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18
A proof of the Kepler conjecture
 Math. Intelligencer
, 1994
"... This section describes the structure of the proof of ..."
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Cited by 112 (11 self)
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This section describes the structure of the proof of
Robustness Analysis of Polynomials with Polynomial Parameter Dependency Using Bernstein Expansion
 IEEE TRANS. AUTOMAT. CONTR
, 1998
"... ..."
Comparison of Interval Methods for Plotting Algebraic Curves
 Comput. Aided Geom. Des
, 2002
"... This paper compares the performance and e#ciency of di#erent function range interval methods for plotting f(x, y) = 0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD. ..."
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Cited by 24 (2 self)
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This paper compares the performance and e#ciency of di#erent function range interval methods for plotting f(x, y) = 0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD.
Solving strict polynomial inequalities by Bernstein expansion
 In: Symbolic Methods in Control System Analysis and Design
, 1999
"... Introduction Many interesting control system design and analysis problems can be recast as systems of inequalities for multivariate polynomials in real variables. In particular, for linear timeinvariant systems, important control issues such as robust stability and robust performance can be reduce ..."
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Cited by 20 (1 self)
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Introduction Many interesting control system design and analysis problems can be recast as systems of inequalities for multivariate polynomials in real variables. In particular, for linear timeinvariant systems, important control issues such as robust stability and robust performance can be reduced to such systems. Typically, the variables in the (multivariate) polynomials come from plant (controlled system) and compensator (controller) parameters. In this chapter, we describe a method for solving such systems of inequalities. By solving we mean that we end up with a collection of axisparallel boxes in the parameter space whose union provides an inner approximation of the solution set, i.e., the polynomial inequalities are fulfilled for each parameter vector taken from such a box. This method is based on the expansion of a multivariate polynomial into Bernstein polynomials. It provides an alternative to symbolic methods like quantifier elimination whose application to control
Application of Bernstein Expansion to the Solution of Control Problems
 University of Girona
, 1999
"... We survey some recent applications of Bernstein expansion to robust stability, viz. checking robust Hurwitz and Schur stability of polynomials with polynomial parameter dependency by testing determinantal criteria and by inspection of the value set. Then we show how Bernstein expansion can be used t ..."
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Cited by 7 (0 self)
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We survey some recent applications of Bernstein expansion to robust stability, viz. checking robust Hurwitz and Schur stability of polynomials with polynomial parameter dependency by testing determinantal criteria and by inspection of the value set. Then we show how Bernstein expansion can be used to solve systems of strict polynomial inequalities.
Investigation of a Subdivision Based Algorithm for Solving Systems of Polynomial Equations
, 2000
"... A method for enclosing all solutions of a system of polynomial equations inside a given box is investigated. This method relies on the expansion of a multivariate polynomial into Bernstein polynomials and constitutes a domainsplitting approach. After a pruning step, a collection of subboxes rem ..."
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Cited by 6 (0 self)
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A method for enclosing all solutions of a system of polynomial equations inside a given box is investigated. This method relies on the expansion of a multivariate polynomial into Bernstein polynomials and constitutes a domainsplitting approach. After a pruning step, a collection of subboxes remain which undergo an existence test provided by Miranda's theorem. In this paper, the complexity of this test is reduced from O(n!) to nearly O(n 2 ). Also, some observations on the effects of preconditioning of the system and results on the application of Bernstein expansion to the mean value form are presented. Keywords: Polynomial equations, Bernstein polynomials, Miranda Theorem, mean value form 1 Introduction Roughly speaking, methods for solving systems of polynomial equations can be divided into three classes: techniques based on elimination theory, e.g., Ch. 8 in Winkler [1], continuation, e.g., Allgower and Georg [2, 3], and subdivision. The first two classes frequently giv...
Bounds for the Range of a Bivariate Polynomial over a Triangle
 Reliable Computing
, 1998
"... this paper we consider the following PROBLEM. Let a bivariate polynomial p(x; y) = ..."
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Cited by 5 (3 self)
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this paper we consider the following PROBLEM. Let a bivariate polynomial p(x; y) =
Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation
"... This paper addresses the problem of finding tight affine lower bound functions for multivariate polynomials, which may be employed when global optimisation problems involving polynomials are solved with a branch and bound method. These bound functions are constructed by using the expansion of the gi ..."
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Cited by 5 (3 self)
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This paper addresses the problem of finding tight affine lower bound functions for multivariate polynomials, which may be employed when global optimisation problems involving polynomials are solved with a branch and bound method. These bound functions are constructed by using the expansion of the given polynomial into Bernstein polynomials. The coefficients of this expansion over a given box yield a control point structure whose convex hull contains the graph of the given polynomial over the box. We introduce a new method for computing tight affine lower bound functions based on these control points, using a linear least squares approximation of the entire control point structure. This is demonstrated to have superior performance to previous methods based on a linear interpolation of certain specially chosen control points. The problem of how to obtain a verified affine lower bound function in the presence of uncertainty and rounding errors is also considered. Numerical results with error bounds for a series of randomlygenerated polynomials are given. Key words: Constrained global optimisation, relaxation, affine bound functions, Bernstein polynomials, linear least squares 1
Towards More Efficient Interval Analysis: Corner Forms and a Remainder Interval Newton Method
, 2005
"... c ○ 2005 ..."
A.P.: Solving linear systems with polynomial parameter dependency
 Preprint No.1/2009, Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Department of Biomathematics, Sofia (2009) http://www.math.bas.bg/ ∼ epopova/papers/09PreprintGPS.pdf Linear Systems with Polynomial Parameter Dependency 17
"... Abstract We give a short survey on methods for the enclosure of the solution set of a system of linear equations where the coefficients of the matrix and the right hand side depend on parameters varying within given intervals. Then we present a hybrid method for finding such an enclosure in the case ..."
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Cited by 4 (4 self)
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Abstract We give a short survey on methods for the enclosure of the solution set of a system of linear equations where the coefficients of the matrix and the right hand side depend on parameters varying within given intervals. Then we present a hybrid method for finding such an enclosure in the case that the dependency is polynomial or rational. A generalpurpose parametric fixedpoint iteration is combined with efficient tools for range enclosure based on the Bernstein expansion of multivariate polynomials. We discuss applications of the generalpurpose parametric method to linear systems obtained by standard finite element analysis of mechanical structures and illustrate the efficiency of the new parametric solver. 1