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19
A Symbolic Approach to Bernstein Expansion for Program Analysis and Optimization
 In 13th International Conference on Compiler Construction, CC 2004
, 2004
"... Several mathematical frameworks for static analysis of programs have been developed in the past decades. Although these tools are quite useful, they have still many limitations. In particular, integer multivariate polynomials arise in many situations while analyzing programs, and analysis system ..."
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Several mathematical frameworks for static analysis of programs have been developed in the past decades. Although these tools are quite useful, they have still many limitations. In particular, integer multivariate polynomials arise in many situations while analyzing programs, and analysis systems are unable to handle such expressions. Although some dedicated methods have already been proposed, they only handle some subsets of such expressions. This paper presents an original and general approach to Bernstein expansion which is symbolic. Bernstein expansion allows bounding the range of a multivariate polynomial over a box and is generally more accurate than classic interval methods.
Lower Bound Functions for Polynomials
 J. Computational and Applied Mathematics
, 2003
"... This paper addresses the construction of relaxations for problems involving multivariate polynomials. The major goal is to show how nonconvex multivariate polynomial terms can be replaced by affine and convex lower bound functions which are computed by using Bernstein coefficients. These bound func ..."
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Cited by 11 (4 self)
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This paper addresses the construction of relaxations for problems involving multivariate polynomials. The major goal is to show how nonconvex multivariate polynomial terms can be replaced by affine and convex lower bound functions which are computed by using Bernstein coefficients. These bound functions may be used in any relaxation method described in the above literature, whenever these approaches do not deliver satisfactory results for polynomial terms of higher degree. Moreover, several properties of these bound functions are discussed. For properties of Bernstein polynomials the reader is referred to Cargo and Shisha [5], Farin [7], Garloff [11], Garloff, Jansson and Smith [12], and Zettler and Garloff [30]. By using Bernstein coefficients, bounds for the range of a multivariate polynomial over a box can be computed. It was shown by Stahl [28] that in the univariate case these bounds are often tighter than bounds which are obtained by applying interval computation techniques (cf. Neumaier [21], Ratschek and Rokne [23]). In [19] a method is presented by which piecewise linear lower (and equally linear upper) bound functions for multivariate polynomials can be obtained. This leads to tight enclosures of the given polynomials which are important, e.g., in intersection testing. The construction is presented there in detail in the univariate and bivariate cases. However, these lower bound functions are in general not convex. So the convex envelope of the piecewise linear lower bound functions has to be taken, requiring additional effort. The paper is organised as follows. In the next section some basic definitions and properties of Bernstein polynomials are given. Affine and convex lower bound functions based on the Bernstein expansion are presented in Section 3. An error bound for the...
Solving nonlinear systems by constraint inversion and interval arithmetic
 IN INT. CONF. ON ARTIFICIAL INTELLIGENCE AND SYMBOLIC COMPUTATION (AISC’2000), LNAI
, 2000
"... A reliable symbolicnumeric algorithm for solving nonlinear systems over the reals is designed. The symbolic step generates a new system, where the formulas are different but the solutions are preserved, through partial factorizations of polynomial expressions and constraint inversion. The numeric s ..."
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A reliable symbolicnumeric algorithm for solving nonlinear systems over the reals is designed. The symbolic step generates a new system, where the formulas are different but the solutions are preserved, through partial factorizations of polynomial expressions and constraint inversion. The numeric step is a branchandprune algorithm based on interval constraint propagation to compute a set of outer approximations of the solutions. The processing of the inverted constraints by interval arithmetic provides a fast and efficient method to contract the variables' domains. A set of experiments for comparing several constraint solvers is reported.
Greedy algorithms for optimizing multivariate horner schemes
 SIGSAM Bull
, 2004
"... For univariate polynomials f(x1), Horner’s scheme provides the fastest way to compute a value. For multivariate polynomials, several different version of Horner’s scheme are possible; it is not clear which of them is optimal. In this paper, we propose a greedy algorithm, which it is hoped will lead ..."
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For univariate polynomials f(x1), Horner’s scheme provides the fastest way to compute a value. For multivariate polynomials, several different version of Horner’s scheme are possible; it is not clear which of them is optimal. In this paper, we propose a greedy algorithm, which it is hoped will lead to good computation times. The univariate Horner scheme has another advantage: if the value x1 is known with uncertainty, and we are interested in the resulting uncertainty in f(x1), then Horner scheme leads to a better estimate for this uncertainty that many other ways of computing f(x1). The second greedy algorithm that we propose tries to find the multivariate Horner scheme that leads to the best estimate for the uncertainty in f(x1,..., xn). 1
Taylor Forms  Use and Limits
 Reliable Computing
, 2002
"... This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 19 ..."
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This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch and Wittwer, and independently studied and popularized since 1996 by Berz, Makino and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001, although the details given are not sufficient to check the validity of their claims.
Solving interval constraints by linearization in computeraided design. Reliable Computing
, 2006
"... Abstract. Current parametric CAD systems require geometric parameters to have fixed values. Specifying fixed parameter values implicitly adds rigid constraints on the geometry, which have the potential to introduce conflicts during the design process. This paper presents a soft constraint representa ..."
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Abstract. Current parametric CAD systems require geometric parameters to have fixed values. Specifying fixed parameter values implicitly adds rigid constraints on the geometry, which have the potential to introduce conflicts during the design process. This paper presents a soft constraint representation scheme based on nominal interval. Interval geometric parameters capture inexactness of conceptual and embodiment design, uncertainty in detail design, as well as boundary information for design optimization. To accommodate underconstrained and overconstrained design problems, a doubleloop GaussSeidel method is developed to solve linear constraints. A symbolic preconditioning procedure transforms nonlinear equations to separable form. Inequalities are also transformed and integrated with equalities. Nonlinear constraints can be bounded by piecewise linear enclosures and solved by linear methods iteratively. A sensitivity analysis method that differentiates active and inactive constraints is presented for design refinement. 1.
Towards More Efficient Interval Analysis: Corner Forms and a Remainder Interval Newton Method
, 2005
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Continuous amortization: A nonprobabilistic adaptive analysis technique
 Electronic Colloquium on Computational Complexity (ECCC
, 2009
"... Let f be a univariate polynomial with real coefficients, f ∈ R[X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the roots of f in a given interval. In this paper, we consider subdivision algorithms based on purely numerical pri ..."
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Let f be a univariate polynomial with real coefficients, f ∈ R[X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the roots of f in a given interval. In this paper, we consider subdivision algorithms based on purely numerical primitives such as function evaluation. Such methods have adaptive complexity, are local, and are also applicable when f is transcendental. The complexity analysis of adaptive algorithms is a new challenge for computer science. In this paper, we introduce a form of continuous amortization for adaptive complexity. Our analysis is applied to an evaluationbased root isolation algorithm called EVAL. EVAL is based on an algorithm of Mitchell and can also be seen as a 1dimensional analogue of algorithms by Plantinga and Vegter for meshing curves and surfaces. The algorithm itself is simple, but its complexity analysis is not. Our main result is an O(d3(log d+L)) bound on the subdivisiontree size of EVAL for the benchmark problem of isolating all real roots of a squarefree integer polynomial f of degree d and logarithmic height L. Our proof introduces several novel techniques: First, we provide an adaptive upper bound on the complexity of EVAL using an integral, analogous to integral bounds provided by Ruppert in a different context. Such integrals can be viewed as a form of continuous amortization. In addition, we use two algebraic amortization techniques: one is based on the standard MahlerDavenport root bounds, but the other, based on evaluation bounds, is new.
A New Strategy For Selecting Subdivision Point In The Bernstein Approach To Polynomial Optimization ∗
"... In the Bernstein approach to polynomial range finding, we propose a new rule for selection of the point where the domain subdivision is to be done. For a given direction of subdivision, instead of subdividing a box at its midpoint (as is done in the existing literature), we propose to subdivide the ..."
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In the Bernstein approach to polynomial range finding, we propose a new rule for selection of the point where the domain subdivision is to be done. For a given direction of subdivision, instead of subdividing a box at its midpoint (as is done in the existing literature), we propose to subdivide the box at a point where the partial derivative of the polynomial (in the direction of subdivision) equals zero. The location of this point is estimated using the variation diminishing property of the derivative polynomial in the Bernstein form. We then compare the performance of the proposed rule for subdivision point with that of the existing midpoint subdivision rule, on nine polynomial problems of different dimensionsvarying from two to eight dimensions. We evaluate both the rules using three different existing subdivision direction selection rules, and find the proposed rule to be overall more efficient in computational time and number of subdivisions.
A Matrix Method for Efficient Computation of Bernstein Coefficients ∗
"... We propose a generalized method, called as the Matrix method, for computationofthecoefficientsofmultivariateBernsteinpolynomials. The proposed Matrix method involves only matrix operations such as multiplication, inverse, transpose and reshape. For a general boxlike domain, the computational comple ..."
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We propose a generalized method, called as the Matrix method, for computationofthecoefficientsofmultivariateBernsteinpolynomials. The proposed Matrix method involves only matrix operations such as multiplication, inverse, transpose and reshape. For a general boxlike domain, the computational complexity of the proposed method is O(n l+1) in contrast to O(n 2l) for existing methods. We conduct numerical experiments to compute the Bernstein coefficients for eleven polynomial problems (with dimensions varying from three to seven) defined over a unit box domain as well as a general box domain, with the existing methods and the proposed Matrix method. A comparison of the results shows the proposed algorithm to yield significant reductions in computational time for ‘larger’ number of Bernstein coefficients.