乙烯深度氧化反应动力学方程的非线性最小二乘回归

介绍了采用非线性最小二乘方法回归乙烯深度氧化反应动力学方程。用高斯-牛顿法求解非线性最小二乘方程组,并推导出简单、适用的迭代公式。针对Gauss-Newton法的局部收敛问题,提出:(1)以任意组合的实验值代入回归方程,求得回归参数作为迭代初值;(2)对不同迭代初值所得的结果择优,作为最终结果。实例计算表明,该方法可以有效地回归Langmuir- Hinshelwood型动力学方程,为乙烯深度氧化反应的研究提供依据。     

一类求解非线性奇异方程组的牛顿改进算法

受一个求解非线性奇异方程组迭代格式的启示,将两种牛顿改进算法推广成一般形式,并将其发展为一类求解具有奇异雅可比矩阵的非线性方程组的牛顿改进算法.首先,描述这类新算法的迭代格式,并导出其收敛阶,该新格式每步迭代仅需计算一次函数值和一次导函数值;然后,对测试函数进行检验,并与牛顿算法及其他奇异牛顿算法进行比较,从而验证该算法的快速收敛性;最后,通过两个实际问题验证所提出算法的有效性.     

拟牛顿法求解化工过程数学模型

使用不需求取偏导数的拟牛顿法,求解化工过程模拟中产生的非线性方程组形式的数学模型。当未知数各分量间绝对值相差较大时,提出了改善收敛性的几种方法,即：（1）加入阻尼因子,以减少迭代值的震荡、（2）将方程适当降价;（3）将差商的绝对步长改为相对步长;（4）新迭代值超来其物理意义范围时,强制其回至其初始值。计算结果表明,与牛顿－拉夫森法相比,拟牛顿法不需求偏导数,对初值要求低,较雅可比迭代法收敛速度快,可用于求解化工过程的非线性方程组。     

有向图在求解大型非线性方程组中的应用

为了提高大维数非线性方程组的求解品质,该文提出了有向图法,将参数关系转化为有向图,利用有向图求圈算法来精简迭代参数,从而提高计算速度,降低算法发散的可能性。这种方法适用于求解工程化的大维数非线性方程组,可以用计算机自动实现。     

非线性微分方程组两点边值问题的参数微分——牛顿迭代解法

引言 在利用变分法或最大值原理求解最优控制问题时,常常最后归结为求解一个微分方程组的两点边值问题,对于线性常微分方程组的两点边值问题,可采用共轭函数、补足函数等法通过一次积分来求解,而对于非线性常微分方程组,则要运用上法或与牛顿法联合使用通过迭代来求解,而其中以牛顿片比较有效。在牛顿迭代程序中,偏导数矩阵     

一个求解非线性方程组的区间检验算法

引言非线性方程组的数值求解一直是计算数学的中心任务之一，越来越受到人们的重视．以Newton迭代法为代表的点迭代方法一直是被广泛运用的求解方法．但是，点迭代方法难以进行可靠的误差估计，且对迭代初值有较为苛刻的要求．本世纪七十年代以后兴起的区间分析方法可用于求解非线性方程组，特别在解的误差估计和解的存在性检验方面是卓有成效的．但是，区间方法的致命缺点是计算速度慢，运算量大；而且由于区间运算的复杂性，其程序的实现难度很大，有时甚至难以用于实际计算．这些缺点严重地限制了区间方法的应用和发展．因此，人们自然…     

竞选优化算法求解非线性方程组的应用研究

针对非线性方程组的求解在工程上具有广泛的实际意义,经典的数值求解方法存在其收敛性依赖于初值而实际计算中初值难确定的问题,将复杂非线性方程组的求解问题转化为函数优化问题,引入竞选优化算法进行求解。同时竞选优化算法求解时无需关心方程组的具体形式,可方便求解几何约束问题。通过对典型非线性测试方程组和几何约束问题实例的求解,结果表明了竞选优化算法具有较高的精确性和收敛性,是应用于非线性方程组求解的一种可行和有效的算法。     

基于PCA法向量计算和投影法的圆锥拟合

针对规则曲面特征重构中圆锥曲面重构拟合的问题,提出了一种基于PCA法向量计算和投影法的圆锥拟合方法。为了实现锥面的参数化重构,首先以距离函数的逼近函数模型作为目标函数,确定待计算的锥面拟合参数,然后求解锥面参数初值。计算参数初值时,通过PCA主成分分析法计算点云法向量,生成轴线后采用投影法计算底面圆心,再求解锥面其他参数初值,最后结合Levenberg-Marquardt法迭代重构圆锥面。通过分析锥面散乱点云,使用该方法得到的参数初值进行迭代,迭代次数7次,迭代最终平均误差为1.8804e-04。该方法在散乱点云中应用良好,降低了稀疏数据对初值的影响,改善了迭代初值造成的迭代不收敛性,进而求得更加精确的圆锥参数。     

拟牛顿法在航空发动机特性仿真中的应用

航空发动机特性仿真中常用牛顿迭代法求解非线性方程组,牛顿法每一步迭代计算都需要计算Jacobi矩阵,这需要多次发动机气动热力过程计算.因此避免大量重复计算Jacobi矩阵可以减少发动机计算整机的气动热力计算次数,从而提高发动机特性计算的速度.文中采用基于Broyden原理的拟牛顿法求解发动机非线性方程组,这种方法可以直接求出第一步迭代后的Jacobi矩阵,从而大幅度提高计算速度.将拟牛顿法应用于某型涡喷和涡扇发动机特性计算,通过分析计算结果,证明了采用拟牛顿法可以提高发动机特性模拟的计算速度.     

动态中子输运方程迭代初值的选取方法

研究离散纵标动态中子输运方程迭代求解时,迭代初值的不同选取方法,设计合理的迭代初值可以适当放宽对时同步长的限制,缩短计算时间.设计四种迭代初值并应用于数值求解中的等比格式和菱形格式,其中等比格式形成非线性离散方程,菱形格式形成线性离散方程.考察不同迭代初值的计算效率,分别对物理量变化平缓以及变化剧烈的问题进行考察.数值算例表明构造的基于物理量随时间走势的预估值作为迭代初值优势明显,这在保证计算精度的前提下提高了数值计算效率.     

Some implementation issues associated with multidimensional interval Newton methods

This paper presents the development of an optimal interval Newton method for systems of nonlinear equations. The context of solving such systems of equations is that of optimization of nonlinear mathematical programs. The modifications of the interval Newton method presented in this paper provide computationally effective enhancements to the general interval Newton method. The paper demonstrates the need to compute an optimal step length in the interval Newton method in order to guarantee the generation of a sequence of improving solutions. This method is referred to as the optimal Newton method and is implemented in multiple dimensions. Secondly, the paper demonstrates the use of the optimal interval Newton method as a feasible direction method to deal with non-negativity constraints. Also, included in this implementation is the use of a matrix decomposition technique to reduce the computational effort required to compute the Hessian inverse in the interval Newton method. The methods are demonstrated on several problems. Included in these problems are mathematical programs with perturbations in the problem coefficients. The numerical results clearly demonstrate the effectiveness and efficiency of these approaches.     

On the use of a modified Newton method for nonlinear finite element analysis

In this paper, the usefulness of modified Newton methods for solving certain minimization problems arising in nonlinear finite element analysis is investigated. The application considered is nonlinear elasticity, in particular geometrically nonlinear shells. On a test problem, it is demonstrated that a particular implementation of a modified Newton method using both descent directions and directions of negative curvature is able to identify a minimizer, whereas an unmodified Newton method and modified Newton methods using only descent directions fail to converge to the minimizer. The use of modified Newton methods is suggested as a useful complement to the present continuation methods used for nonlinear finite element analysis.     

Structural optimization using the Newton modified barrier method

The Newton Modified Barrier Method (NMBM) is applied to structural optimization problems with large a number of design variables and constraints. This nonlinear mathematical programming algorithm was based on the Modified Barrier Function (MBF) theory and the Newton method for unconstrained optimization. The distinctive feature of the NMBM method is the rate of convergence that is due to the fact that the design remains in the Newton area after each Lagrange multiplier update. This convergence characteristic is illustrated by application to structural problems with a varying number of design variables and constraints. The results are compared with those obtained by optimality criteria (OC) methods and by the ASTROS program.     

多标签分类器准确性评估方法的研究

分类是数据挖掘领域研究的核心技术之一,分类器性能评估方法也是众多学者的研究热点之一。以往的分类器性能评估方法一般针对于单标签数据集,对于多标签问题并未涉及。文中主要针对多标签分类问题中的单实例情况,提出了一种多标签分类器准确性评估方法（EMOSIML）。该方法的思路是：如果分类器对一个多标签对象预测的类别标签是其属于的多个类别标签中的任何一个,则分类结果都是正确的。该方法用C#编程实现,并对朴素贝叶斯分类器进行分类器性能评估实验,实验结果表明,EMOSIML评估方法较传统的准确率评估方法更合理。     

插管式多层地温测量传感器设计

介绍了四线制引线Pt100电阻器用于实现多层地温测量传感器的具体设计方案,该方案主要包括AD采样电路、多路切换测量电路、牛顿迭代法计算温度值以及插管式结构设计,并进行算法验证和实际测量对比,经过验证,采用牛顿迭代法计算的温度与查表法得到的温度误差在0.015℃以内,实际测量与标准温度误差在0.2℃以内,该方案完全满足目前对地温测量的要求。     

Numerical experience with newton-like methods for nonlinear algebraic systems

In this paper we present an extensive computational experience with several Newton-like methods, namely Newton’s method, the ABS Huang method, the ABS row update method and six Quasi-Newton methods. The methods are first tested on 31 families of problems with dimensionsn=10, 50, 100 and two starting points. Newton’s method appears to be the best in terms of number of solved problems, followed closely by the ABS Huang method. Broyden’s “bad” method and Greenstadt’s second method show a very poor performance. The other four Quasi-Newton methods perform similarly, strongly suggesting that Greenstadt’s first method and Martínez’ column update method are locally and superlinearly convergent, a result that has yet to be proven theoretically. Thomas’ method appears to be marginally more robust and fast and provides moreover a better approximation to the Jacobian. An interesting and somewhat unexpected observation is that the number of iterations for satisfying the convergence test increases very little with the dimension of the problem. In a second set of experiments we look at the structure of the regions of convergence/nonconvergence by starting the methods from all nodes of a regular grid and assigning to each node a number according to the outcome of the iteration. The obtained regions have clearly a fractal type structure, which, on the two tested problems, is much simpler for Newton’s method than for the other methods. Newton’s method also is the one with the smallest nonconvergence region. Among the Quasi-Newton methods Thomas’ method shows a definitely smaller nonconvergence region.     

On the solution of a rational matrix equation arising in G-networks

We consider the problem of solving a rational matrix equation arising in the solution of G-networks. We propose and analyze two numerical methods: a fixed point iteration and the Newton–Raphson method. The fixed point iteration is shown to be globally convergent with linear convergence rate, while the Newton method is shown to have a local convergence, with quadratic convergence rate. Numerical experiments show the effectiveness of the proposed methods.     

Observer design for nonlinear systems with discrete-timemeasurements

This paper focuses on the development of asymptotic observers for nonlinear discrete-time systems. It is argued that instead of trying to imitate the linear observer theory, the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations. In particular, it is shown that the discrete Newton method, properly interpreted, yields an asymptotic observer for a large class of discrete-time systems, while the continuous Newton method may be employed to obtain a global observer. Furthermore, it is analyzed how the use of Broyden's method in the observer structure affects the observer's performance and its computational complexity. An example illustrates some aspects of the proposed methods; moreover, it serves to show that these methods apply equally well to discrete-time systems and to continuous-time systems with sampled outputs     

Modified online Newton step based on elementwise multiplication

The second-order method using a Newton step is a suitable technique in online learning to guarantee a regret bound. The large data are a challenge in the Newton method to store second-order matrices such as the hessian. In this article, we have proposed a modified online Newton step that stores first- and second-order matrices of dimension m (classes) by d (features). We have used elementwise arithmetic operations to maintain the size of matrices. The modified second-order matrix size results in faster computations. Also, the mistake rate is on par with respect to popular methods in the literature. The experimental outcome indicates that proposed method could be helpful to handle large multiclass datasets on common desktop machines using second-order method as the Newton step.     

Convergence of the modified SOR–Newton method for non-smooth equations

Motivated by Chen [On the convergence of SOR methods for nonsmooth equations. Numer. Linear Algebra Appl. 9 (2002), pp. 81–92], in this paper, we further investigate a modified SOR–Newton (MSOR–Newton) method for solving a system of nonlinear equations F(x)=0, where F is strongly monotone and locally Lipschitz continuous but not necessarily differentiable. The convergence interval of the parameter in the MSOR–Newton method is given. Compared with that of the SOR–Newton method, this interval can be enlarged. Furthermore, when the B-differential of F(x) is difficult to compute, a simple replacement can be used, which can reduce the computational load. Numerical examples show that at the same cost of computational complexity, this MSOR–Newton method can converge faster than the corresponding SOR–Newton method by choosing a suitable parameter.