An algorithm for sums of squares of real polynomials

Powers, Victoria; Wörmann, Thorsten

doi:10.1016/s0022-4049(97)83827-3

Cited by 179 publications

(108 citation statements)

References 2 publications

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“…, t δ . Using this combination and applying Descartes' rule of signs of the characteristic polynomial, Powers and Wörmann [13] proposed an algorithm for checking whether a polynomial is sos. The minimal rank of X hence is determined for some δ−tuple (t 1 , .…”

Section: Proposition 1 Let Fmentioning

confidence: 99%

“…The system of nonlinear equations (13) is solved by using the Levenberg-Marquardt method [11] implemented by "fsolve". In this sense, the system of nonlinear equations (13) is equivalent to the problem of finding an (ω × r )−matrix H with rank(H ) = r and (H H T ) αβ = r l=1 h lα h lβ satisfying (13). In the algorithm, we need an upper bound of the residual as follows.…”

Section: Algorithm For Finding the Pythagoras Numbermentioning

confidence: 99%

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The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials

Sorber

Barel

2012

Calcolo

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In this paper, we conjecture a formula for the value of the Pythagoras number for real multivariate sum of squares polynomials as a function of the (total or coordinate) degree and the number of variables. The conjecture is based on the comparison between the number of parameters and the number of conditions for a corresponding low-rank representation. This is then numerically verified for a number of examples. Additionally, we discuss the Pythagoras number of (complex) multivariate Laurent polynomials that are sum of square magnitudes of polynomials on the n-torus. For both types of polynomials, we also propose an algorithm to numerically compute the Pythagoras number and give some numerical illustrations.

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Section: Proposition 1 Let Fmentioning

confidence: 99%

Section: Algorithm For Finding the Pythagoras Numbermentioning

confidence: 99%

The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials

Sorber

Barel

2012

Calcolo

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“…In order to solve this problem, iteration algorithms and sub-optimization problems are proposed [18]. Specifically, with regard to the problem (11), we start with a selected polynomial Lyapunov function (i.e., (8) holds), and find an admissible controller K such that the estimate of V(c) is maximized, i.e., we aim at finding…”

Section: B Problem Formulationmentioning

confidence: 99%

Control synthesis for non-polynomial systems: A domain of attraction perspective

Han

Althoff

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Abstract-This paper studies a control synthesis problem to enlarge the domain of attraction (DA) for non-polynomial systems by using polynomial Lyapunov functions. The basic idea is to formulate an uncertain polynomial system with parameter ranges obtained from the truncated Taylor expansion and the parameterizable remainder of the non-polynomial system. A strategy for searching a polynomial output feedback controller and estimating the lower bound of the largest DA is proposed via an optimization of linear matrix inequalities (LMIs). Furthermore, in order to check the tightness of the lower bound of the largest estimated DA, a necessary and sufficient condition is given for the proposed controller. Lastly, several methods are provided to show how the proposed strategy can be extended to the case of variable Lyapunov functions. The effectiveness of this approach is demonstrated by numerical examples.

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“…Then, we need to solve 2n + 1 equations in (n+1)(n+2) 2 variables (the parameters b ij ) corresponding to the monomials in H. According to [15], this can be done by taking (n+1)(n+2) 2 − (2n + 1) = n 2 −n 2 of the b ij as unknowns which can be given appropriate values that are obtained using (2), i.e., B must be positive semidefinite. This can be done by computing the characteristic polynomial det(zI n+1 − B) = n i=0 c i z i of B and requiring its roots to be non-negative [15]. They show that this can be achieved by imposing…”

Section: Theorem 2 [7]mentioning

confidence: 99%

Using Representation Theorems for Proving Polynomials Non-negative

Lucas

2014

Artificial Intelligence and Symbolic Computation

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Abstract. Proving polynomials non-negative when variables range on a subset of numbers (e.g., [0, +∞)) is often required in many applications (e.g., in the analysis of program termination). Several representations for univariate polynomials P that are non-negative on [0, +∞) have been investigated. They can often be used to characterize the property, thus providing a method for checking it by trying a match of P against the representation. We introduce a new characterization based on viewing polynomials P as vectors, and find the appropriate polynomial basis B in which the non-negativeness of the coordinates [P ]B representing P in B witnesses that P is non-negative on [0, +∞). Matching a polynomial against a representation provides a way to transform universal sentences ∀x ∈ [0, +∞) P (x) ≥ 0 into a constraint solving problem which can be solved by using efficient methods. We consider different approaches to solve both kind of problems and provide a quantitative evaluation of performance that points to an early result by Pólya and Szegö's as an appropriate basis for implementations in most cases.

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An algorithm for sums of squares of real polynomials

Cited by 179 publications

References 2 publications

The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials

The Pythagoras number of real sum of squares polynomials and sum of square magnitudes of polynomials

Control synthesis for non-polynomial systems: A domain of attraction perspective

Using Representation Theorems for Proving Polynomials Non-negative

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