Optimal Newton–Secant like methods without memory for solving nonlinear equations with its dynamics

We construct two optimal Newton–Secant like iterative methods for solving nonlinear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane.     

一种适用于多机器人搜索动态目标的改进粒子群算法

粒子群算法引导机器人搜索跟踪动态目标时,在迭代后期易出现收敛停滞现象。为了改善上述情况,本文提出了结合牛顿法的改进粒子群算法。为了结合粒子群算法与牛顿法,我们在算法中引入了马尔科夫链,这使得机器人在每一次迭代时以一定的概率随机选择牛顿法或粒子群算法搜索跟踪目标。为了模拟机器人搜索动态目标的真实环境,本文还利用了通信项使机器人以一定的方式努力与基站保持通信,用来实时更新目标信息。仿真结果表明,改进的粒子群算法能有效的寻找并跟踪动态目标。     

Optimal Coding of Quantized Laplacian Sources for Predictive Image Compression

This paper discusses the optimal coding of uniformly quantized Laplacian sources. The techniques known for designing optimal codes for sources with infinite alphabets are used for the quantized Laplacian sources which have probability mass functions with two geometrically decaying tails. Due to the simple parametric model of the source distribution the Huffman iterations are possible to be carried on analytically, using the concept of reduced source, and the final codes are obtained as a sequence of very simple arithmetic operations, avoiding the need to store coding tables. Comparing three uniform quantizers, we find one which consistently outperforms the others in the rate-distortion sense. We foresee for the newly introduced codes an important area of applications in low complexity lossy image coding, since similar codes, designed for two-sided geometrical sources, became the basic tools used in JPEG-LS lossless image compression.     

Fourth-order convergence theorem by using majorizing functions for super-Halley method in Banach spaces

In this paper, we study the semilocal convergence of a multipoint fourth-order super-Halley method for solving nonlinear equations in Banach spaces. We establish the Newton–Kantorovich-type convergence theorem for the method by using majorizing functions. We also get the error estimate. In comparison with the results obtained in Wang et al. [X.H. Wang, C.Q. Gu, and J.S. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces, Numer. Algorithms 56 (2011), pp. 497–516], we can provide a larger convergence radius. Finally, we report some numerical applications to demonstrate our approach.     

A family of chebyshev type methods in banach spaces

A family of third order iterative processes, that includes Chebyshev method, is studied in Banach Spaces. Results on convergence and uniqueness of solution are given, as well as error estimates.     

A new iterative method for large sparse saddle point problems

In this short note, the convergence of a new iterative method for the Saddle Point Problem is presented.     

A simple algorithm for the solution of the multiple dosing problem

A system theoretical approach to the general multiple dosing problem is discussed [1], which is in turn reduced to a classical parameter optimization problem. An efficient numerical algorithm based on the cyclic coordinate search method to solve the latter is explained. A flexible and user-friendly program written in standard FORTRAN which is suitable for this problem is introduced. Finally, the problem is applied to the dopaminergic substitution problem in Parkinson's disease, and the necessary input data as well as the resulting output is displayed in a sample run.     

On one numerical method for solving some self-similarityproblems of gas-dynamics on a multiprocessor

In this paper we consider mathematical models of some problems of natural science, for example, self-similarity problems of gas-dynamics giving rise to boundary problems of first order ordinary differential equations (ODE) with one parameter. The boundary problems of first order ODE with one parameter are considered in [1, 2], where iterative methods based on the implementation of Newton's Method, are presented. Next, an iterative method for solving the boundary value problem of the first order system of ODE with one parameter on a multiprocessor type SIMD [3] is shown. The convergence of this process is proved and the speed of convergence is estimated. The feasibility of this method is illustrated for the one dimensional instability movement of gas arising from the movement of the piston in presence of a volume source (volume channel) of mass, impulse and energy in gas. Finally the results are given.     

New iterative method for solving non-linear equations with fourth-order convergence

In this work, we introduce an extension of the classical Newton's method for solving non-linear equations. This method is free from second derivative. Similar to Newton's method, the proposed method will only require function and first derivative evaluations. The order of convergence of the introduced method for a simple root is four. Numerical results show that the new method can be of practical interest.     

粒子群算法在求解方程组中的应用

提出了采用粒子群算法求解线性方程组和非线性方程组的智能算法。采用粒子群算法求解方程组具有形式简单、收敛迅速和容易理解等特点,且能在一次计算中多次发现方程组的解,可以解决非线性方程组多解的求解问题,为线性方程组和非线性方程组的求解提供了一种新的方法。     

Numerical solution of homogeneous Smoluchowski's coagulation equation

Smoluchowski's equation is widely applied to describe the time evolution of the cluster-size distribution during aggregation processes. Analytical solutions for this equation, however, are known only for a very limited number of kernels. Therefore, numerical methods have to be used to describe the time evolution of the cluster-size distribution. A numerical technique is presented for the solution of the homogeneous Smoluchowski's coagulation equation with constant kernel. In this paper, we use Taylor polynomials and radial basis functions together to solve the equation. This method converts Smoluchowski's equation to a system of nonlinear equations that can be solved for unknown parameters. A numerical example with known solution is included to demonstrate the validity and applicability of the technique.     

基于正四棱锥结构的机器人声源定位系统研究

声源定位成为机器人智能研究的重要方向。针对当前声源定位精度不理想、实时性不佳等问题,提出了一种正四棱锥麦克风阵列声源定位结构。采用时间延迟估计的声源定位方法,并提出时延值的快速搜索策略;推导了该结构的基于信号时延的时空映射关系,建立了声源目标位置的几何计算模型,并依据正四棱锥结构特点及冗余的时延值对值域划分,缩小求解范围,运用迭代算法得到声源的位置坐标,并通过双重筛选机制剔除错误的定位结果。实验结果证明了该结构及定位算法在提高系统定位精度和实时性能的有效性,能满足机器人应用中对声源定位的需求。     

Optimization method for the inverse problem of reconstructing the source term in a parabolic equation

This work investigates the inverse problem of reconstructing a spacewise dependent heat source in the parabolic heat equation using a final temperature measurement. Such problem has important application in a large field of applied science. On the basis of the optimal control framework, the existence and necessary condition of the minimizer for the cost functional are established. The global uniqueness and stability of the minimizer are deduced from the necessary condition. The Landweber iteration algorithm is applied to the inverse problem and some numerical results are presented for various typical test examples.     

Numerical solution of time-dependent component with sparse structure of source term for a time fractional diffusion equation

We consider an inverse time-dependent component of source term with sparse structure for the time fractional diffusion equation in the present paper. We prove the uniqueness of the inverse problem with nonlocal observation data by Laplace transform technique. Concerning the sparsity of the source term, we transform the inverse source problem into an elastic-net regularization optimization problem. The semi-smooth Newton method is adopted to solve the optimization problem and the superconvergence of the semi-smooth Newton algorithm is proven. Several numerical examples are tested to verify the efficiency of the algorithm.     

On a third order family of methods for solving nonlinear equations

In this article we present a third-order family of methods for solving nonlinear equations. Some well-known methods belong to our family, for example Halley's method, method (24) from [M. Basto, V. Semiao, and F.L. Calheiros, A new iterative method to compute nonlinear equations, Appl. Math. Comput. 173 (2006), pp. 468–483] and the super-Halley method from [J.M. Gutierrez and M.A. Hernandez, An acceleration of Newton's method: Super-Halley method, Appl. Math. Comput. 117 (2001), pp. 223–239]. The convergence analysis shows the third order of our family. We also give sufficient conditions for the stopping inequality |x _n+1?α|≤|x _n+1?x _n | for this family. Comparison of the family members shows that there are no significant differences between them. Several examples are presented and compared.     

基于Hamilton回路的车辆巡逻问题优化算法

一类车辆巡逻问题可以归结为赋权Hamilton回路最小化问题。该文采用一种局部优化的单点切割方法,优化了业已求得的Hamilton回路经典启发式算法,给出了算法基础定理的数学证明,通过算例说明了算法的实现过程。该算法改进了经典启发式算法的性能,在实践中取得了良好的效果。     

Simulation of thin film thickness distribution for thermal evaporation process using a scanning linear source

Thermal evaporation process is the main process involved in the production of OLED displays and with the trends toward larger substrate size and display resolution, film thickness uniformity must be carefully controlled in order to implement exact pixel data. To secure stable film thickness uniformity on the substrate area, thin films are deposited on large‐area glass substrates via thermal evaporation process using a linear source. We designed a linear source and mathematical model was developed to describe the system with a focus on the linear source. Then, system parameters were determined to guarantee uniform thickness using computer‐based simulation, replacing wasteful actual experiments, followed by carrying out experiments based on the determined parameters. After the deposition process, data from the mathematical model and experiments was compared and the resulting agreement was good, verifying the validity of the proposed method. Consequently, by applying the proposed method, display manufacturing process related to thermal evaporation can be controlled within a tight tolerance in order to maximize the production yield rate.     

非线性迭代学习控制问题的延拓修正牛顿法

对于非线性迭代学习控制问题,提出基于延拓法和修正Newton法的具有全局收敛性的迭代学习控制新方法.由于一般的Newton型迭代学习控制律都是局部收敛的,在实际应用中有很大局限性.为拓宽收敛范围,该方法将延拓法引入迭代学习控制问题,提出基于同伦延拓的新的Newton型迭代学习控制律,使得初始控制可以较为任意的选择.新的迭代学习控制算法将求解过程分成N个子问题,每个子问题由换列修正Newton法利用简单的递推公式解出.本文给出算法收敛的充分条件,证明了算法的全局收敛性.该算法对于非线性系统迭代学习控制具有全局收敛和计算简单的优点.     

彩色图像颜色量化问题的求解方法

针对K-means聚类算法在彩色图像颜色量化问题中对初始条件依赖性较强而易陷入局部最优的缺点,以及传统智能优化算法在寻优时只考虑了种群层内个体的相互竞争而忽略种群层间相互协作的问题,提出了一种基于K-means的金字塔结构演化策略（PES）彩色图像量化算法。首先,将K-means聚类算法中的聚类损失函数作为新算法的适应度函数;其次,运用PES对色彩进行种群初始化、分层、探索、加速以及聚类等操作;最后,利用新算法对4幅标准彩色测试图像进行不同色彩量化级的量化。实验结果表明,所提算法能够改善K-means聚类算法以及传统智能算法的上述缺陷,在类内均方误差评判准则下,图像的平均失真率比基于PES的算法低12.25%,比差分进化算法低15.52%,比粒子群优化（PSO）算法低58.33%,比K-means算法低15.06%,且随着色彩量化级的减少,算法量化后的图像失真率比其他算法降低更多,此外,算法量化图像的视觉效果优于其他算法。