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 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.

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 .

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。

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.

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.

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.

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.

New Fourth Order Iterative Methods Second Derivative Free

In a recent paper, Noor and Khan [M. Aslam Noor, & W. A. Khan, (2012) New Iterative Methods for Solving Nonlinear Equation by Using Homotopy Perturbation Method, Applied Mathematics and Computation, 219, 3565-3574], suggested a fourth-order method for solving nonlinear equations. Per iteration in this method requires two evaluations of the function and two of its first derivatives;therefore, the efficiency index is 1.41421 as Newton’s method. In this paper, we modified this method and obtained a family of iterative methods for appropriate and suitable choice of the parameter. It should be noted that per iteration for the new methods requires two evaluations of the function and one evaluation of its first derivatives, so its efficiency index equals to 1.5874. Analysis of convergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods.

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.

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.

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.

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.

非精确Newton方法中线性迭代收敛判据研究

在计算流体力学中,采用隐式时间推进方法时通常需要采用Newton类迭代方法求解大型非线性离散系统。每步非线性迭代需求解由非线性系统Jacobian矩阵组成的大型线性方程组,其中线性方程组求解误差会对非线性系统的收敛性产生显著影响,然而对存在Jacobian矩阵误差情况下的线性迭代收敛判据缺乏深入的研究。本文针对上述问题,首先给出了存在Jacobian矩阵误差和线性迭代误差情况下Newton迭代式的形式,并通过数值测试验证了Jacobian矩阵误差对迭代产生较大影响的可能性;其次对常见的两种不同类型的线性迭代收敛判据进行了数值测试,重点研究了存在Jacobian矩阵误差情况下容易产生的过度求解问题;最后,结合上述两类判据的特点发展了一种新的线性迭代收敛判据,结果表明:新提出的迭代收敛判据能够有效缓解过度求解问题,从而提高计算效率。

基于MATLAB的民航发动机涡轮叶片寿命建模

针对民航飞机维修工作中发动机涡轮叶片普遍存在故障率较高的问题,收集某型民用航空发动机涡轮叶片的可靠性数据,并运用三参数Weibull分布建立该型发动机涡轮叶片的可靠性寿命模型。在模型求解的数值计算过程中,为保证模型的计算精度,采用经典的牛顿迭代法及三参数相关系数优化法对涡轮叶片的寿命数据进行拟合分析及计算;同时为减少计算工作量及提高数值计算过程中人工智能的参与度,基于MATLAB软件对上述数值计算方法编写计算程序,最后对计算结果进行K-S假设检验。结果表明:对涡轮叶片寿命数据进行可靠性分析时,运用Weibull分布建立的数学模型符合客观规律;同时牛顿迭代法、三参数相关系数优化法及MATLAB计算程序的有效利用,保证了计算结果的精度。

针对十参数标定模型的导航避障相机影像精确校正

十参数标定模型是近景摄影测量中一种经典的内参标定模型。在影像畸变校正中,如果利用此模型采用直接重采样方式会产生空洞条纹,而常规的间接重采样方式虽无空洞产生,但在畸变较大的非中心区域同样无法得到较好的校正效果。针对此问题,本文提出了一种针对十参数标定模型的影像牛顿迭代精确校正算法。首先利用十参数标定模型将像点理想值和实际值进行重组,构建以像点实际值为待求量的非线性方程组;然后采用牛顿迭代法对影像中畸变部分进行局部线性逼近求解;最后采用间接重采样方法实现精确校正的目的。以移动机器人导航和避障相机作为研究对象,试验结果表明,牛顿迭代法校正可以得到高精度的优质去畸变影像,对避障相机的处理效果尤为显著。本文算法适用于移动机器人导航避障相机影像的畸变精确去除,为后续立体匹配环节生成优质的双目核线影像提供了可靠数据源。