution are presented. They can be found iteratively by one-dimension integral on a finite interval. The new algorithm is very convenient and with high accuracy. It can meet the practical engineering need excellently. However, the primitive algorithm is rather cumbersome. By the way, the very close approximate limits with a simple algorithm are derived. They can be applied immediately to engineering. Otherwise, they can also be used as a search interval to find the root of equation for the exact limits.