一种同时求多项式零点的加速迭代法

本文讨论了在无重根情况下,利用改进的Newton迭代法对一种同时求多项式零点的并行迭代法进行加速,得到了一种新的加速迭代法。首先证明了该方法是收敛的,并且理论证明出收敛阶至少是5阶;其次,分析了该方法的计算效率;最后通过实际的数值算例表明:计算收敛阶和定理结论是一致的,且本算法具有较高的计算效率。     

鞍点问题的等价模型及其求解

本文将鞍点问题转化为一个具有对称正定系数矩阵的等价模型.在同等条件下,将求解鞍点问题的SOR-like方法与等价模型的SOR方法进行了对比,发现等价模型效果更好.此外,我们还提出了一种新的修正Chebyshev加速迭代方法,它的参数是由优化模型而不是Chebyshev多项式产生,并讨论了修正的Chebyshev加速迭代方法的收敛性.最后,通过数值例子比较各种算法的收敛速度和迭代次数,验证了修正的Chebyshev加速迭代方法的收敛性优势.     

广义预条件对称–反对称分裂迭代法的收敛性分析?

为了高效地求解大型稀疏鞍点问题,在白中治等人提出的预条件对称–反对称分裂迭代法(PHSS)的基础上,本文通过引入新的待定参数对原有迭代算法进行加速的思想,提出了一种解鞍点问题的具有三个待定参数的广义预条件对称–反对称分裂迭代法(GPHSS),并在每一步迭代过程中采用直接法和内迭代相结合计算,给出了该算法收敛性的条件.理论上证明了算法的收敛性,数值算例表明算法是有效可行的.     

求解奇异问题加速迭代格式的构造

构造了一类求解奇异问题加速迭代格式，给出了收敛性定理及误差估计。     

分数阶电报方程的Chebyshev多项式数值解法研究

分数阶电报方程作为通信工程中的一类重要方程,在实际应用中往往很难求得解析解,因而对其进行数值求解就显得至关重要．为了求得分数阶电报方程的数值解,本文借助Chebyshev多项式函数构造相应的微分算子矩阵,并结合Tau方法将待求方程转化为非线性代数方程组,然后对该方程组进行数值离散求解,最后给出的数值算例也验证了该方法的可行性及有效性．     

三角域上双变量Chebyshev多项式及其与Bernstein基的转换

为了更好的解决三角域上的B6zier曲面在CAGD中的最佳一致逼近问题,构造出了三角域上的双变量Chebyshev正交多项式,研究了与单变量Chebyshev多项式相类似的性质,并且给出了三角域上双变量Chebyshev基和Bernstein基的相互转换矩阵。通过实例比较双变量Chebyshev多项式与双变量Bernstein多项式以及双变量Jacobi多项式的最小零偏差的大小,阐述了双变量Chebyshev多项式的最小零偏差性。     

求复多项式全部零点的Kunn－球形组合迭代法

本文结合应用球形迭代法的几何估计技巧与ｋｕｎｎ算法的整体收敛特性，提出了求复多项式全部单零点的一种组合迭代法。大量数值实验说明，该组合算法较之于ｋｕｎｎ算法有更高的计算效率和更小的计算复杂性，因而值得推荐。     

求多项式全部零点的快速Halley算法

在Halley圆盘迭代法的基础上,用圆盘算术构造了一种求多项式全部零点的快速Halley算法,并在与Halley迭代法相同的条件下建立了它的收敛性定理,该算法取得了七阶收敛速度。数值结果表明该算法是十分有效的。     

一类求解波动方程的加速Schwarz波形松弛方法

波动方程在声学、电磁学和流体动力学等领域上有着广泛的应用.本文针对波动方程,研究了一类新的Schwarz波形松弛方法.经典Schwarz波形松弛方法是一种迭代方法,在求解波动方程时,特别是当子区域间的重叠量特别小的情形下,迭代次数往往较多,计算量较大.而本文构造的加速Schwarz波形松弛方法,即AitkenSchwarz波形松弛方法与Steffensen Schwarz波形松弛方法,是一种直接方法,它通过构造子区域边界信息的映射矩阵,很大程度地提升了计算性能.文中分别分析了这两种方法的收敛性,并且验证了新方法对于波动方程的可行性.数值算例证实了方法的有效性.     

基于自适应多项式拟合的涡流测量校正方法

针对管状基体在常规误差校正方法中往往采用的高斯函数、高阶多项式等拟合方法复杂化的问题,分析曲面曲率半径对涡流信号的影响,从涡流测量值与真实值的关系角度出发,提出基于自适应参数的二阶多项式拟合的校正方法,根据多项式参数与曲率半径的自适应关系,通过自适应参数求解,系统性地建立多项式参数与曲率半径的表达式.通过实验进行可行性...     

A New GSOR Method for Generalised Saddle Point Problems

A novel generalised successive overrelaxation (GSOR) method for solving generalised saddle point problems is proposed, based on splitting the coefficient matrix. The proposed method is shown to converge under suitable restrictions on the iteration parameters, and we present some illustrative numerical results.     

A Block Diagonal Preconditioner for Generalised Saddle Point Problems

A lopsided alternating direction iteration (LADI) method and an induced block diagonal preconditioner for solving block two-by-two generalised saddle point problems are presented. The convergence of the LADI method is analysed, and the block diagonal preconditioner can accelerate the convergence rates of Krylov subspace iteration methods such as GMRES. Our new preconditioned method only requires a solver for two linear equation sub-systems with symmetric and positive definite coefficient matrices. Numerical experiments show that the GMRES with the new preconditioner is quite effective.     

求解大型稀疏线性方程组的不完全SAOR预条件共轭梯度法

预条件共轭梯度法是求解大型稀疏线性方程组的有效方法之一,SSOR预条件方法是基于矩阵分裂的较有效的预条件共轭梯度法。通过矩阵分裂,本文讨论不完全SAOR预条件方法,研究此方法的预条件因子及系数矩阵的预条件数,并证明了此方法的预条件数小于SSOR预条件方法的预条件数。最后通过求解离散化波松(Poisson)方程组表明了该方法的有效性。     

大规模点云选择及精简

点云选择与精简是三维扫描系统中应对背景数据、冗余采样、分布不均匀等问题的必要后处理步骤.针对定制低成本三维扫描系统的需求,传统方法仍有很多局限性.这是由于研究领域未提供支持套索UI接口的点云选择算法;传统点云精简方法侧重曲率自适应分布,无法保证平坦区域的均衡分布.论文提出一种支持套索UI接口的点云选择算法,通过构建套索形状矩形覆盖与点云八叉树剔除大部分点在多边形内的判断;提出一种基于Poisson-disk采样的均衡分布的点云精简算法,并以采样点邻域球布尔交运算来定义曲面上的圆盘半径度量,具有保持尖锐边特征及边界的性质.实验结果表明,论文方法能够较好满足低成本三维扫描系统中点云删减处理的需求.     

The Use of Bivariate Polynomial Factorization Algorithms in Two-dimensional Phase Problems

In this paper is demonstrated the possibility of using a bivariate factorization algorithm in phase retrieval when a finite degree polynomial model is adopted to describe the (at least twice over-sampled) intensity function.     

Iterative Algorithms for the Solution of Quasistatic Unilateral Problems of Thermal Friction

We present the differential and variational statements of unilateral quasistatic problems of thermal friction for anisotropic thermoelastic bodies within the framework of the generalized model of linear dissipative mechanism of sliding in the case of imperfect thermal contact in the region of actual interaction. It is shown that these statements are equivalent for sufficiently smooth solutions. We also construct an alternative iterative algorithm for the solution of the problem, obtain estimates of the rate of its convergence similar to those established earlier for the constructive algorithm, and determine sufficient conditions for the convergence of both these algorithms in the corresponding norms to the solution of the analyzed problem (in particular, to the unique solution).     

Algorithms for Computing Sparse Shifts for Multivariate Polynomials

In this paper, we investigate the problem of finding t-sparse shifts for multivariate polynomials. Given a polynomial f∈ℱ[x ₁, x ₂, …, x _n ] of degree d, and a positive integer t, we consider the problem of representing f(x) as a ?-linear combination of the power products of u _i where u _i = x _i −b _i for some b _i ∈?, an extension of ℱ, for i = 1, …, n, i.e., f = ∑ _j F _j u ^αj , in which at most t of the F _j are non-zero. We provide sufficient conditions for uniqueness of sparse shifts for multivariate polynomials, prove tight bounds on the degree of the polynomial being interpolated in terms of the sparsity bound t and a bound on the size of the coefficients of the polynomial in the standard representation, and describe two new efficient algorithms for computing sparse shifts for a multivariate polynomial. Received: January 30, 1996; revised version: January 15, 2000     

近似锥-次类凸集值优化问题严有效解的广义鞍点刻画

本文研究集值优化问题严有效解的广义鞍点刻画问题。利用广义鞍点的性质和凸集分离定理,得到了广义鞍点的一个集分离性质,并且在近似锥-次类凸假设下建立了集值优化问题严有效解为广义鞍点的充分条件和必要条件。     

Fast Algorithms for Zero-Dimensional Polynomial Systems using Duality

Many questions concerning a zero-dimensional polynomial system can be reduced to linear algebra operations in the quotient algebra A=k[X ₁ ,,X _n ]/, where is the ideal generated by the input system. Assuming that the multiplicative structure of the algebra A is (partly) known, we address the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A. We present new formulæ for the rational parametrizations, extending those of Rouillier, and algorithms extending ideas introduced by Shoup in the univariate case. Our approach is based on the A-module structure of the dual space . An important feature of our algorithms is that we do not require to be free and of rank 1. The complexity of our algorithms for computing the minimal polynomial and the rational parametrizations are O(2 ^{nD 5/2} ) and O(n2 ^{nD 5/2} ) respectively, where D is the dimension of A. For fixed n, this is better than algorithms based on linear algebra except when the complexity of the available matrix product has exponent less than 5/2.     

广义混合变分不等式的稳定迭代算法

本文提出并研究一类广义混合变分不等式的某些迭代算法的稳定性.由于在实际计算中不可避免的存在舍入误差,这种稳定性在实际应用中具有十分重要的意义.