直角坐标网格下 LSFD方法的显式公式和性能研究

重点讨论了 LSFD(least square-based finite difference)方法和传统的FD(finite difference)方法在性能上的对比问题.对于传统的中心差分格式,一阶导数和二阶导数在二维情况的数值格式基架点有9个点,三维情况有27个点.在同样的基架点下,给出了LSFD方法近似一阶导数和二阶导数的显式公式,并指出LSFD方法在这种情况下实质上就是在不同网格线上的传统中心差分格式的组合.在数值模拟中,LSFD方法达到收敛所需要的迭代步数比传统差分格式少,并且x和y方向的网格纵横尺度比在 LSFD方法中是一个非常重要的参数,对计算的稳定性有重要影响.     

基于连分式重新推导的Newton迭代公式及其变体

基于Thiele连分式,重新建立了求解非线性方程的经典的Newton迭代公式.为了避免求导数运算,采用差商可以近似代替导数的办法,得到Newton迭代方法的几个变体并给出了其收敛的阶数.最后,数值实例证实了这些迭代格式是有效的.     

一类不带导数的牛顿迭代格式

构造了一类新型的不带导数的牛顿迭代格式,通过建立误差方程,证明了该迭代格式至少是4阶收敛,同时获得了该迭代格式对应参数所满足的条件.     

一种加速收敛的Halley迭代修正格式（英文）

提出了一种求解非线性方程的加速收敛的Halley迭代修正格式,该格式不需要计算二阶导数,每步迭代只需要计算三个函数值和一个一阶导数值,该方法的效率指数为46~(1/2)≈1.565.数值实验结果表明,与已有文献[Appl.Math.Comput.,2010,217(6):2448-2455]和[J.Comput.Appl.Math.,2010,233(9):2278-2284]中最优八阶迭代格式相比,该修正格式具有更大的收敛半径,有效改善了最优八阶迭代格式对初值的苛刻要求,并且扩展计算指数大于最优八阶迭代格式的扩展计算指数,显示了其计算优势.     

带凸约束的非线性方程组的无导数记忆法

基于无导数线搜索技术和投影方法,本文提出了一种新的求解带凸约束的非线性方程组的无导数记忆法.该方法在每步迭代时不需要计算和贮存任何矩阵,因而适合求解大规模非线性方程组问题.在较弱条件下,该算法具有全局收敛性.数值试验结果及其相关的比较表明该算法是比较有效的.     

无记忆拟牛顿方法的收敛性

本讨论了无约束最优化问题的无记忆拟牛顿方法的收敛性,给出了对于非凸目标函数,在非精确线搜索条件下,无记忆拟牛顿方法收敛性的几个充分性条件。     

一类具有导数型非线性记忆项的半线性双波动方程解的爆破研究

研究了一类具有导数型非线性记忆项的半线性双波动方程在次临界情况下解的爆破问题.应用测试函数和泛函分析方法得到了其解的第一下界和迭代序列.然后运用迭代方法推出了其全局解的非存在性和生命跨度的上界估计.进一步补充了有关高阶波动方程柯西问题解的爆破研究.     

求解PageRank问题的多步幂法修正的内外迭代法

引用两种加速计算PageRank的算法,分别为内外迭代法和两步分裂迭代算法.从这两种方法中,得到多步幂法修正的内外迭代方法.首先,详细介绍了算法实施过程.然后,对此算法的收敛性进行证明,并且将此算法的谱半径与两步分裂迭代算法的谱半径进行比较.最后,数值试验说明该算法的计算速度比两步分裂迭代法要快.     

分数阶常微分方程的改进精细积分法

基于Mittag-Leffler函数的定义式,构造Mittag-Leffler矩阵函数的精细迭代计算格式.与常规指数函数的迭代格式相比,迭代递推中多了修正项,其表达式与分数阶导数的阶次有关.对于以Caputo分数导数定义的动力学分数阶常微分方程,使用基于Mittag-Leffler函数的精细积分法可计算方程解在各时间段端点对应函数值.算例表明了所提计算方法的有效性,其精度可由所增加修正项的阶次控制.     

求解非线性方程根四阶收敛的迭代格式

提出了求解非线性方程根新的四阶收敛迭代方法,新方法每次迭代只需要两次函数计算,一次一阶导数值计算,效能指数达到1.587.通过几个数值算例来解释该方法的有效性.     

Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs

A general class of multi-step iterative methods for finding approximate real or complex solutions of nonlinear systems is presented. The well-known technique of undetermined coefficients is used to construct the first method of the class while the higher order schemes will be attained by a frozen Jacobian. The point of attraction theory will be taken into account to prove the convergence behavior of the main proposed iterative method. Then, it will be observed that an m-step method converges with 2m-order. A discussion of the computational efficiency index alongside numerical comparisons with the existing methods will be given. Finally, we illustrate the application of the new schemes in solving nonlinear partial differential equations.     

Derivative free iterative methods with memory of arbitrary high convergence order

In this article we present three derivative free iterative methods with memory to solve nonlinear equations. With the process developed, we can obtain n-step derivative free iterative methods with memory of arbitrary high order. Numerical examples are provided to show that the new methods have an equal or superior performance, on smooth and nonsmooth equations, compared to classical iterative methods as Steffensen’s and Newton’s methods and other derivative free methods with and without memory with high order of convergence.     

Nonmonotone derivative-free methods for nonlinear equations

In this paper we study nonmonotone globalization techniques, in connection with iterative derivative-free methods for solving a system of nonlinear equations in several variables. First we define and analyze a class of nonmonotone derivative-free linesearch techniques for unconstrained minimization of differentiable functions. Then we introduce a globalization scheme, which combines nonmonotone watchdog rules and nonmonotone linesearches, and we study the application of this scheme to some recent extensions of the Barzilai–Borwein gradient method and to hybrid stabilization algorithms employing linesearches along coordinate directions. Numerical results on a set of standard test problems show that the proposed techniques can be of value in the solution of large-dimensional systems of equations.     

关于非定常不可压Navier-Stokes方程的时间高精度隐式差分方法

The incompressible Navier-Stokes equations,upon spatial discretization,become a system of differential algebraic equations,formally of index2.But due to the special forms of the discrete gradient and disrete divergence,its index can be regarded as 1.Thus,in this paper,a systematic approach following the ODE theory and methods is presented for the construction of high-order time-accurate implicit schemes for the incompressible Navier-Stokes equations,with projection methods for efficiency of numerical solution.The 3rd order 3-step BDF with componentconsistent pressure-correction projection method is a first attempt in this direction;the related iterative solution of the auxiliary velocyty,the boundary conditions and the stability of the algorithm are discussed.Results of numerical tests on the incompressible Navier-Stokes equations with an exact solution are presented,confirming the accureacy,stability and component-consistency of the proposed method.     

On a simultaneous method of Newton-Weierstrass’ type for finding all zeros of a polynomial

Combining a suitable two-point iterative method for solving nonlinear equations and Weierstrass’ correction, a new iterative method for simultaneous finding all zeros of a polynomial is derived. It is proved that the proposed method possesses a cubic convergence locally. Numerical examples demonstrate a good convergence behavior of this method in a global sense. It is shown that its computational efficiency is higher than the existing derivative-free methods.     

A globally convergent derivative-free method for solving large-scale nonlinear monotone equations

In this paper, we propose two derivative-free iterative methods for solving nonlinear monotone equations, which combines two modified HS methods with the projection method in Solodov and Svaiter (1998) [5]. The proposed methods can be applied to solve nonsmooth equations. They are suitable to large-scale equations due to their lower storage requirement. Under mild conditions, we show that the proposed methods are globally convergent. The reported numerical results show that the methods are efficient.     

Weak order stochastic Runge–Kutta methods for commutative stochastic differential equations

A new explicit stochastic Runge–Kutta scheme of weak order 2 is proposed under a commutativity condition, which is derivative-free and which attains order 4 for ordinary differential equations. The weak order conditions are derived by utilizing multi-colored rooted tree analysis and a solution is found in a transparent way. The scheme is compared with other derivative-free and weak second order schemes in numerical experiments.     

Conjugate gradient path method without line search technique for derivative-free unconstrained optimization

In this paper, we propose conjugate gradient path method for solving derivative-free unconstrained optimization. The iterative direction is obtained by constructing and solving quadratic interpolation model of the objective function with conjugate gradient methods. The global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, the numerical results are reported to show the effectiveness of the proposed algorithm.     

A class of generalized Newton iterative schemes

We study an N-step iterative scheme which generalizes several Newton-type schemes that have appeared in the literature. We show that, under generalized Zabrejko-Nguen conditions, the iterative scheme converges whenever 1 ≤ N ≤ ∞. This proves in a unified context the convergence of an infinite number of iterative schemes which include as special cases the classical Newton scheme, the classical chord scheme, and the generalized Newton scheme.     

Introducing memory to a family of multi-step multidimensional iterative methods with weight function

In this paper, we construct a derivative-free multi-step iterative scheme based on Steffensen’s method. To avoid excessively increasing the number of functional evaluations and, at the same time, to increase the order of convergence, we freeze the divided differences used from the second step and use a weight function on already evaluated operators. Therefore, we define a family of multi-step methods with convergence order $2m$, where $m$ is the number of steps, free of derivatives, with several parameters and with dynamic behaviour, in some cases, similar to Steffensen’s method. In addition, we study how to increase the convergence order of the defined family by introducing memory in two different ways: using the usual divided differences and the Kurchatov divided differences. We perform some numerical experiments to see the behaviour of the proposed family and suggest different weight functions to visualize with dynamical planes in some cases the dynamical behaviour.