A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots

Corless, Robert M.; Gianni, Patrizia; Trager, Barry M.

doi:10.1145/258726.258767

Cited by 81 publications

(103 citation statements)

References 12 publications

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“…Following the strategy of Corless, Gianni and Trager [20], one can find such h by choosing the c i 's at random. Then, with high probability, h(v) (v ∈ V C (I)) are pairwise distinct.…”

Section: Example 210 Consider the Idealmentioning

confidence: 99%

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Sums of Squares, Moment Matrices and Optimization Over Polynomials

Laurent

2008

The IMA Volumes in Mathematics and Its Applications

642

721

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Abstract. We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zero-dimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.

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“…Following the strategy of Corless, Gianni and Trager [20], one can find such h by choosing the c i 's at random. Then, with high probability, h(v) (v ∈ V C (I)) are pairwise distinct.…”

Section: Example 210 Consider the Idealmentioning

confidence: 99%

“…, x n -to recover the components of v from the vector ζ B,v . According to [20], if we choose h as a random linear combination of the monomials x 1 , . .…”

Section: Sums Of Squares Moments and Polynomial Optimization 73mentioning

confidence: 99%

Sums of Squares, Moment Matrices and Optimization Over Polynomials

Laurent

2008

The IMA Volumes in Mathematics and Its Applications

642

721

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“…It is called Stickelberger theorem in textbook [20], and it is credited to Stetter and Müller in [4], see also the recent work [9]. The method can be sketched as follows.…”

Section: Solving the Polynomial System Of Equationsmentioning

confidence: 99%

“…In order to compute these eigenvalues, we follow [4] and build a random combination of multiplication matrices…”

Section: Common Eigenvaluesmentioning

confidence: 99%

Detecting Global Optimality and Extracting Solutions in GloptiPoly

Henrion¹,

Lasserre²

2005

Lecture Notes in Control and Information Sciences

261

296

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GloptiPoly is a Matlab/SeDuMi add-on to build and solve convex linear matrix inequality (LMI) relaxations of non-convex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sum-of-squares decompositions of positive polynomials, the theory of moments allows to detect global optimality of an LMI relaxation and extract globally optimal solutions. In this report, we describe and illustrate the numerical linear algebra algorithm implemented in GloptiPoly for detecting global optimality and extracting solutions. We also mention some related heuristics that could be useful to reduce the number of variables in the LMI relaxations.

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“…For the first two we knew already that the roots were of multiplicity one in the generic case, ie each eigenspace was of dimension one, this enabled us to compute the roots just by computing the eigenvectors of only one generic M f . But in the case of extremal radii the trouble was that generically the system had multiplicities and we had to consider another more sophisticated and costly method to recover the roots and their multiplicities (see for instance [6] for further details). Figure 8 shows the six cylinders solutions for the set of points corresponding to the regular tetrahedron (the real solutions are displayed with pov-ray).…”

Section: Theoremmentioning

confidence: 99%