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19
Interval arithmetic: From principles to implementation
 J. ACM
"... We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithme ..."
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Cited by 76 (12 self)
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We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithmetic. Next, we define interval arithmetic operations on intervals with IEEE 754 floating point endpoints to be sound and optimal approximations of the real interval operations and we show that the IEEE standard’s specification of operations involving the signed infinities, signed zeros, and the exact/inexact flag are such as to make a correct and optimal implementation more efficient. From the resulting theorems we derive data that are sufficiently detailed to convert directly to a program for efficiently implementing the interval operations. Finally we extend these results to the case of general intervals, which are defined as connected sets of reals that are not necessarily closed. 1
Universally Quantified Interval Constraints
 PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2000
"... Nonlinear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifierfree equivalent form by means of Cylindrical Algebraic Decompo ..."
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Cited by 46 (0 self)
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Nonlinear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifierfree equivalent form by means of Cylindrical Algebraic Decomposition (CAD). However, CAD restricts its input to be conjunctions and disjunctions of polynomial constraints with rational coefficients, while some applications such as camera control involve systems with arbitrary forms where time is the only universally quantified variable. In this paper, the handling of universally quantified variables is first related to the computation of innerapproximation of real relations.
Comparing Partial Consistencies
, 1999
"... Global search algorithms have been widely used in the constraint programming framework to solve constraint systems over continuous domains. This paper precisely states the relations among the different partial consistencies which are main emphasis of these algorithms. The ..."
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Cited by 22 (4 self)
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Global search algorithms have been widely used in the constraint programming framework to solve constraint systems over continuous domains. This paper precisely states the relations among the different partial consistencies which are main emphasis of these algorithms. The
Interval Constraints
, 1999
"... ; vn g and a set of floatingpoint intervals fI 1 ; : : : ; I n g representing the variables' domains of possible values, to isolate a set of fngary canonical boxes (Cartesian products of I i s subintervals whose bounds are either equal or consecutive floatingpoint numbers) approximating the cons ..."
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Cited by 22 (0 self)
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; vn g and a set of floatingpoint intervals fI 1 ; : : : ; I n g representing the variables' domains of possible values, to isolate a set of fngary canonical boxes (Cartesian products of I i s subintervals whose bounds are either equal or consecutive floatingpoint numbers) approximating the constraint system solution space. To compute such a set, a search procedure navigates through the Cartesian product I 1 \Theta : : : \Theta I n alternating pruning and branching steps. The pruning step uses a relational form of interval arithmetic [Moo66], [AH83]. Given a set of constraints over the reals, interval arithmetic is used to compute local approximations of the solution space for a given constraint. This approximation results in the elimination of
Consistency Techniques in Ordinary Differential Equations
, 2000
"... This paper takes a fresh look at the application of interval analysis to ordinary differential equations and studies how consistency techniques can help address the accuracy problems typically exhibited by these methods, while trying to preserve their efficiency. It proposes to generalize interval t ..."
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Cited by 16 (1 self)
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This paper takes a fresh look at the application of interval analysis to ordinary differential equations and studies how consistency techniques can help address the accuracy problems typically exhibited by these methods, while trying to preserve their efficiency. It proposes to generalize interval techniques intoatwostep process: a forward process that computes an enclosure and a backward process that reduces this enclosure. Consistency techniques apply naturally to the backward (pruning) step but can also be applied to the forward phase. The paper describes the framework, studies the various steps in detail, proposes a number of novel techniques, and gives some preliminary experimental results to indicate the potential of this new research avenue.
Solving Constraints over FloatingPoint Numbers
, 2001
"... This paper introduces a new framework for tackling constraints over the floatingpoint numbers. An important application area where such solvers are required is program analysis (e.g., structural test case generation, correctness proof of numeric operations). Albeit the floatingpoint numbers are a ..."
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Cited by 11 (1 self)
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This paper introduces a new framework for tackling constraints over the floatingpoint numbers. An important application area where such solvers are required is program analysis (e.g., structural test case generation, correctness proof of numeric operations). Albeit the floatingpoint numbers are a finite subset of the real numbers, classical CSP techniques are ine ective due to the huge size of the domains. Relations that hold over the real numbers may not hold over the floatingpoint numbers. Moreover, constraints that have no solutions over the reals may hold over the floats. Thus, intervalnarrowing techniques, which are used in numeric CSP, cannot safely solve constraints systems over the floats.
Accelerating Filtering Techniques for Numeric CSPs
, 2002
"... Search algorithms for solving Numeric CSPs (Constraint Satisfaction Problems) make an extensive use of filtering techniques. In this paper we show how those filtering techniques can be accelerated by discovering and exploiting some regularities during the filtering process. Two kinds of regularit ..."
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Cited by 11 (3 self)
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Search algorithms for solving Numeric CSPs (Constraint Satisfaction Problems) make an extensive use of filtering techniques. In this paper we show how those filtering techniques can be accelerated by discovering and exploiting some regularities during the filtering process. Two kinds of regularities are discussed, cyclic phenomena in the propagation queue and numeric regularities of the domains of the variables. We also present in this paper an attempt to unify numeric CSPs solving methods from two distinct communities, that of CSP in artificial intelligence, and that of interval analysis. 2002 Elsevier Science B.V. All rights reserved.
RealPaver: An Interval Solver using Constraint Satisfaction Techniques
 ACM TRANS. ON MATHEMATICAL SOFTWARE
, 2006
"... RealPaver is an interval software for modeling and solving nonlinear systems. Reliable approximations of continuous or discrete solution sets are computed, using Cartesian products of intervals. Systems are given by sets of equations or inequality constraints over integer and real variables. Moreove ..."
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Cited by 11 (0 self)
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RealPaver is an interval software for modeling and solving nonlinear systems. Reliable approximations of continuous or discrete solution sets are computed, using Cartesian products of intervals. Systems are given by sets of equations or inequality constraints over integer and real variables. Moreover, they may have different natures, being square or non square, sparse or dense, linear, polynomial or involving transcendental functions. The modeling language permits stating constraint models and tuning parameters of solving algorithms, which efficiently combine interval methods and constraint satisfaction techniques. Several consistency techniques (box, hull, 3B) are implemented. The distribution includes C sources, executables for different machine architectures, documentation and benchmarks. The portability is ensured by the GNU C compiler.
A rigorous global filtering algorithm for quadratic constraints
 Constraints
, 2005
"... Abstract. This article introduces a new filtering algorithm for handling systems of quadratic equations and inequations. Such constraints are widely used to model distance relations in numerous application areas ranging from robotics to chemistry. Classical filtering algorithms are based upon local ..."
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Cited by 10 (2 self)
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Abstract. This article introduces a new filtering algorithm for handling systems of quadratic equations and inequations. Such constraints are widely used to model distance relations in numerous application areas ranging from robotics to chemistry. Classical filtering algorithms are based upon local consistencies and thus, are often unable to achieve a significant pruning of the domains of the variables occurring in quadratic constraint systems. The drawback of these approaches comes from the fact that the constraints are handled independently. We introduce here a global filtering algorithm that works on a tight linear relaxation of the quadratic constraints. The Simplex algorithm is then used to narrow the domains. Since most implementations of the Simplex work with floating point numbers and thus, are unsafe, we provide a procedure to generate safe linearizations. We also exploit a procedure provided by Neumaier and Shcherbina to get a safe objective value when calling the Simplex algorithm. With these two procedures, we prevent the Simplex algorithm from removing any solution while filtering linear constraint systems. Experimental results on classical benchmarks show that this new algorithm yields a much more effective pruning of the domains than local consistency filtering algorithms. Keywords: global constraints, quadratic constraints, safe linearizations 1.
Progress in the Solving of a Circuit Design Problem
 JOURNAL OF GLOBAL OPTIMIZATION
, 2001
"... A new branchandprune algorithm for globally solving nonlinear systems is proposed. The pruning technique combines a multidimensional interval Newton method with the constraint satisfaction algorithm HC4 [1]. The main contributions of this paper are the finegrained interaction between both algori ..."
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Cited by 9 (2 self)
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A new branchandprune algorithm for globally solving nonlinear systems is proposed. The pruning technique combines a multidimensional interval Newton method with the constraint satisfaction algorithm HC4 [1]. The main contributions of this paper are the finegrained interaction between both algorithms which avoids some unnecessary computation,and the description of HC4 in terms of a chain rule for constraints’ projections. Our algorithm is experimentally compared with two global methods from Ratschek and Rokne [17] and from Puget and Van Hentenryck [16] on Ebers and Moll’ circuit design problem [6]. An interval enclosure of the solution with a precision of twelve significant digits is computed in four minutes, providing an improvement factor of five on the same machine.