【作者】 李喆；

【导师】 张树功；

【作者基本信息】 吉林大学 ， 计算数学， 2011， 博士

【摘要】 为了统一研究多项式插值问题,人们提出了理想插值的概念.多项式插值问题构成一个理想插值当且仅当其插值条件由一簇插值节点,以及每个节点上一组由有限维微分闭多项式子空间所定义的微商条件所确定.每个理想插值都可以诱导一个理想投影算子,理想投影算子是多项式空间到自身的线性幂零算子,其核恰为一理想.目前理想插值问题的理论研究主要包括两个方面的内容：一方面是探讨多元理想投影算子与多元Hermite投影算子之间的关系;另一方面是研究理想投影算子的误差公式.在具体实际中,给定满足理想插值的一组插值条件,可以确定相应的各种形式的插值基.求插值基的方法可以是精确的,也可以是近似的.求近似插值基的问题就涉及到近似消逝理想计算.本文利用代数几何和数值计算工具来研究理想插值中的若干问题.主要工作如下：第二章研究具有唯一的相伴单项商环基的点集.具有唯一的相伴单项商环基的点集是一类特殊点集,其消逝理想具有唯一的单项商环基.利用代数几何工具,给出了具有唯一的单项商环基的零维理想的判别准则,从而得到了具有唯一的相伴单项商环基的点集的代数特征.设(?)i为关于变元xi的消去序,则理想I(?)F[x]具有唯一的单项商环基当且仅当对每个i=1,...,d,理想I相应于单项序-(?)i的Grobner escalier都相等,其中F[x]：=F[x1,...,xd]表示特征为零的数域F上的d元多项式环.进一步,结合Cartesian点集的判别方法,证明了Cartesian点集具有唯一的相伴单项商环基,并揭示了Cartesian点集与具有唯一的相伴单项商环基的点集之间的关系.在二元情形中,Cartesian点集与具有唯一的相伴单项商环基的点集等价.对于三元及三元以上情形,都存在非Cartesian而具有唯一的相伴单项商环基的点集.第三章给出一类具有“好”误差公式的理想投影算子.一元理想投影算子的误差公式可以表示为在节点处取零值的多项式乘以一个函数的形式,即f-Pf=C(Dnf)h,其中f为某一多项式,P为理想投影算子,h为理想投影算子核的理想基,C为多项式空间到自身的线性映射,D为微分算符.因此,人们期望多元理想投影算子也具有与一元情形结构相同的误差公式,即存在齐次多项式Hj和线性映射Cj使得误差公式可以写成∑jm=1Cj(Hj(D)f)hj,其中Hj(D)为微分算子,且满足条件Hj(D)hk=δj,k,{H1,...,hm}为理想投影算子核的理想基de Boor称具有上述形式的误差公式为理想投影算子“好”误差公式.我们考虑一类特殊的理想投影算子,其核空间为具有唯一的单项商环基的零维理想,像空间由核的唯一的单项商环基张成.利用代数几何工具,证明了此类理想投影算子具有“好”误差公式,并证明了此类理想投影算子的像空间为满足插值条件的极小次数插值空间.第四章讨论一类Hermite投影算子的离散化.在一元情形中,所有的理想投影算子都是Lagrange投影算子的逐点极限形式.这促使de Boor定义Hermite投影算子为一歹Lagrange投影算子的逐点极限形式,并猜想在多元情形中所有的复理想投影算子都是Hermite投影算子.但Shekhtman已证明了该猜想仅在二元情形成立.针对该猜想,Shekhtman提出了更深层次且有实际意义的两个问题：对于给定的理想投影算子,如何给出判定其是否为Hermite投影算子的可行性方法;对于给定的Hermite投影算子,如何将其离散为一歹Lagrange投影算子.我们就两类由不同特定结构多项式张成的微分闭子空间进行了讨论,证明了这两类微分闭子空间相应的理想投影算子为Hermite投影算子,并给出逐点逼近其的Lagrange投影算子列.设ξ(1),...,ξ(μ),ξ(μ+1),...,ξ(μ+v)∈Fd为互异节点.对每个k=1,...,μ,设A(k)(?)Nd为lower集,设d个单位向量ρ1(k),...,ρd(k)∈Fd线性无关,记ρ(k)=(ρ1(k)+,...,ρd(k)).对每个l=1,...,v,引入以下记号.设a(l)=(a0(l),a1(l),...,an(l)(l))为各个分量都为正整数的n(l)+1-元组,其中n(L)≥1且a0(l)=1,a1(l)>...>an(l)(l)≥2.设ci(l)=(ci,0,(l),ci,1(l),...,ci,n(l)(l))∈Fn(l)+1,i=1,...,d,其中c1,0,(l),c2,0(l),...,cd,0(l)不全为零.定义映射其中γi=(γi,0,...,γi,n(l))∈Nn(l)+1,i=1,...,d.设利用矩阵计算工具,证明了当h趋于0时,插值条件集合为的理想投影算子是插值条件集合为的Lagrange投影算子的逐点极限形式,其中δξ表示在点ξ∈Fd处的赋值泛函,对于第五章给出一类Lagrange投影算子列逐点收敛的充分条件.在一元情形中,对于Lagrange投影算子而言,当其中一些插值节点重合时,Lagrange投影算子一定逐点收敛到Hermite投影算子.然而,在多元情形中,这个结论并不总是成立.我们考虑给定插值条件集合为{δξ(k)+ha：α∈A(k),k=1,...,μ}的Lagrange投影算子列Ph,0<|h|<η,其中η由算法5.2.1所确定,证明了若Ph的像空间由核相应于字典序的Grobner escalier张成,则当h趋于0时,Ph,0<|h|<η,逐点收敛到插值条件集合为{δξ(k)οDα：α∈A(k),k=1,...,μ}的Hermite投影算子,其中第六章提出了基于约束总体最小二乘的近似消逝理想算法和低次超曲面拟合算法.众所周知,将点集中的点的坐标做一个微小摄动,点集消逝理想的Grobner基结构可能会发生本质的变化.然而,在工程计算中,点集往往是在实际应用中获取的,那么点的坐标不可避免的存在误差.此时,需要用一个多项式集合来刻画实验点集的近似几何分布,即计算实验点集的近似消逝理想.给定实验点集(?)ε,基于约束总体最小二乘的近似消逝理想算法输出序理想(?)和多项式集合(?).当(?)中单项的个数等于(?)ε的基数时,(?)即为(?)ε的近似消逝理想基.该算法充分考虑赋值向量的扰动之间的内在联系,因此在关注向量的数值相关性方面,要优于目前其它同类算法.给定实验点集(?)ε,基于约束总体最小二乘的低次超曲面算法输出一个多项式g和误差向量e.在忽略二阶截断误差的前提下,(?)所对应的曲面为通过实验点集(?)ε的一个容许点集的低次代数曲面,(?)(e)为相应的容许点集.更多还原

【Abstract】 The definition of ideal interpolation was put forward to unify the study of polynomial interpolation. The problem of polynomial interpolation is ideal interpolation if and only if its interpolation conditions consist of a family of in-terpolation points, and for each point, a family of derivative conditions induced by a finite-dimensional D-invariant polynomial subspace. Ideal interpolation is defined by an ideal projector. An ideal projector is a linear idempotent operator on the polynomial space whose kernel is an ideal. Theoretical study of ideal interpolation mainly includes two aspects:Discussing the relation be-tween ideal projectors and Hermite projectors; Investigating the error formu-las for ideal projectors. In the actual application, given a set of interpolation conditions satisfying ideal interpolation, one can determine various forms of interpolation bases. The method of computing interpolation bases can be not only exact, but also approximate. The computation of approximate interpo-lation bases involves the computation of approximate vanishing ideals. With the aid of algebraic geometry and numerical computation, we discuss several problems on the study of ideal interpolation. The main results of this paper are as follows:Chapter 2 analyzes the point sets with unique associated monomial quo-tient bases. The point sets with unique associated monomial quotient bases belong to a special class of point sets, whose vanishing ideals have unique monomial quotient bases. Using algebraic geometry, we give a useful criterion for a zero-dimensional ideal with a unique monomial quotient basis. Let-?i be an elimination order for xi, then a zero-dimensional ideal I C F[x] has a unique monomial quotient basis if and only if the Grobner escaliers of I w.r.t. -<i, i=1,...,d, are identical, where F[x]:=F[x1,...,xd]is the polynomial ring in d variables over the field F, and F is a field of characteristic zero. By the criteria for Cartesian sets and the zero-dimensional ideals with unique monomial quotient bases, we verify that Cartesian sets have the unique mono-mial quotient bases, and reveal the relationship between Cartesian sets and the point sets with unique associated monomial quotient bases. A finite set of distinct pointsΞC F2 is Cartesian if and only if it has a unique associated monomial quotient basis. For every d≤3, there exists a non-Cartesian point set in Fd which has a unique associated monomial basis.Chapter 3 presents a class of ideal projectors with "good" error formu-las. The error formulas for univariate ideal projectors can be expressed by f-Pf = C(Dnf)h, where f is a polynomial, P is an ideal projector, h is an ideal basis for kerP, C is a linear operator on the polynomial space, and D is the differentiation symbol. Therefore, people expect that multivariate ideal projectors have the error formulas which are the same as univariate ideal pro-jectors, i.e., there exist homogeneous polynomials Hj and linear operators Cj, j=1,...,m, such that f-Pf=∑jm=1Cj{Hj(D)f)hj, where Hj(D)hk=δj,k, and{h1,..., hm} is an ideal basis for kerP. de Boor called such error formulas as "good" error formulas. We consider a special class of ideal projectors, whose kernel are the zero-dimensional ideals with unique monomial quotient bases, whose ranges are spanned by the unique monomial quotient bases. Using al-gebraic geometry, we prove that for this special class of ideal projectors, there exist "good" error formulas. By the property of the interpolation conditions matched by this class of ideal projectors, we show that the ranges of this class of ideal projectors are the minimal degree interpolation spaces with regard to their associated interpolation conditions.Chapter 4 discusses the discretization for a class of Hermite projectors. In the univariate case, every ideal projector is the pointwise limit of Lagrange pro-jectors. This prompted de Boor to define the Hermite projector as the point-wise limit of Lagrange projectors, and conjecture that multivariate complex ideal projectors are all Hermite projectors.However,Shekhtman have showed that this conjecture only holds for bivariate complex projectors.Furthermore, Shekhtman proposed two more useful problems.Given an ideal projector, how can we determine whether it is an Hermite proj ector?Given an Hermite projector,how can we discrete it as a sequence of Lagrange projectors? We focus on two classes of D-invariant polynomial subspaces,and constructively prove that every ideal projector with each D-invariant subspace belonging to either the first class or the second is the pointwise limit of Lagrange projec-tors.Letξ(1),…,,ξ(μ),∈(μ+1),…,ξ(μ+v)∈Fd be distinct points. For each k, k=1,…,μ,let A(k)c Nd be a lower set and p1(k),p2(k),…,pd(k)∈Fd be F-linearly independent unit vectors.Remark that p(k)=(p1(k’,p2(k),...,pd(k)). For each l,l=l,...,v,the following notation is introduced..Let a(t)=(a0(l),a1(l),…,an(l)(l))with n(l)≥1 be ann(l)+1-tuple of positive integers satisfying.Let ci(l)=(ci,0(l),ci,1(l),…,c,n(l)(l))∈Fn(l)+1,i=...,d,satisfy that c1,0,(l),c(2,0,(l),...,cd,0(l) are not all zero..Define the mapτ(l)as follows: whereτi=(τi,0,τi1,...,τi,n(l))∈Nn(l)+1,i=1,...,d..Let Using the tools of matrix computation, we show that the ideal projector with the set of interpolation conditionsis the pointwise limit of Lagrange projectors with the set of interpolation con-ditionswhereδξdenotes the evaluation functional at the pointξ,Chapter 5 provides a sufficient condition for the pointwise convergence of a class of Lagrange projectors. In the univariate case, when some of the interpolation points coalesce, the sequence of Lagrange projectors converges pointwise to an Hermite projector. In general, this does not hold true in two and more variables. We discuss the sequence of Lagrange projectors Ph,0<▏h▕<η, with the sets of interpolation conditionswhereηis computed by Algorithm 5.2.1. If the range of Ph is spanned by the Grobner escalier of ker Ph w.r.t. lexicographic order, then this sequence Ph, 0<▕h▕<η, converges pointwise to an Hermite projector P, as h tends to zero, where the set of interpolation conditions matched by P is Chapter 6 gives the algorithms of the approximate vanishing ideal and low-order hypersurface fitting based on constrained total least squares tech-nique. It is well known that when a point set does a small perturbation, the Grobner basis for the vanishing ideal of this point set may change greatly. IfΞis a set of empirical points, representing real-world measurements, then typi-cally the coordinates are known only imprecisely. Now, one wants to compute a set of polynomials to determine geometrical configurations simultaneously characterizing the setΞand all its equivalent point setsΞ. Given a set of empirical pointsΞε, the algorithm of the approximate vanishing ideal based on constrained total least squares technique outputs an order ideal O and a set of polynomials g. If# O=#Ξε, then g forms a basis for the approxi-mate vanishing ideal ofΞε. Since this algorithm pays sufficient attention to the relationship among the perturbations of the evaluation vectors, it gives a better performance than other similar algorithms in the numerical dependence. Given a set of empirical pointsΞε, the algorithm of the low-order hypersurface fitting based on constrained total least squares technique outputs a polyno-mial g and an error indeterminates e. g is the low-order polynomial which vanishes at an admissible perturbation point setΞ(e) ofΞε, up to an error term asymptotically equal toε2.更多还原