基于插值的Bernstein多项式复合及其曲线曲面应用

在曲线曲面造型中,Bernstein多项式复合被广泛用于许多几何操作,因而具有重要的理论和实际意义.基于多项式插值和符号计算的思想,研究了Bernstein多项式函数复合问题, 并将其应用于曲线曲面的情形.与两种已有方法相比,新方法具有速度快、易于编程实现、占用存储空间少的特点,但数值精度低于基于广义de Casteljau算法的多项式复合结果.     

多项式循环程序的秩函数探测

秩函数法是循环程序终止性分析的主流方法.针对一类多分支多项式循环程序,这类程序的秩函数计算问题被证明可归结为单形上正定多项式的探测问题,从而便于利用线性规划工具Simplex去计算这类程序的秩函数.不同于现有基于柱形代数分解的量词消去算法,该方法能够在可接受的时间内计算更为复杂的多项式秩函数.     

基于SVM的多项式循环程序秩函数生成

程序终止性问题是自动程序验证领域中的一个研究热点.秩函数探测是进行终止性分析的主要方法.针对单重无条件分支的多项式循环程序,将其秩函数计算问题归结为二分类问题,从而可利用支持向量机（SVM）算法来计算程序的秩函数.与基于量词消去技术的秩函数计算方法不同,该方法能在可接受的时间范围内探测到更为复杂的秩函数.     

几种复合型数值方法的算法模拟与性能比较

浅水波问题的数值模拟一直是计算数学、计算流体力学的研究热点之一,采用低阶方法和高阶方法相复合的数值方法引起了人们的注意,并在水力学的数值模拟中取得了很大的成功。文中对三种复合型的数值方法,即Lax-Wendroff(LW)格式与Lax-Friedrichs(LF)格式的复合算法,Upwind格式与Lax-Wendroff(LW)格式的复合算法,WENO格式与LW格式的复合算法,进行了分析比较和改进,并就计算流体力学中的一维浅水波方程的两个算例分别做了数值对比试验,在解的光滑性、锐利性,计算速度等几个方面做了比较,模拟结果表明三种方法均能准确捕捉激波又不产生非物理震荡。     

几种复合型数值方法的算法模拟与性能比较

浅水波问题的数值模拟一直是计算数学、计算流体力学的研究热点之一,采用低阶方法和高阶方法相复合的数值方法引起了人们的注意,并在水力学的数值模拟中取得了很大的成功.文中对三种复合型的数值方法,即Lax-Wendroff(LW)格式与Lax-Friedrichs(LF)格式的复合算法,Upwind格式与Lax-Wendroff(LW) 格式的复合算法,WENO格式与LW格式的复合算法,进行了分析比较和改进,并就计算流体力学中的一维浅水波方程的两个算例分别做了数值对比试验,在解的光滑性、锐利性,计算速度等几个方面做了比较,模拟结果表明三种方法均能准确捕捉激波又不产生非物理震荡.     

一类非线性动态系统的非参数GFRF模型辨识

本文对一类用多项式描述的非线性动态系统提出使用ＧＦＲＦ模型类的非参数辨识算法，这种算法的显著特点是需要很小的计算量和存储空间，而且辨识精度较高，仿真结果表明这种非参数模型辨识算法是有效的，而且由辨识方法获得的模型一般具有很好的泛化能力，因而是一种具有重要应用前景的实用方法 。     

一种快速计算Zernike矩的改进q-递归算法

提出了一种快速计算Zernike矩的改进q-递归算法,该方法通过同时降低核函数中Zernike多项式和Fourier函数的计算复杂度以提高Zernike矩的计算效率。采用 q-递归法快速计算Zernike多项式以避免复杂的阶乘运算,再利用x轴、y轴、x=y和x=-y 4条直线将图像域分成8等分。计算Zernike矩时,仅计算其中1个区域的核函数的值,其他区域的值可以通过核函数关于4条直线的对称性得到。该方法不仅减少了核函数的存储空间,而且大大降低了Zernike矩的计算时间。试验结果表明,与现有方法相比,改进q-递归算法具有更好的性能。     

高分辨率数值计算研究

高分辨率计算是高置信度计算中一个极其重要而复杂的研究问题。相对传统的数值计算,高分辨率计算对计算机系统和应用程序(物理建模、参数、计算方法和算法等)提出了很高的要求。并行计算机的发展为大规模科学计算,特别是数值计算分辨率的提高提供了条件。同时,数值计算分辨率的提高也对计算机的计算能力、计算方法、物理建模和参数等提出了新的、更高的要求。本文以一个二维流体力学程序计算平面爆轰问题为例,研究在计算分辨率提高时初始起爆区域、时间步长、网格构造、人为粘性、计算机模拟误差、计算量增长等方面出现的问题,提出了相应的解决办法,提高了计算的精确度。     

指数函数多项式的实根分离算法

针对超越函数多项式的实根分离问题，提出了一种指数函数多项式的区间分离算法exRoot，将非多项式型实函数的实根分离问题转化为多项式正负性判定问题进而对其求解。首先，利用泰勒替换法构造目标函数的多项式区间套；然后，将指数函数的求根问题转化为多项式在区间内正负性的判定问题；最后，给出综合算法，并且试探性地应用于实特征值线性系统的可达性判定问题。所提算法在Maple中实现，输出的结果可读，且高效易行。区别于HSOLVER和数值计算方法fsolve，exRoot回避了直接讨论根的存在性问题，理论上具有终止性和完备性，且可达到任意精度，应用于最优化问题时可避免数值解带来的系统误差。     

Zernike矩在图像旋转与抗噪声中的应用

Zernike矩是基于称作Zernike多项式的正交化函数。尽管同几何矩和Legendre矩相比其计算更加复杂,但Zernike矩在图像的旋转和低的噪声敏感度方面是有较大的优越性。特别是本文采取了Zernike矩的一种快速算法,使计算量大大减少,并且采取了方-圆变换的方法,对矩形图像仍能起到同样的作用。     

基于密切多项式近似的多项式插值算法框架

多项式插值技术是近似理论中一种常见的近似方法,被广泛用于数值分析、信号处理等领域。但传统的多项式插值技术大多是基于数值分析与实验结果相结合得到的,没有统一的理论描述和规律性的解决方案。为此,根据密切多项式近似理论为图像的多项式插值算法提出一个统一的理论框架。密切多项式近似的理论框架包括采样点数目、密切阶数和导数近似规则三个部分,它既可以用于分析现有的多项式插值算法,也可以用于开发新的多项式插值算法。分析了主流多项式插值技术在密切多项式近似理论框架下的表现形式,并以四点二阶密切多项式插值算法为例详细描述了利用密切多项式插值的理论框架开发新的多项式插值算法的一般流程。理论分析和数值实验表明大多数主流插值算法都属于密切多项式插值算法,它们的处理效果与采样点数目、密切阶数和导数近似规则有紧密的关系。     

Calculating polynomial zeros on a local memory parallel computer

We consider the calculation, on a local memory parallel computer, of all the zeros of an n th degree polynomial P_n(x) which has real coefficients. We describe a generic parallel algorith, which approximates all the zeros simultaneously and we give three specific examples of this algorithm which have orders of convergence two, three and four. We report extensive numerical tests of the algorithms; the fourth order algorithm is not robust, with many failures to convergence, whereas the other two algorithms are reliable and display very respectable parallel speedups for higher degree polynomials.     

Parallel computation of determinants of matrices with polynomial entries

An algorithm for computing the determinant of a matrix whose entries are multivariate polynomials is presented. It is based on classical multivariate Lagrange polynomial interpolation, and it exploits the Kronecker product structure of the coefficient matrix of the linear system associated with the interpolation problem. From this approach, the parallelization of the algorithm arises naturally. The reduction of the intermediate expression swell is also a remarkable feature of the algorithm.     

基于多尺度边缘表示的图像增强快速算法

低对比度结构广泛存在于各种数字图像之中，研究如何通过后期处理增强数字图像的对比度是很有意义的。灰度图像对比度的高低总是与图像灰度梯度幅值的大小相联系，受这种思想的启发，提出了一种基于图像多尺度边缘表示的，利用对信号小波变换模极大值的拉伸和Hermite插值多项式实现的图像增强快速算法。此算法可以实现对噪声的抑制和对图像中不同尺度特征的增强。数值实验结果表明，该算法增强效果明显，运算速度快，是一种实用性较强的图像对比度增强算法。     

Approximate polynomial GCD over integers

Symbolic numeric algorithms for polynomials are very important, especially for practical computations since we have to operate with empirical polynomials having numerical errors on their coefficients. Recently, for those polynomials, a number of algorithms have been introduced, such as approximate univariate GCD and approximate multivariate factorization for example. However, for polynomials over integers having coefficients rounded from empirical data, changing their coefficients over reals does not remain them in the polynomial ring over integers; hence we need several approximate operations over integers. In this paper, we discuss computing a polynomial GCD of univariate or multivariate polynomials over integers approximately. Here, “approximately” means that we compute a polynomial GCD over integers by changing their coefficients slightly over integers so that the input polynomials still remain over integers.     

Approximating of conic sections by DP curves with endpoint interpolation

Conic sections have many applications in industrial design, however, they cannot be exactly represented in polynomial form. Hence approximating conic sections with polynomials is a challenging problem. In this paper, we use the monomial form of Delgado and Peña (DP) curves and present a matrix representation for them. Using the matrix form and the least squares method, we propose a simple and efficient algorithm for approximating conic sections by DP curves of arbitrary degree with endpoint interpolation. Finally, we test and compare the proposed algorithm on some numerical examples which validates and confirms efficiency of it.     

Robust identification and interpolation in H∞

We consider system identification in H_∞ in the framework proposed by Helmicki, Jacobson and Nett. An algorithm using the Jackson polynomials is proposed that achieves an exponential convergence rate for exponentially stable systems. It is shown that this, and similar identification algorithms, can be successfully combined with a model reduction procedure to produce low-order models. Connections with the Nevanlinna-Pick interpolation problem are explored, and an algorithm is given in which the identified model interpolates the given noisy data. Some numerical results are provided for illustration. Finally, the case of unbounded random noise is discussed and it is shown that one can still obtain convergence with probability 1 under natural assumptions.     

Matrix pencil methodologies for computing the greatest common divisor of polynomials: hybrid algorithms and their performance

The computation of the greatest common divisor (GCD) of several polynomials is a problem that emerges in many fields of applications. The GCD computation has a non-generic nature and thus its numerical computation is a hard problem. In this paper we examine the family of matrix pencil methods for GCD computation and investigate their performance as far as their complexity, error analysis and their effectiveness for evaluating approximate solutions. The relative merits of the various variants of such methods are examined for the different cases of sets of polynomials with varying number of elements and degree. The developed algorithms combine symbolical and numerical programming and this is what we define here as hybrid computations. The combination of numerical operations with symbolical programming can improve the nature of the methods and guarantees the stability of the algorithm. Furthermore, it emphasizes the significance of hybrid computations in complex problems such as the computation of GCD. All methods are tested thoroughly for several sets of polynomials and the results are presented in tables.     

On multivariate interpolation by generalized polynomials on subsets of grids

This note may be regarded as a complement to a paper of H. Werner [17] who has carried over Newton's classical interpolation formula to Hermite interpolation by algebraic polynomials of several real variables on certain subsets of grids. Here generalized polynomials of several real or complex variables are treated. Recursive procedures are presented showing that interpolation by generalized multivariate polynomials is performed nearly as simply as interpolation by algebraic polynomials. Having in general the same approximation power, generalized polynomials may be better adapted to special situations. In particular, the results of this note can be used for constructing nonpolynomial finite elements since in that case the interpolation points usually are rather regular subsystems of grids. Though the frame is more general than in [17] some of our proofs are simpler. As an alternative method to evaluate multivariate generalized interpolation polynomials for rectangular grids a Neville-Aitken algorithm is presented.