Implementation of LDA+ Gutzwiller with Newton's method

In order to calculate the electronic structure of correlated materials, we propose implementation of the LDA＋Gutzwiller method with Newton＇s method. The self-consistence process, efficiency and convergence of calculation are improved dramatically by using Newton＇s method with golden section search and other improvement approaches.We compare the calculated results by applying the previous linear mix method and Newton＇s method. We have applied our code to study the electronic structure of several typical strong correlated materials, including SrVO3, LaCoO3, and La2O3Fe2Se2. Our results fit quite well with the previous studies.

INEXACT DAMPED NEWTON METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS

In this paper, we propose an inexact damped Newtonmethod for solving nonlinear complementarity problems based on the equivalent B differentiable equations.Global convergence and locally quadratic convergence are obtained,and numerical results are given.

The Formulas to Compare the Convergences of Newton’s Method and the Extended Newton’s Method (Tsuchikura-Horiguchi Method) and the Numerical Calculations

This paper gives the extension of Newton’s method, and a variety of formulas to compare the convergences for the extension of Newton’s method (Section 4). Section 5 gives the numerical calculations. Section 1 introduces the three formulas obtained from the cubic equation of a hearth by Murase (Ref. [1]). We find that Murase’s three formulas lead to a Horner’s method (Ref. [2]) and extension of a Newton’s method (2009) at the same time. This shows originality of Wasan (mathematics developed in Japan) in the Edo era (1603-1868). Suzuki (Ref. [3]) estimates Murase to be a rare mathematician in not only the history of Wasan but also the history of mathematics in the world. Section 2 gives the relations between Newton’s method, Horner’s method and Murase’s three formulas. Section 3 gives a new function defined such as .

Gauss-Newton法的半局部收敛性

设f:Rn→Rm 是Frechet可微的 ,m≥n .则非线性最小二乘问题可描述为下面的极小化问题 :minF(x) :=12 f(x) Tf(x) .Gauss Newton法是求解非线性最小二乘问题的最基本的方法之一 ,其n + 1步迭代定义为 :xn + 1=xn - f′(xn) Tf′(x) -1f′(xn) Tf(xn) .本文主要研究解非线性最小二乘问题的Gauss Newton法的半局部收敛性 .假设f(x)在B(x0 ,r)内连续可导且f′(x0 )满秩 ,若f的导数满足Lipschitz连续F′(x) -f′(x′)≤γx -x′ , x ,x′∈B(x0 ,r) .在一个关于初始点x0 的判断准则c =f(x0 ) ,β =f′T(x0 )f′(x0 ) -1f′(x0 ) T ,β2 cγ <1 1 0下 ,Gauss Newton法产生的序列 {xn}收敛到一个驻点x ,从而给出了Gauss Newton法的半局部收敛性 .

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.

On the Fourier approximation method for steady water waves

A computational method for steady water waves is presented on the basis of potential theory in the physical plane with spatial variables as independent quantities. The finite Fourier series are applied to approximating the free surface and potential function. A set of nonlinear algebraic equations for the Fourier coefficients are derived from the free surface kinetic and dynamic boundary conditions. These algebraic equations are numerically solved through Newton＇s iterative method, and the iterative stability is further improved by a relaxation technology. The integral properties of steady water waves are numerically analyzed, showing that （1） the set-up and the set-down are both non-monotonic quantities with the wave steepness, and （2） the Fourier spectrum of the free surface is broader than that of the potential function. The latter further leads us to explore a modification for the present method by approximating the free surface and potential function through different Fourier series, with the truncation of the former higher than that of the latter. Numerical tests show that this modification is effective, and can notably reduce the errors of the free surface boundary conditions.

Novel Newton’s learning algorithm of neural networks

Newton＇s learning algorithm of NN is presented and realized. In theory, the convergence rate of learning algorithm of NN based on Newton＇s method must be faster than BP＇s and other learning algorithms, because the gradient method is linearly convergent while Newton＇s method has second order convergence rate. The fast computing algorithm of Hesse matrix of the cost function of NN is proposed and it is the theory basis of the improvement of Newton＇s learning algorithm. Simulation results show that the convergence rate of Newton＇s learning algorithm is high and apparently faster than the traditional BP method＇s, and the robustness of Newton＇s learning algorithm is also better than BP method＇ s.

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.

Hybrid Steffensen’s Method for Solving Nonlinear Equation

In this paper, we are going to present a class of nonlinear equation solving methods. Steffensen’s method is a simple method for solving a nonlinear equation. By using Steffensen’s method and by combining this method with it, we obtain a new method. It can be said that this method, due to not using the function derivative, would be a good method for solving the nonlinear equation compared to Newton’s method. Finally, we will see that Newton’s method and Steffensen’s hybrid method both have a two-order convergence.

The Convergences Comparison between the Halley’s Method and Its Extended One Based on Formulas Derivation and Numerical Calculations

The purpose of this paper is that we give an extension of Halley’s method (Section 2), and the formulas to compare the convergences of the Halley’s method and extended one (Section 3). For extension of Halley’s method we give definition of function by variable transformation in Section 1. In Section 4 we do the numerical calculations of Halley’s method and extended one for elementary functions, compare these convergences, and confirm the theory. Under certain conditions we can confirm that the extended Halley’s method has better convergence or better approximation than Halley’s method.

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.

Least Square Finite Element Method for Viscous Splitting of Unsteady Incompressible Navier–Stokes Equations

In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split into diffusive and convective parts in each time step. The diffusive part is discretized by the backward difference method in time and discretized by the standard Galerkin method in space. The convective part is a first-order nonlinear equation.After the linearization of the nonlinear part by Newton’s method, the convective part is also discretized by the backward difference method in time and discretized by least square scheme in space. C0-type element can be used for interpolation of the velocity and pressure in the present model. Driven cavity flow and flow past a circular cylinder are conducted to validate the present model. Numerical results agree with previous numerical results, and the model has high accuracy and can be used to simulate problems with complex geometry.

MEMS传感器的惯性测量模块的设计与初始校准

设计了一种基于微机电系统(MEMS)传感器的惯性测量模块,它包括三轴加速度计、三轴陀螺和三轴磁阻传感器。这种惯性测量模块具有体积小、功耗低、成本低的优点,可以方便地用在微小机器人定位系统以及空中机器人控制系统中。详细分析了模块的误差来源,提出了模块中三轴加速度计的非正交误差模型。并运用基于Newton迭代的方法实现了一种自动初始校准算法。实验结果表明:这种自动初始校准算法可以有效地消除该模块的固定偏差、比例误差和非正交误差。

基于NURBS插补算法的汽轮机叶片数控加工

为了提高汽轮机叶片的加工精度,文章通过分析汽轮机叶片的结构特点,采用一种NURBS曲线插补算法对其进行插补仿真。文章所采用算法由两部分组成:速度规划和参数计算。首先速度规划采用简化的五段S曲线加减速控制方法,保证了叶片在高速加工过程中加速度的连续,使机床运行平稳,然后利用牛顿迭代法来计算插补参数,得到更精确的插补参数,进一步提高了叶片的加工精度和加工速度。汽轮机叶片的插补仿真表明,该算法有高的稳定性和运算精度,并且使机床振动减小,速度波动小,保证了叶片的高质量加工。

一族三阶收敛的Newton型迭代法

基于中点处的导数值构造了一个三阶收敛的Newton型迭代法,与[10]中的方法相比,每步迭代计算相同的函数值,但是数值实例表明该方法的迭代效果更好。此外,在该方法的基础上构造了一族三阶收敛的迭代方法。

NURBS插补中的速度规划与参数计算

为满足数控机床高速度、高质量加工的需求,提出一种新的非均匀有理B样条曲线插补算法。该算法包括速度规划和参数计算两部分。速度规划部分采用五段S曲线加减速控制方法,能够保证高速运行过程中加速度的连续,使机床运行平稳,避免产生激烈的震颤;参数计算部分应用抛物线插值结合牛顿迭代的方法计算插补参数,将实时插补时产生的进给速度波动控制到理想水平,从而进一步减小机床震颤。仿真实验表明,该算法能够减小机床振动,实现高质量加工。

一类奇摄动非线性边值问题激波解的Sinc-Galerkin方法

讨论了一类非线性奇摄动方程的激波问题.利用Sinc-Galerkin方法,构造出边值问题的激波解,并由Newton法得到其近似解.

重根情形的Newton迭代法收敛性加速

基于Newton迭代法对于求重根具有线性收敛性,给出了加速其收敛的方法以及迭代公式,收敛速度得到了有效的提高。最后从数值实验加以比较,此算法是可行的。

Simpson牛顿公式的一种改进

文章基于牛顿公理给出非线性方程求根的一种三阶方法,证明了该迭代格式三阶收敛到单根,计算效能高于其他同类迭代法;在方程根的重数m已知和未知的情形下,分别给出了该方法的改进公式,并指出了它们的收敛阶;最后给出数值试验并与其他方法进行比较,结果显示该方法非常有效,具有一定的理论价值和应用价值。

一族带有两参数的修正型Chebyshev-Halley迭代方法（英文）

应用(2,1)阶Padé逼近方法,得到不需要计算二阶导数求解非线性方程的修正型Chebyshev-Halley方法的新两参数族,证明该族方法是至少三阶收敛。该族方法的每步迭代需要计算两个函数和一个一阶导数,数值实验表明,该族迭代方法与其它方法相比,在许多方面得到了更好的数值结果。