Numerical Polynomial Algebra

Stetter, Hans J.

doi:10.1137/1.9780898717976

Cited by 207 publications

(198 citation statements)

References 2 publications

Supporting

Mentioning

197

Contrasting

Order By: Relevance

“…the monographs of Basu, Pollack and Roy [2], Dickenstein and Emiris [9], Mora [27,28], Elkadi and Mourrain [10], Stetter [43], Sturmfels [44]. We do not attempt a complete description of all existing methods, but instead we only try to give a coarse classification.…”

Section: Existing Methodsmentioning

confidence: 99%

“…In particular, algorithms have been proposed for constructing border bases of I leading to general (connected to 1) bases of R[x]/I (see [9,Chap. 4], [14], [29], [43]); these objects are introduced below. The moment matrix approach for computing real roots presented in this chapter leads naturally to the computation of such general bases.…”

Section: Border Bases and Normal Formsmentioning

confidence: 99%

See 1 more Smart Citation

The Approach of Moments for Polynomial Equations

Laurent

Rostalski

2011

International Series in Operations Research &Amp; Management Science

View full text Add to dashboard Cite

Summary. In this chapter we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of polynomials and the dual theory of moments. A crucial ingredient is a semidefinite characterization of the real radical ideal, consisting of all polynomials with the same real zero set as the system of polynomials to be solved. Combining this characterization with ideas from commutative algebra, (numerical) linear algebra and semidefinite optimization yields a new class of real algebraic algorithms. This chapter sheds some light on the underlying theory and the link to polynomial optimization.

show abstract

Section: Existing Methodsmentioning

confidence: 99%

Section: Border Bases and Normal Formsmentioning

confidence: 99%

The Approach of Moments for Polynomial Equations

Laurent

Rostalski

2011

International Series in Operations Research &Amp; Management Science

View full text Add to dashboard Cite

show abstract

“…Some strategies to render the problem well-posed can be formulated along the lines in [43], see also [44]. Given a first approximation of the structure at infinity or the null-space of A(s), we can think in some kind of iterative refinement over some manifold where this structure is invariant, and hence over which the problem is well-posed.…”

Section: Accuracy and Row Pivotingmentioning

confidence: 99%

An improved Toeplitz algorithm for polynomial matrix null-space computation

Anaya

Henrion

2009

Applied Mathematics and Computation

View full text Add to dashboard Cite

In this paper we present an improved algorithm to compute the minimal null-space basis of polynomial matrices, a problem which has many applications in control and systems theory. This algorithm takes advantage of the block Toeplitz structure of the Sylvester matrix associated with the polynomial matrix. The analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well-known pencil (state-space) algorithms found in the literature.

show abstract

“…[19]), or more general symbolic/numeric methods (e.g. [16] or [18], see also the monograph [23]). For instance, Verschelde [24] proposes a numerical algorithm via homotopy continuation methods (cf.…”

Section: Related Literaturementioning

confidence: 99%

“…Other techniques have been proposed for producing bases of the ideal I and of the vector space R[x]/I, which do not depend on a specific monomial ordering. In particular, algorithms have been proposed for constructing border bases of I leading to general (stable by division) bases of R[x]/I (see [6,Chapter 4], [8] and [23]). Another normal form algorithm is proposed by Mourrain [14] (see also [15,17]) leading to more general (namely, connected to 1) bases of R[x]/I.…”

Section: Multiplication Operators Given a Polynomial H ∈ R[x]mentioning

confidence: 99%

A Unified Approach to Computing Real and Complex Zeros of Zero-Dimensional Ideals

Lasserre

Laurent

Rostalski

2008

Emerging Applications of Algebraic Geometry

View full text Add to dashboard Cite

Abstract. In this paper we propose a unified methodology for computing the setWe show how moment matrices, defined in terms of a given set of generators of the ideal I, can be used to (numerically) find not only the real variety V R (I), as shown in the authors' previous work, but also the complex variety V C (I), thus leading to a unified treatment of the algebraic and real algebraic problem. In contrast to the real algebraic version of the algorithm, the complex analogue only uses basic numerical linear algebra because it does not require positive semidefiniteness of the moment matrix and so avoids semidefinite programming techniques. The links between these algorithms and other numerical algebraic methods are outlined and their stopping criteria are related.

show abstract

Numerical Polynomial Algebra

Cited by 207 publications

References 2 publications

The Approach of Moments for Polynomial Equations

The Approach of Moments for Polynomial Equations

An improved Toeplitz algorithm for polynomial matrix null-space computation

A Unified Approach to Computing Real and Complex Zeros of Zero-Dimensional Ideals

Contact Info

Product

Resources

About