关于Sturm定理的一点注记

经典的S turm定理用于判定多项式在给定区间上不同的实根个数,但是并不能刻画重根的情况.在这里定义了推广的S turm序列,将S turm定理进行一定地延拓,给出区间上多项式的所有实根均是偶重根或奇重根的充要条件.作为应用,讨论了多项式正(负)半定的判定问题.     

关于多项式与多项式函数相等的注记

给出了域上两个多项式作为多项式相等与作为多项式函数恒等的充要条件,引入有限域上约化多项式的概念,给出了有限域上多项式函数重根的定义及判定法则.     

计算结式的一种方法

我们知道,结式在代数中有着许多重要应用。利用结式能有效地解决两个一元多项式以及两个二元多项式的公共零点问题。我们还知道,判别式在多项式理论中占有重要的地位。根据判别式不但可以判定一个多项式是否有重根,而且还可以根据判别式的符号判定实系数多项式的根的情况。而判別式恰与结式有密切联系,前者往往通过后者进行计算。有关结式的计算,在一般高等代数教程中大致有以下两种方法,其一是行列式法(见本文(3)),共二是公式法(见本文公式(5))。本文给出另一种计算结式的方法。这种方法在计算结式时只须对所给两个一元多项式进行有限次带余除法(即辗转相除)就可以了。这种方法的优点在于:它既可以避免高阶行列式的复杂计算,又可以避开求多项式的所有根的困难。实践表明,就连普通的中学生也可以根据本文所给出的方法计算结式。     

单圈图的伴随多项式的极小根（英文）

引入伴随多项式是为了从补图的角度研究色多形式,图的伴随多项式的极小根可用于判定色等价图.β（G）表示图G的伴随多项式的极小根.n表示n个顶点的单圈图的集合.分别确定了具有max{β（G）｜G∈Ωn}和min{β（G）｜G∈Ωn}的所有单圈图.     

多项式判别矩阵的若干性质及其应用

具有文字系数的多项式f(x)，其判别矩阵是f与f′的Sylvester矩阵通过添加一行一列而得，已经知道，判别矩阵的偶数阶主子式的符号确定了f(x)的相异根(实根、复根)的数目，这里介绍如何将奇数阶与偶数阶主子式相结合用以判定该多项式的相异负根或正根的数目，并进一步判定其在区间上的实根数，本文还研究了与判别矩阵相关的一些实用性质，并应用这些性质给出了4次键合多项式不能正分解的一组简洁的充分必要条件。     

多项式矩阵根的研讨

引用源根研讨多项式的矩阵根，获得了一个矩阵M为多项式矩阵根的充要条件，并给出了求多项式矩阵根的简便方法。     

时滞系统的鲁棒性与边界定理

本文首先注意到滞后型拟多项式与中立型拟多项式零点分布的根本区别，并指出一些通常的事实：如特征根对时滞的连续依赖性，所有特征根位于开左半复平面与稳定性的等价关系等对于中立型系统都不成立，然后建立了中立型拟多项式胞鲁棒稳定性的边界定理．依此将可公度时滞中立型系统族的鲁棒稳定性化为相应拟多项式胞所有边的Ｈｕｒｗｉｔｚ稳定性与其中立型项构成的子胞所有边的Ｓｃｈｕｒ稳定性的判定．最后给出了检验中立型胞鲁棒稳定性的一个有效的图解法．     

Tarski模型外的一类机器可判定问题

利用对称多项式的降维方法和证明代数不等式的胞腔分解方法,给出了一个实用的算法, 用于判定一类变元个数也是变量的多项式正性命题.这是一类在Tarski模型外的机器可判定问题.在Maple平台上,根据该算法设计的程序nprove,可以快速实现判定目标.     

一元复系数多项式实根的判定

对于一元实系数多项式的实根问题,运用斯图姆(Sturm)方法不仅可以确定其实根的个数以及正负根的个数,而且对于任意给定的区间(a,b)可以确定这个多项式在此区间内实根的个数,但是对于一元复系数多项式呢?本文给出一般方法把一元复系数多项式的实根问题归     

一元多项式整根的判定

由高中代数第三册第一章的内容可知,若整系数多项式f(x)=a厂+a,:一l了一’+…+alx+吻有因式二一冬(其中p I’定是首项系数a,,的约数,q是互质的整数)那么P ?一定是末项系数内的约数.当户二1时,因式即成为x一q.为了判定x一q是否为f(x)的因式,对于q的可能值要经过检验,这是够麻烦的。下面的整根判定定理可以帮你减轻部分劳动量,特别是在判断当a,=1时f(x)有无有理因式方面有独到的功效. 盆根判定定理:一个整系数的多项式f(x),若f(o)和f(l)均为奇数,则当x不管为任何整数时,f(x)手0,(即多项式f(x)无整根). 证明:设f(x)二a,了+a,一;广一’+…+。x…     

用二分法求解一元实系数多项式方程的全部实根

运用二分法,结合实系数多项式零点的界定理及Sturm定理,给出了一个求解一元实系数多项式方程全部实根的实用数值方法.     

An approach to solving nonlinear algebraic systems. 2

New methods of solving nonlinear algebraic systems in two variables are suggested, which make it possible to find all zero-dimensional roots without knowing initial approximations. The first method reduces the solution of nonlinear algebraic systems to eigenvalue problems for a polynomial matrix pencil. The second method is based on the rank factorization of a two-parameter polynomial matrix, allowing, us to compute the GCD of a set of polynomials and all zero-dimensional roots of the GCD. Bibliography: 10 titles. Translated by V. N. Kublanovskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 71–96     

An effective implementation of a modified Laguerre method for the roots of a polynomial

Numerical Algorithms - Two common strategies for computing all roots of a polynomial with Laguerre’s method are explicit deflation and Maehly’s procedure. The former is only a...     

The computation of multiple roots of a polynomial

This paper considers structured matrix methods for the calculation of the theoretically exact roots of a polynomial whose coefficients are corrupted by noise, and whose exact form contains multiple roots. The addition of noise to the exact coefficients causes the multiple roots of the exact form of the polynomial to break up into simple roots, but the algorithms presented in this paper preserve the multiplicities of the roots. In particular, even though the given polynomial is corrupted by noise, and all computations are performed on these inexact coefficients, the algorithms ‘sew’ together the simple roots that originate from the same multiple root, thereby preserving the multiplicities of the roots of the theoretically exact form of the polynomial. The algorithms described in this paper do not require that the noise level imposed on the coefficients be known, and all parameters are calculated from the given inexact coefficients. Examples that demonstrate the theory are presented.     

求复数根的路斯法

<正> 我们要研究的问题是求实系数代数方程的根.为了解决这个问题,首先应当求出根的近似值.求出充分好的近似根后,刚已有多种有效的方法使近似根逐步地精确化.设该代数方程仅有实根,则求近似根的问题并不困难.设该代数方程有虚根(非实数的复根),用路斯法可以逐步地定出该虚根的实部的近似值.如何求出与该实部近似值对应的虚根的虛部近似值,至今还没有很简单的方法.在本文内作者将证明在用路期法定出虚根的实部的近似值后,就可以从路斯列表计算法的表格上的已算出的数字毫不费力地算出该虚根的虚部的近似值.即使同一实部对应着两对或更多对虚根吋,定出这些虚根的各部也没虚有困难.当虚根的虚部很小时及     

On the number of roots of self-inversive polynomials on the complex unit circle

We present a sufficient condition for a self-inversive polynomial to have a fixed number of roots on the complex unit circle. We also prove that these roots are simple when that condition is satisfied. This generalizes the condition found by Lakatos and Losonczi for all the roots of a self-inversive polynomial to lie on the complex unit circle.     

一种求多项式方程根的参数并行加速迭代法

推广了一种在无重根情况下,利用Newton类迭代法对同时求多项式零点的加速的迭代法.讨论了该方法的收敛性和收敛阶;最后给出数值算例表明:计算收敛阶和定理结论是一致的,且本算法具有较大的收敛范围.     

求解多项式重根时修正的Ehrlich法之改进

1引言次数大于4的多项式的根已没有一般的公式解法,但多项式求根有很多应用背景,因此有不少文献讨论多项式根的迭代解法,如文献[1-9]。对多项式f(x)=sum from j=0 to na_jx~j(a_n=1),(1.1)若其根全是单根,则已有一些收敛快效率高的迭代解法,如著名的Newton法,Durand- Kerner算法。特别是Ehrlich L W在文[1]中提出的同时决定多项式(1.1)的全部单根     

Quadratically Convergent Method for Simultaneously Approaching the Roots of Polynomial Solutions of a Class of Differential Equations: Application to Orthogonal Polynomials

This paper investigates the application of the method introduced by L. Pasquini (1989) for simultaneously approaching the zeros of polynomial solutions to a class of second-order linear homogeneous ordinary differential equations with polynomial coefficients to a particular case in which these polynomial solutions have zeros symmetrically arranged with respect to the origin. The method is based on a family of nonlinear equations which is associated with a given class of differential equations. The roots of the nonlinear equations are related to the roots of the polynomial solutions of differential equations considered. Newton's method is applied to find the roots of these nonlinear equations. In (Pasquini, 1994) the nonsingularity of the roots of these nonlinear equations is studied. In this paper, following the lines in (Pasquini, 1994), the nonsingularity of the roots of these nonlinear equations is studied. More favourable results than the ones in (Pasquini, 1994) are proven in the particular case of polynomial solutions with symmetrical zeros. The method is applied to approximate the roots of Hermite–Sobolev type polynomials and Freud polynomials. A lower bound for the smallest positive root of Hermite–Sobolev type polynomials is given via the nonlinear equation. The quadratic convergence of the method is proven. A comparison with a classical method that uses the Jacobi matrices is carried out. We show that the algorithm derived by the proposed method is sometimes preferable to the classical QR type algorithms for computing the eigenvalues of the Jacobi matrices even if these matrices are real and symmetric.     

Polynomial solutions of q-Heun equation and ultradiscrete limit

We study polynomial-type solutions of the q-Heun equation, which is related with quasi-exact solvability. The condition that the q-Heun equation has a non-zero polynomial-type solution is described by the roots of the spectral polynomial, whose variable is the accessory parameter E. We obtain sufficient conditions that the roots of the spectral polynomial are all real and distinct. We consider the ultradiscrete limit to clarify the roots of the spectral polynomial and the zeros of the polynomial-type solution of the q-Heun equation.