Linearized ridge-path method for function minimization

A modification based on a linearization of a ridge-path optimization method is presented. The linearized ridge-path method is a nongradient, conjugate direction method which converges quadratically in half the number of search directions required for Powell's method of conjugate directions. The ridge-path method and its modification are compared with some basic algorithms, namely, univariate method, steepest descent method, Powell's conjugate direction method, conjugate gradient method, and variable-metric method. The assessment indicates that the ridge-path method, with modifications, could present a promising technique for optimization.This work was in partial fulfillment of the requirements for the MS degree of the first author at Cairo University, Cairo, Egypt. The authors would like to acknowledge the helpful and constructive suggestions of the reviewer.     

Finding roots of a real polynomial simultaneously by means of Bairstow's method

Aberth's method for finding the roots of a polynomial was shown to be robust. However, complex arithmetic is needed in this method even if the polynomial is real, because it starts with complex initial approximations. A novel method is proposed for real polynomials that does not require any complex arithmetic within iterations. It is based on the observation that Aberth's method is a systematic use of Newton's method. The analogous technique is then applied to Bairstow's procedure in the proposed method. As a result, the method needs half the computations per iteration than Aberth's method. Numerical experiments showed that the new method exhibited a competitive overall performance for the test polynomials.     

Stokes方程的投影／方向分裂谱方法

我们提出和分析了一种求解Stokes方程的数值方法．新方法基于空间上的Legendre谱离散,时间上则采用投影／方向分裂格式．更确切地说,时间离散的出发点是旋度形式的压力校正投影法,在此基础上进一步应用方向分裂法,把速度和压力方程分裂为一系列一维的椭圆型子问题．然后生成的这些一维子问题用Legendre谱方法进行空间离散．另外,我们证明了全离散格式的稳定性．一些数值实验验证了收敛性和方法的有效性．     

对称正定矩阵的多级迭代法

本文主要研究解对称正定矩阵的多级迭代法,并对其收敛性进行证明。最后用数值实验验证此方法的有效性．多级迭代法特别适用于并行计算,并且可以被理解为古典迭代法的扩展,或共轭梯度法的预处理子。     

从常步长梯度方法的视角看不可微凸优化增广Lagrange方法的收敛性

增广Lagrange方法是求解非线性规划的一种有效方法.从一新的角度证明不等式约束非线性非光滑凸优化问题的增广Lagrange方法的收敛性.用常步长梯度法的收敛性定理证明基于增广Lagrange函数的对偶问题的常步长梯度方法的收敛性,由此得到增广Lagrange方法乘子迭代的全局收敛性.     

Rational Krylov for Nonlinear Eigenproblems,an Iterative Projection Method

In recent papers Ruhe suggested a rational Krylov method for nonlinear eigenproblems knitting together a secant method for linearizing the nonlinear problem and the Krylov method for the linearized problem. In this note we point out that the method can be understood as an iterative projection method. Similarly to the Arnoldi method the search space is expanded by the direction from residual inverse iteration. Numerical methods demonstrate that the rational Krylov method can be accelerated considerably by replacing an inner iteration by an explicit solver of projected problems.     

A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger–Poisson equations with discontinuous potentials

In this paper, we propose a high order Fourier spectral-discontinuous Galerkin method for time-dependent Schrödinger–Poisson equations in 3-D spaces. The Fourier spectral Galerkin method is used for the two periodic transverse directions and a high order discontinuous Galerkin method for the longitudinal propagation direction. Such a combination results in a diagonal form for the differential operators along the transverse directions and a flexible method to handle the discontinuous potentials present in quantum heterojunction and supperlattice structures. As the derivative matrices are required for various time integration schemes such as the exponential time differencing and Crank Nicholson methods, explicit derivative matrices of the discontinuous Galerkin method of various orders are derived. Numerical results, using the proposed method with various time integration schemes, are provided to validate the method.     

Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces

In this paper, we introduce a new iteration method based on the hybrid method in mathematical programming and the descent-like method for finding a common element of the set of solutions for a variational inequality and the set of fixed points for a nonexpansive mapping in Hilbert spaces. Our method modifies and improves some methods in literature.     

A hybrid inexact Logarithmic-Quadratic Proximal method for nonlinear complementarity problems

Inspired by the Logarithmic-Quadratic Proximal method [A. Auslender, M. Teboulle, S. Ben-Tiba, A logarithmic-quadratic proximal method for variational inequalities, Comput. Optim. Appl. 12 (1999) 31-40], we present a new prediction-correction method for solving the nonlinear complementarity problems. In our method, an intermediate point is produced by approximately solving a nonlinear equation system based on the Logarithmic-Quadratic Proximal method; and the new iterate is obtained by convex combination of the previous point and the one generated by the improved extragradient method at each iteration. The proposed method allows for constant relative errors and this yields a more practical Logarithmic-Quadratic Proximal type method. The global convergence is established under mild conditions. Preliminary numerical results indicate that the method is effective for large-scale nonlinear complementarity problems.     

解线性互补问题的预处理加速模Gauss-Seidel迭代方法

本文提出了解线性互补问题的预处理加速模系Gauss-Seidel迭代方法,当线性互补问题的系统矩阵是M-矩阵时证明了方法的收敛性,并给出了该预处理方法关于原方法的一个比较定理.数值实验显示该预处理迭代方法明显加速了原方法的收敛.     

Formalization of implication based fuzzy reasoning method

Fuzzy reasoning includes a number of important inference methods for addressing uncertainty. This line of fuzzy reasoning forms a common logical foundation in various fields, such as fuzzy logic control and artificial intelligence. The full implication triple I method (a method only based on implication, TI method for short) for fuzzy reasoning is proposed in 1999 to improve the popular CRI method (a hybrid method based on implication and composition). The current paper delves further into the TI method, and a sound logical foundation is set for the TI method based on the monoidal t-norm based logical system MTL.     

Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problems

In this letter, a new numerical method is proposed for solving second order linear singularly perturbed boundary value problems with left layers. Firstly a piecewise reproducing kernel method is proposed for second order linear singularly perturbed initial value problems. By combining the method and the shooting method, an effective numerical method is then proposed for solving second order linear singularly perturbed boundary value problems. Two numerical examples are used to show the effectiveness of the present method.     

An Liu-Storey-Type Method for Solving Large-Scale Nonlinear Monotone Equations

An algorithm for solving nonlinear monotone equations is proposed, which combines a modified Liu-Storey conjugate gradient method with hyperplane projection method. Under mild conditions, the global convergence of the proposed method is established with a suitable line search method. The method can be applied to solve large-scale problems for its lower storage requirement. Numerical results indicate that our method is efficient.     

On preconditioned Uzawa methods and SOR methods for saddle-point problems

This paper studies convergence analysis of a preconditioned inexact Uzawa method for nondifferentiable saddle-point problems. The SOR-Newton method and the SOR-BFGS method are special cases of this method. We relax the Bramble-Pasciak-Vassilev condition on preconditioners for convergence of the inexact Uzawa method for linear saddle-point problems. The relaxed condition is used to determine the relaxation parameters in the SOR-Newton method and the SOR-BFGS method. Furthermore, we study global convergence of the multistep inexact Uzawa method for nondifferentiable saddle-point problems.     

一般单调变分不等式的一个改进的预估-校正算法

最近何炳生等提出了解大规模单调变分不等式的一种预估-校正算法,然而,这个方法在计算每一个试验点时需要一次投影运算,因而计算量较大．为了克服这个缺点,我们提出了一个解一般大规模g-单调变分不等式的新的预估-校正算法,该方法使用了一个非常有效的预估步长准则,每个步长的选取只需要计算一次投影,这将大大减少计算量．数值试验说明我们的算法比最新文献中出现的投影类方法有效．     

求解一类非线性方程的改进的平均法

本文研究确定非线性方程一致有效近似解的平均法,得到B方法(Krylov-Bogoliubov method)与KBM方法(Krylov-Bogoliubov-Mitropolski method)的改进形式,通过两个例子与多尺度方法的比较,说明改进的平均法的有效性,从而拓广了平均法的应用范围。     

一个解凸二次规划的预测-校正光滑化方法

本文为凸二次规划问题提出一个光滑型方法,它是Engelke和Kanzow提出的解线性规划的光滑化算法的推广。其主要思想是将二次规划的最优性K-T条件写成一个非线性非光滑方程组,并利用Newton型方法来解其光滑近似。本文的方法是预测-校正方法。在较弱的条件下,证明了算法的全局收敛性和超线性收敛性。     

Optimality criteria of the disk modulo allocation method for Cartesian product files

Cartesian product (CP) files have been shown to be very effective for partial match retrieval. Further the Disk Modulo (DM) allocation method has been shown to be a simple and effective method for allocating CP files onto multiple independently accessible disks to further facilitate partial match retrieval. In this paper, a useful criterion for the DM method to be the best allocation method for a given CP file is presented. The presented criterion is much more general than that suggested previously which makes the DM allocation method much more applicable for real applications.     

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.     

对多等级质量产品（以σ衡量）抽样数n值的计算及分极标准的制定方法

本文利用抽样分布理论，对以σ衡量质量的多等级产品质量检验问题，提出了一个具体的处理方法，该方法给出了计算抽样数的公式，也给出了确定各等级之间距离的计算方法，揭示了抽样置信度、抽样数ｎ、等级距离三者之间的关系。并按本文的方法，提出了对广东商检局出口桑蚕丝原抽样方案的修改意见。