Symbolic–numeric sparse interpolation of multivariate polynomials

We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact Ben-Or/Tiwari sparse interpolation algorithm and the classical Prony’s method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony’s method. We analyse the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications.     

Improved error bound for multivariate Chebyshev polynomial interpolation

ABSTRACT

Chebyshev interpolation is a highly effective, intensively studied method and enjoys excellent numerical properties which provides tremendous application potential in mathematical finance. The interpolation nodes are known beforehand, implementation is straightforward and the method is numerically stable. For efficiency, a sharp error bound is essential, in particular for high-dimensional applications. For tensorized Chebyshev interpolation, we present an error bound that improves existing results significantly.     

自适应图像缩放的切触有理混合插值算法

分析了切触混合有理插值的基本特性,同时研究了图像缩放时边缘区域产生模糊的原因,并考虑到数字图像实时传输的要求,给出了一类新的自适应图像插值算法。由于采用颜色分段的处理方法,根据不同类型的颜色区域,分别采用Salzer连分式和扩展的Newton多项式逼近Sinc函数。提出的算法尽可能保持了边缘像素原有特征。数值模拟与仿真显示该方法比传统方法有更清晰的边界。     

A new algorithm for sparse interpolation of multivariate polynomials

To reconstruct a black box multivariate sparse polynomial from its floating point evaluations, the existing algorithms need to know upper bounds for both the number of terms in the polynomial and the partial degree in each of the variables. Here we present a new technique, based on Rutishauser’s qd$qd$-algorithm, in which we overcome both drawbacks.     

On the decomposition of rational functions

Let f:=p/q∈K(x)$f:=p/q\in \mathbb{K}\left(x\right)$ be a rational function in one variable. By Lüroth’s theorem, the collection of intermediate fields K(f)?L?K(x)$\mathbb{K}\left(f\right)?\mathbb{L}?\mathbb{K}\left(x\right)$ is in bijection with inequivalent proper decompositions f=g°h$f=g°h$, with g,h∈K(x)$g,h\in \mathbb{K}\left(x\right)$ of degrees ≥2$\ge 2$. In [Alonso, Cesar, Gutierrez, Jaime, Recio, Tomas, 1995. A rational function decomposition algorithm by near-separated polynomials. J. Symbolic Comput. 19, 527–544] an algorithm is presented to calculate such a function decomposition. In this paper we describe a simplification of this algorithm, avoiding expensive solutions of linear equations. A MAGMA implementation shows the efficiency of our method. We also prove some indecomposability criteria for rational functions, which were motivated by computational experiments.     

函数分段有理三次Bézier插值

根据函数的几何性质,对函数进行适当分段。定义了函数的分段三角形凸包,提出了一种控制顶点和权因子的确定方案。详细地讨论了函数的分段有理三次Bézier插值算法,定义了一种便于计算的新型误差。插值函数保持了原始函数的重要几何性质,如单调性、凹凸性、G1连续性。最后以数值实验结果表明了算法的有效性和可行性,该算法提供了函数近似表示的一条有效途径。     

二元切触有理插值公式

降低切触有理插值的次数和解决切触有理插值函数的存在性是有理插值的一个重要问题。利用牛顿插值承袭性的思想和分段组合方法,构造出一种二元切触有理插值算法并推广到向量值有理插值,既解决了有理插值的存在性问题,又降低了切触有理插值函数的次数。相比于其他方法,算法的可行性是无条件的,有理插值函数次数较低,算法具有承袭性、计算量低、便于实际应用的特点。     

Modeling orientation fields of fingerprints with rational complex functions

In this paper, a novel model is proposed for the orientation field of fingerprints, which can be expressed as the argument of a rational complex function. It is suitable for all types of fingerprints. Experimental results show that it performs much better than the previous works.     

Analytical properties of bivariate fractal interpolation functions with vertical scaling factor functions

Based on the construction of bivariate fractal interpolation functions (FIFs), a class of FIFs with vertical scaling factor functions are presented and the analytical properties of smoothness and stability are proved.     

An algorithm for interpolation with positive rational functions on the imaginary axis

A procedure is shown to construct a rational transfer function that is real-valued and positive on the imaginary axis. The degree of the solution is lowered with respect to solutions based on existing algorithms. The results can be applied to problems such as the robust strictly positive real problem.     

Using multivariate resultants to find the implicit equation of a rational surface

Given a parametrization of a rational surface, the absence of base points is shown to be a necessary and sufficient condition for the auxiliary resultant to be a power of the implicit polynomial. The method of resultants also reveals other important properties of rational surface representations, including the coefficients of the implicit equation, the relationship between the implicit and parametric degrees, the degree of each coordinate variable of the implicit equation, and the number of correspondence of the parametrization.     

基于混合有理插值的图像渐变算法*

针对现有的图像渐变方法只考虑两个图像间渐变的情况,提出一种非线性的多幅图像间渐变的新方法,即一元混合有理插值方法。将多幅图像间相同位置的像素点建立对应关系,按照该关系建立一元混合有理插值函数,对插值函数进行重采样,得到一系列的渐变中间图像。实验表明,新算法在反映空间数据的分布特性、保证图像纹理特征方面均优于其他算法,具有计算精度高、适应性强、易于编程实现等优点,是一种较实用的算法。     

Global asymptotic stabilisation of rational dynamical systems based on solving BMI

In this paper, the global asymptotic stabiliser design of rational systems is studied in detail. To develop the idea, the state equations of the system are transformed to a new coordinate via polynomial transformation and the state feedback control law. This in turn is followed by the satisfaction of the linear growth condition (i.e. Lipschitz at zero). Based on a linear matrix inequality solution, the system in the new coordinate is globally asymptotically stabilised and then, leading to the global asymptotic stabilisation of the primary system. The polynomial transformation coefficients are derived by solving the bilinear matrix inequality problem. To confirm the capability of this method, three examples are highlighted.     

Pattern discrimination using ellipsoidally symmetric multivariate density functions

A brief review of ellipsoidally symmetric density functions is done. For the case of monotonic functional forms and distributions with common covariance matrices, a lower bound on the probability of correct classification is calculated in terms of either an incomplete beta or gamma integral, for a class of common functional forms. The lower bound is a monotonically increasing function of the Mahalanobis distance for all monotonic ellipsoidally symmetric forms.     

可变参数的有理分形插值曲线建模

为了有效地处理复杂真实现象中的不规则数据,提出一种利用有理分形插值进行分形曲线建模的方法.首先,基于传统的具有形状参数的有理样条,构造了一类具有函数尺度因子的有理迭代函数系统,并定义了有理分形插值曲线.然后,研究了有理分形曲线的一些重要性质,包括光滑性、稳定性以及收敛性.最后,估计了有理分形曲线计盒维数的上下界.提出的...     

Geometric Hermite interpolation with circular precision

We present several Hermite-type interpolation methods for rational cubics. In case the input data come from a circular arc, the rational cubic will reproduce it.     

基于Hermite插值的SVM研究

在传统SVM的分类求解算法中,由于严格凸的无约束最优化问题中单变量函数x+是不可微的,不能使用通常的最优化的算法进行求解。三次Hermite插值多项式光滑的支持向量机模型采用的是一种多项式光滑技术,用三次Hermite插值多项式代替单变量函数x+,将原来不可微的模型变为可微的模型,并且给出了三次Hermite插值多项式光滑化单变量函数x+的推导过程。使用UCI机器学习数据集中的数据,通过实验验证了该模型的有效性。     

The class of HDT0L sequences is closed with respect to rational functions

We show that the class of HDT0L sequences is closed with respect to total rational functions.     

Adaptive sampling applied to multivariate,multiple output rational interpolation models with application to microwave circuits

A fast and efficient adaptive sampling algorithm for multivariate, multiple output rational interpolation models is presented, which is based on convergents of Thiele type branched continued fractions. The multiple output interpolation model consists of a set of rational interpolants, and each interpolant models one of the output parameters. A single global error function is defined that incorporates all the output parameters, and it is used for the selection of the same set of support points for all the interpolants. The technique is evaluated on several passive microwave structures and compared to previously published results. © 2002 Wiley Periodicals, Inc. Int J RF and Microwave CAE 12: 332–340, 2002. Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mmce10032     

一种有理二次插值函数的凸性分析

在给定的插值数据条件下,利用一种带参数的分母为二次的有理二次插值方法,通过调整插值函数中的参数,给出了插值曲线的保凸方法和该方法得以实现的充分必要条件。这种条件是对参数的简单的线性的不等式约束,容易在计算机辅助设计中得到实际应用。