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CLP(Intervals) Revisited
, 1994
"... The design and implementation of constraint logic programming (CLP) languages over intervals is revisited. Instead of decomposing complex constraints in terms of simple primitive constraints as in CLP(BNR), complex constraints are manipulated as a whole, enabling more sophisticated narrowing procedu ..."
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Cited by 121 (18 self)
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The design and implementation of constraint logic programming (CLP) languages over intervals is revisited. Instead of decomposing complex constraints in terms of simple primitive constraints as in CLP(BNR), complex constraints are manipulated as a whole, enabling more sophisticated narrowing procedures to be applied in the solver. This idea is embodied in a new CLP language Newton whose operational semantics is based on the notion of boxconsistency, an approximation of arcconsistency, and whose implementation uses Newton interval method. Experimental results indicate that Newton outperforms existing languages by an order of magnitude and is competitive with some stateoftheart tools on some standard benchmarks. Limitations of our current implementation and directions for further work are also identified.
Solving Polynomial Systems Using a Branch and Prune Approach
 SIAM Journal on Numerical Analysis
, 1997
"... This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in ..."
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Cited by 101 (7 self)
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This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called boxconsistency, which approximates the notion of arcconsistency wellknown in artificial intelligence. Boxconsistency is parametrized by an interval extension of the constraint and can be instantiated to produce the HansenSegupta's narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with stateoftheart continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.
Interval constraint logic programming
 CONSTRAINT PROGRAMMING: BASICS AND TRENDS, VOLUME 910 OF LNCS
, 1995
"... Abstract. In this paper, we present anoverview on the use of interval arithmetic to process numerical constraints in Constraint Logic Programming. The main principle is to approximate nary relations over IR with Cartesian products of intervals whose bounds are taken in a nite subset of I R.Variabl ..."
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Cited by 47 (5 self)
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Abstract. In this paper, we present anoverview on the use of interval arithmetic to process numerical constraints in Constraint Logic Programming. The main principle is to approximate nary relations over IR with Cartesian products of intervals whose bounds are taken in a nite subset of I R.Variables represent real values whose domains are intervals de ned in the same manner. Narrowing operators are de ned from approximations. These operators compute, from an interval and a relation, aset included in the initial interval. Sets of constraints are then processed thanks to a local consistency algorithm pruning at each stepvalues from initial intervals. This algorithm is shown to be correct and to terminate, on the basis of a certain number of properties of narrowing operators. We focus here on the description of the general framework based on approximations, on its application to interval constraint solving over continuous and discrete quantities, we establish a strong link between approximations and local consistency notions and show that arcconsistency is an instance of the approximation framework. We nally describe recentwork on di erent variants of the initial algorithm proposed by John Cleary and developed by W. Older and A. Vellino which havebeen proposed in this context. These variants address four particular points: generalization of the constraint language, improvement of domain reductions, e ciency of the computation and nally, cooperation with other solvers. Some open questions are also identi ed. 1
Automatic Differentiation Of Advanced CFD Codes For Multidisciplinary Design
 Journal on Computing Systems in Engineering
, 1992
"... This paper addresses one such synergism for computa ..."
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Cited by 22 (16 self)
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This paper addresses one such synergism for computa
Surface Intersection Using Affine Arithmetic
 In Graphics Interface
, 1996
"... We describe a variant of a domain decomposition method proposed by Gleicher and Kass for intersecting and trimming parametric surfaces. Instead of using interval arithmetic to guide the decomposition, the variant described here uses affine arithmetic, a tool recently proposed for range analysis. Aff ..."
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Cited by 18 (7 self)
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We describe a variant of a domain decomposition method proposed by Gleicher and Kass for intersecting and trimming parametric surfaces. Instead of using interval arithmetic to guide the decomposition, the variant described here uses affine arithmetic, a tool recently proposed for range analysis. Affine arithmetic is similar to standard interval arithmetic, but takes into account correlations between operands and subformulas, generally providing much tighter bounds for the computed quantities. As a consequence, the quadtree domain decompositions are much smaller and the intersection algorithm runs faster. keywords: surface intersection, trimming surfaces, range analysis, interval analysis, CAGD.
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.
A software package for the numerical integration of ODE by means of highorder Taylor methods
, 2001
"... This paper revisits the Taylor method for the numerical integration of initial value problems of Ordinary Dierential Equations (ODEs). The main issue is to present a computer program that, given a set of ODEs, produces the corresponding Taylor numerical integrator. The step size control adaptivel ..."
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Cited by 14 (1 self)
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This paper revisits the Taylor method for the numerical integration of initial value problems of Ordinary Dierential Equations (ODEs). The main issue is to present a computer program that, given a set of ODEs, produces the corresponding Taylor numerical integrator. The step size control adaptively selects both order and step size to achieve a prescribed error, and trying to minimize the global number of operations. The package provides support for several extended precision arithmetics, including userdened types.
Newton: Constraint Programming over Nonlinear Constraints
 SCIENCE OF COMPUTER PROGRAMMING
, 1998
"... This paper is an introduction to Newton, a constraint programming language over nonlinear real constraints. Newton originates from an eort to reconcile the declarative nature of constraint logic programming (CLP) languages over intervals with advanced interval techniques developed in numerical analy ..."
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Cited by 8 (3 self)
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This paper is an introduction to Newton, a constraint programming language over nonlinear real constraints. Newton originates from an eort to reconcile the declarative nature of constraint logic programming (CLP) languages over intervals with advanced interval techniques developed in numerical analysis, such as the interval Newton method. Its key conceptual idea is to introduce the notion of boxconsistency, which approximates arcconsistency, a notion wellknown in articial intelligence. Boxconsistency achieves an eective pruning at a reasonable computation cost and generalizes some traditional interval operators. Newton has been applied to numerous applications in science and engineering, including nonlinear equationsolving, unconstrained optimization, and constrained optimization. It is competitive with continuation methods on their equationsolving benchmarks and outperforms the intervalbased methods we are aware of on optimization problems. Key words: Constraint Programming, Nonlinear Programming, Interval Reasoning 1 Introduction Many applications in science and engineering (e.g., chemistry, robotics, economics, mechanics) require nding all isolated solutions to a system of nonlinear real constraints or nding the minimum value of a nonlinear function subject to nonlinear constraints. These problems are dicult due to their inherent computational complexity (i.e., they are NPhard) and due to the numerical issues involved to guarantee correctness (i.e., nding all solutions or the global optimum) and to ensure termination. Preprint submitted to Elsevier Preprint 11 June 2001 Newton is a constraint programming language designed to support this class of applications. It originates from an attempt to reconcile the declarative nature of CLP(Intervals) languag...
Multiple Precision Interval Packages: Comparing Different Approaches
, 2003
"... We give a survey on packages for multiple precision interval arithmetic, with the main focus on three specific packages. One is within a Maple environment, intpakX, and two are C/C++ libraries, GMPXSC and MPFI. We discuss their different features, present timing results and show several application ..."
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Cited by 8 (0 self)
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We give a survey on packages for multiple precision interval arithmetic, with the main focus on three specific packages. One is within a Maple environment, intpakX, and two are C/C++ libraries, GMPXSC and MPFI. We discuss their different features, present timing results and show several applications from various fields, where high precision intervals are fundamental.
The design and implementation of an objectoriented validated ODE solver
, 2002
"... Validated methods for initial value problems (IVPs) for ordinary differential equations (ODEs) produce bounds that are guaranteed to contain the true solution of a problem. These methods use interval techniques to enclose rounding errors, truncation errors, and uncertainties in the model. We describ ..."
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Cited by 8 (1 self)
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Validated methods for initial value problems (IVPs) for ordinary differential equations (ODEs) produce bounds that are guaranteed to contain the true solution of a problem. These methods use interval techniques to enclose rounding errors, truncation errors, and uncertainties in the model. We describe the design and implementation...