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.     

Improvement of Polynomial Methods for Reproducing Functional Dependences

A comparative analysis is made of polynomial methods of reproducing functions. Recommendations on reducing the errors and/or increasing the interpolation interval for Newton polynomials are proposed in a general form and also using the example of the sin function.     

两类问题的最佳逼近

本文研究了矩阵方程约束下的两类最佳逼近问题。利用矩阵的奇异值分解等技术,我们导出了这两类问题的解,并说明了现实中的许多问题是本文所研究问题的特例。     

Fast Algorithms for the Round Trip Location Problem

This paper presents several fast algorithms for the round trip location problem. I show how to simplify the method given in an article by Chan and Hearn in which only the rectilinear case is addressed and get an algorithm with improved complexity for a given accuracy. An exact solution algorithm (the complexity of which is unknown) that solved test data problems even faster is presented and generalized for Euclidean and ℓ_p distances. Computational results are also presented.     

求置换因子循环矩阵的极小多项式和逆的算法

本文引入了任意域上置换因子循环矩阵,利用多项式环的理想的Gr(?)bner基的算法给出了任意域上置换因子循环矩阵的极小多项式和公共极小多项式的算法,同时给出了这类矩阵逆矩阵的两种算法最后,利用Schur补给出了任意域上具有置换因子循环矩阵块的分块矩阵逆的一个算法,在有理数域或模素数剩余类域上,这一算法可由代数系统软件CoCoA4．0实现。     

Fast Algorithms for the Anisotropic LLT Model in Image Denoising

In this paper, we propose a new projection method for solving a general minimization problems with two $L^1$-regularization terms for image denoising. It is related to the split Bregman method, but it avoids solving PDEs in the iteration. We employ the fast iterative shrinkage-thresholding algorithm (FISTA) to speed up the proposed method to a convergence rate $O$($k^-$$^2$). We also show the convergence of the algorithms. Finally, we apply the methods to the anisotropic Lysaker, Lundervold and Tai (LLT) model and demonstrate their efficiency.     

L_p空间中非线性最佳逼近的广义强唯一性

本文讨论Lp（1＜P＜∞）中非线性最佳逼近的广义强唯一性·证明了定理：设G是Lp（1＜p＜∞）中闭集，如对有则G必为凸集．     

对称的广义中心对称矩阵逆特征问题的最佳逼近

在结构设计中,矩阵逼近问题通常用来校正刚度矩阵或质量矩阵,使得它们具有给定谱约束条件.本文基于逆特征值理论讨论了线性流形上的一类对称的广义中心对称矩阵逼近问题,给出了它们的最小二乘解的显示表达式及其最佳逼近,提供了一个数值方法并给出了数值例子.     

Moore-Penrose广义逆和最佳逼近算子的表达式

在空间较弱的几何假定下,给出了A^ 的存在唯一性、极小性,并建立了A^ 为线性算子的充要条件。还给出了一般Banauch空间中线性流形上的最佳逼近算子存在的充要条件,并借助于正规对偶映射得到了相应的最佳逼近算子的表达式。所得的结果推广和改进了他人的相应结果。     

A Survey of Lower Limb Rehabilitation Systems and Algorithms Based on Functional Electrical Stimulation

Functional electrical stimulation is a method of repairing a dysfunctional limbin a stroke patient by using low-intensity electrical stimulation. Currently, it is widelyused in smart medical treatment for limb rehabilitation in stroke patients. In this paper,the development of FES systems is sorted out and analyzed in a time order. Then, theprogress of functional electrical stimulation in the field of rehabilitation is reviewed indetails in two aspects, i.e., system development and algorithm progress. In the systemaspect, the development of the first FES control and stimulation system, the core of thelower limb-based neuroprosthesis system and the system based on brain-computerinterface are introduced. The algorithm optimization for control strategy is introduced inthe algorithm. Asynchronous stimulation to prolong the function time of the lower limbsand a method to improve the robustness of knee joint modeling using neural networks.Representative applications in each of these aspects have been investigated and analyzed.     

Loewner方程组极小范数最小二乘解的快速算法

本文给出了求以m×n阶Loewner矩阵为系数阵的线性方程组极小范数最小二乘解的快速算法。     

Lp[-1,1](1≤p＜∞)空间中复系数多项式的倒数逼近

讨论了Lp[-1,1](1≤p＜∞)空间函数的复系数多项式的倒数逼近问题,证明了如下结论:设f(x)∈Lp[-1,1](1≤p＜∞),则存在一个次数不超过3n的复系数多项式Q3n(x)及一个仅与p有关的常数Mp＞0使得 （f(x)-(1)/(Q3n(x))Lp[-1,1]Mpωφ(f,1/n)L2p）     

函数最佳平方逼近多项式是否唯一

本文对最佳一致逼近多项式与最佳平方逼近多项式相互比较,给出了相同的内积和基函数最佳平方逼近多项式是唯一,并且验证了勒让德展开部分和得到的最佳平方逼近多项式与Hn中的最佳平方逼近多项式是一致的.     

函数最佳平方逼近多项式是否唯一

本文对最佳一致逼近多项式与最佳平方逼近多项式相互比较,给出了相同的内积和基函数最佳平方逼近多项式是唯一,并且验证了勒让德展开部分和得到的最佳平方逼近多项式与Hn中的最佳平方逼近多项式是一致的。