Isolated real zero of a real polynomial system under perturbation
Isolated real zero of a real polynomial system under perturbation
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DOI：
10.1145/2016567.2016591
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发表时间：
2011-07
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通讯作者：
Hiroshi Sekigawa;Kiyoshi Shirayanagi
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作者：
Hiroshi Sekigawa;Kiyoshi Shirayanagi
在求解经验多项式系统时，人们可能会关心零点的多重性和唯一性等性质是否在系数的扰动下保持不变。对于实多项式系统，我们考虑了这样一个问题，并利用Kantorovich定理[2]解决了这个问题。假设我们得到多项式fi∈R[x1，.。。，xn](1≤i≤n)和一个点x(0)∈Rn，使得(I)x(0)是多项式系统f1=···=fn=0或其逼近的孤立单实零点，且(Ii)f‘(x(0))非奇异，其中f=(f1，.。。，fn)：Rn→Rn.当x(0)是实零点的逼近时，我们进一步假设下面Kantorovich定理的一个特例保证牛顿迭代x(m+1)=x(M)−f‘(x(M))−1f(x(M))(m≥0)是明确定义的，并且在x(0)的邻域内收敛到多项式系统f1=……=fn=0的唯一简单实零点。此外，对于每个i(1≤i≤n)，我们给出多项式uij∈R[x1，.。。，xn](1≤j≤mi)，其中ui1，.。。，uimi在R上线性无关。每个uij是任意多项式(不一定是单项式)，并且它的次数可能大于fI的次数。在这种情况下，我们考虑以下问题。问题1计算一个尽可能大的正值，并构造一个x(0)的邻域U，使得对于每个实多项式f̃i=fi+∑mi j=1 ijuij(1≤i≤n)有|ij|≤，多项式系统f̃1=···=f̃n=0在U中有唯一的简单实零点。
When solving a system of empirical polynomials, one might be concerned whether properties of a zero such as multiplicity and uniqueness are preserved under perturbation of the coefficients. We consider such a problem for real polynomial systems and solve the problem by using the Kantorovich theorem [2]. Suppose that we are given polynomials fi ∈ R[x1, . . . , xn] (1 ≤ i ≤ n) and a point x(0) ∈ Rn such that (i) x(0) is an isolated simple real zero of the polynomial system f1 = · · · = fn = 0 or its approximation and (ii) f ′(x(0)) is nonsingular, where f = (f1, . . . , fn) : Rn → Rn. When x(0) is an approximation of a real zero, we further assume that a special case of the Kantorovich theorem below guarantees that the Newton iterates x(m+1) = x(m) − f ′(x(m))−1f(x(m)) (m ≥ 0) are well-defined and converge to a unique simple real zero of the polynomial system f1 = · · · = fn = 0 in a neighborhood of x(0). Furthermore, for each i (1 ≤ i ≤ n), we are given polynomials uij ∈ R[x1, . . . , xn] (1 ≤ j ≤ mi), where ui1, . . . , uimi are linearly independent over R. Each uij is an arbitrary polynomial (not necessarily a monomial) and its degree might be larger than that of fi. In this setting, we consider the following problem. Problem 1 Compute a positive value as large as possible and construct a neighborhood U of x(0) such that for each real polynomial f̃i = fi + ∑mi j=1 ijuij (1 ≤ i ≤ n) with | ij | ≤ , the polynomial system f̃1 = · · · = f̃n = 0 has a unique simple real zero in U .