The paper is concerned with the improvement of the rational representation theory for solving positive-dimensional polynomial systems. The authors simplify the expression of rational representation set proposed by Tan...The paper is concerned with the improvement of the rational representation theory for solving positive-dimensional polynomial systems. The authors simplify the expression of rational representation set proposed by Tan and Zhang(2010), obtain the simplified rational representation with less rational representation sets, and hence reduce the complexity for representing the variety of a positive-dimensional ideal. As an application, the authors compute a "nearly" parametric solution for the SHEPWM problem with a fixed number of switching angles.展开更多
Ideal interpolation is a generalization of the univariate Hermite interpolation. It is well known that every univariate Hermite interpolant is a pointwise limit of some Lagrange interpolants.However, a counterexample ...Ideal interpolation is a generalization of the univariate Hermite interpolation. It is well known that every univariate Hermite interpolant is a pointwise limit of some Lagrange interpolants.However, a counterexample provided by Shekhtman Boris shows that, for more than two variables,there exist ideal interpolants that are not the limit of any Lagrange interpolants. So it is natural to consider: Given an ideal interpolant, how to find a sequence of Lagrange interpolants(if any) that converge to it. The authors call this problem the discretization for ideal interpolation. This paper presents an algorithm to solve the discretization problem. If the algorithm returns "True", the authors get a set of pairwise distinct points such that the corresponding Lagrange interpolants converge to the given ideal interpolant.展开更多
This paper demonstrates the equivalence of two classes of D-invariant polynomial subspaces, i.e., these two classes of subspaces are different representations of the breadth-one D-invariant subspace. Moreover, the aut...This paper demonstrates the equivalence of two classes of D-invariant polynomial subspaces, i.e., these two classes of subspaces are different representations of the breadth-one D-invariant subspace. Moreover, the authors solve the discrete approximation problem in ideal interpolation for the breadth-one D-invariant subspace. Namely, the authors find the points, such that the limiting space of the evaluation functionals at these points is the functional space induced by the given D-invariant subspace, as the evaluation points all coalesce at one point.展开更多
In this paper,the authors first apply the Fitzpatrick algorithm to multivariate vectorvalued osculatory rational interpolation.Then based on the Fitzpatrick algorithm and the properties of an Hermite interpolation bas...In this paper,the authors first apply the Fitzpatrick algorithm to multivariate vectorvalued osculatory rational interpolation.Then based on the Fitzpatrick algorithm and the properties of an Hermite interpolation basis,the authors present a Fitzpatrick-Neville-type algorithm for multivariate vector-valued osculatory rational interpolation.It may be used to compute the values of multivariate vector-valued osculatory rational interpolants at some points directly without computing the interpolation function explicitly.展开更多
In this paper,for a zero-dimensional polynomial ideal I,the authors prove that k[x_(1),x_(2),…,x_(n)]/I is cyclic if and only if the breadth of I is 0 or 1.Furthermore,the authors present a new algorithm to compute p...In this paper,for a zero-dimensional polynomial ideal I,the authors prove that k[x_(1),x_(2),…,x_(n)]/I is cyclic if and only if the breadth of I is 0 or 1.Furthermore,the authors present a new algorithm to compute polynomial univariate representation(PUR)of such an ideal.展开更多
Every univariate Hermite interpolation problem can be written as a pointwise limit of Lagrange interpolants.However,this property is not preserved for the multivariate case.In this paper,the authors first generalize t...Every univariate Hermite interpolation problem can be written as a pointwise limit of Lagrange interpolants.However,this property is not preserved for the multivariate case.In this paper,the authors first generalize the result of P.Gniadek.As an application,the authors consider the discrete approximation problem for a special case when the interpolation condition contains all partial derivatives of order less than n and one nth order differential polynomial.In addition,for the case of n≥3,the authors use the concept of Cartesian tensors to give a sufficient condition to find a sequence of discrete points,such that the Lagrange interpolation problems at these points converge to the given Hermite-type interpolant.展开更多
基金supported by the National Natural Science Foundation of China under Grant No.11671169Scientific Research Fund of Liaoning Provincial Education Department under Grant No.L2014008
文摘The paper is concerned with the improvement of the rational representation theory for solving positive-dimensional polynomial systems. The authors simplify the expression of rational representation set proposed by Tan and Zhang(2010), obtain the simplified rational representation with less rational representation sets, and hence reduce the complexity for representing the variety of a positive-dimensional ideal. As an application, the authors compute a "nearly" parametric solution for the SHEPWM problem with a fixed number of switching angles.
基金supported by the National Natural Science Foundation of China under Grant Nos.11171133 and 11326209
文摘Ideal interpolation is a generalization of the univariate Hermite interpolation. It is well known that every univariate Hermite interpolant is a pointwise limit of some Lagrange interpolants.However, a counterexample provided by Shekhtman Boris shows that, for more than two variables,there exist ideal interpolants that are not the limit of any Lagrange interpolants. So it is natural to consider: Given an ideal interpolant, how to find a sequence of Lagrange interpolants(if any) that converge to it. The authors call this problem the discretization for ideal interpolation. This paper presents an algorithm to solve the discretization problem. If the algorithm returns "True", the authors get a set of pairwise distinct points such that the corresponding Lagrange interpolants converge to the given ideal interpolant.
基金supported by Chinese National Natural Science Foundation under Grant Nos.11601039,11671169,11501051the Open Fund Key Laboratory of Symbolic Computation and Knowledge Engineering(Ministry of Education)under Grant No.93K172015K06the Education Department of Jilin Province,“13th Five-Year”Science and Technology Project under Grant No.JJKH20170618KJ
基金supported by the National Natural Science Foundation of China under Grant Nos.11171133 and 11271156
文摘This paper demonstrates the equivalence of two classes of D-invariant polynomial subspaces, i.e., these two classes of subspaces are different representations of the breadth-one D-invariant subspace. Moreover, the authors solve the discrete approximation problem in ideal interpolation for the breadth-one D-invariant subspace. Namely, the authors find the points, such that the limiting space of the evaluation functionals at these points is the functional space induced by the given D-invariant subspace, as the evaluation points all coalesce at one point.
基金supported by the National Science Foundation of China under Grant No.11171133the Open Fund of Automated Reasoning and Cognition Key Laboratory of Chongqing under Grant No.CARC2014001
文摘In this paper,the authors first apply the Fitzpatrick algorithm to multivariate vectorvalued osculatory rational interpolation.Then based on the Fitzpatrick algorithm and the properties of an Hermite interpolation basis,the authors present a Fitzpatrick-Neville-type algorithm for multivariate vector-valued osculatory rational interpolation.It may be used to compute the values of multivariate vector-valued osculatory rational interpolants at some points directly without computing the interpolation function explicitly.
基金supported by the National Natural Science Foundation of China under Grant No.11671169。
文摘In this paper,for a zero-dimensional polynomial ideal I,the authors prove that k[x_(1),x_(2),…,x_(n)]/I is cyclic if and only if the breadth of I is 0 or 1.Furthermore,the authors present a new algorithm to compute polynomial univariate representation(PUR)of such an ideal.
基金supported by the National Natural Science Foundation of China under Grant Nos.11901402 and 11671169。
文摘Every univariate Hermite interpolation problem can be written as a pointwise limit of Lagrange interpolants.However,this property is not preserved for the multivariate case.In this paper,the authors first generalize the result of P.Gniadek.As an application,the authors consider the discrete approximation problem for a special case when the interpolation condition contains all partial derivatives of order less than n and one nth order differential polynomial.In addition,for the case of n≥3,the authors use the concept of Cartesian tensors to give a sufficient condition to find a sequence of discrete points,such that the Lagrange interpolation problems at these points converge to the given Hermite-type interpolant.
