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28
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.
Guaranteeing the Topology of an Implicit Surface Polygonization for Interactive Modeling
, 1997
"... Morse theory shows how the topology of an implicit surface is affected by its function's critical points, whereas catastrophe theory shows how these critical points behave as the function's parameters change. Interval analysis finds the critical points, and they can also be tracked efficiently durin ..."
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Cited by 99 (9 self)
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Morse theory shows how the topology of an implicit surface is affected by its function's critical points, whereas catastrophe theory shows how these critical points behave as the function's parameters change. Interval analysis finds the critical points, and they can also be tracked efficiently during parameter changes. Changes in the function value at these critical points cause changes in the topology. Techniques for modifying the polygonization to accommodate such changes in topology are given. These techniques are robust enough to guarantee the topology of an implicit surface polygonization, and are efficient enough to maintain this guarantee during interactive modeling. The impact of this work is a topologicallyguaranteed polygonization technique, and the ability to directly and accurately manipulate polygonized implicit surfaces in real time.
Some tests of generalized bisection
 ACM Trans. Math. Software
, 1987
"... This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically co ..."
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Cited by 49 (2 self)
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This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically compares such generalized bisection algorithms to themselves, to continuation methods, and to hybrid steepest descent/quasiNewton methods. A specific algorithm containing novel “expansion ” and “exclusion ” steps is fully described, and the effectiveness of these steps is evaluated. A test problem consisting of a small, highdegree polynomial system that is appropriate for generalized bisection, but very difticult for continuation methods, is presented. This problem forms part of a set of 17 test problems from published literature on the methods being compared; this test set is fully described here.
Robust Process Simulation Using Interval Methods
 Comput. Chem. Eng
, 1996
"... Ideally, for the needs of robust process simulation, one would like a nonlinear equation solving technique that can find any and all roots to a problem, and do so with mathematical certainty. In general, currently used techniques do not provide such rigorous guarantees. One approach to providing suc ..."
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Cited by 31 (19 self)
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Ideally, for the needs of robust process simulation, one would like a nonlinear equation solving technique that can find any and all roots to a problem, and do so with mathematical certainty. In general, currently used techniques do not provide such rigorous guarantees. One approach to providing such assurances can be found in the use of interval analysis, in particular the use of interval Newton methods combined with generalized bisection. However, these methods have generally been regarded as extremely inefficient. Motivated by recent progress in interval analysis, as well as continuing advances in computer speed and the availability of parallel computing, we consider here the feasibility of using an interval Newton/generalized bisection algorithm on process simulation problems. An algorithm designed for parallel computing on an MIMD machine is described, and results of tests on several problems are reported. Experiments indicate that the interval Newton/generalized bisection method works quite well on relatively small problems, providing a powerful method for finding all solutions to a problem. For larger problems, the method performs inconsistently with regard to efficiency, at least when reasonable initial bounds are not provided.
Motivations for an arbitrary precision interval arithmetic and the MPFI library
 Reliable Computing
, 2002
"... Nowadays, computations involve more and more operations and consequently errors. The limits of applicability of some numerical algorithms are now reached: for instance the theoretical stability of a dense matrix factorization (LU or QR) is ensured under the assumption that n 3 u < 1, where n is the ..."
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Cited by 30 (7 self)
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Nowadays, computations involve more and more operations and consequently errors. The limits of applicability of some numerical algorithms are now reached: for instance the theoretical stability of a dense matrix factorization (LU or QR) is ensured under the assumption that n 3 u < 1, where n is the dimension of the matrix and u = 1 + − 1, with 1 + the smallest floatingpoint larger than 1; this means that n must be less than 200,000, which is almost reached by modern simulations. The numerical quality of solvers is now an issue, and not only their mathematical quality. Let us cite studies performed by the CEA (French Nuclear Agency) on the simulation of nuclear plant accidents and also softwares controlling and possibly correcting numerical programs, such as Cadna [10] or Cena [20]. Another approach consists in computing with certified enclosures, namely interval arithmetic [21, 2, 18]. The fundamental principle of this arithmetic consists in replacing every number by an interval enclosing it. For instance, π cannot be exactly represented using a binary or decimal arithmetic, but it
Robust phase stability analysis using interval methods
 In AIChE Symp. Ser
, 1995
"... Conventional equation solving and optimization techniques for solving the phase stability problem may fail to converge or may converge to an incorrect result. A technique for solving the problem with mathematical certainty is needed. One approach to providing such assurance can be found in the use o ..."
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Cited by 22 (21 self)
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Conventional equation solving and optimization techniques for solving the phase stability problem may fail to converge or may converge to an incorrect result. A technique for solving the problem with mathematical certainty is needed. One approach to providing such assurance can be found in the use of interval methods. An interval Newton/generalized bisection technique is applied here to solve the phase stability problem. Results for two models of liquidphase systems, using several different feed compositions, indicate that the technique used is reliable and very efficient.
A Constraint Satisfaction Approach to a Circuit Design Problem
, 1998
"... A classical circuitdesign problem from Ebers and Moll [6] features a system of nine nonlinear equations in nine variables that is very challenging both for local and global methods. This system was solved globally using an interval method by Ratschek and Rokne [23] in the box [0; 10] 9 . Their ..."
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Cited by 21 (1 self)
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A classical circuitdesign problem from Ebers and Moll [6] features a system of nine nonlinear equations in nine variables that is very challenging both for local and global methods. This system was solved globally using an interval method by Ratschek and Rokne [23] in the box [0; 10] 9 . Their algorithm had enormous costs (i.e., over 14 months using a network of 30 Sun Sparc1 workstations) but they state that "at this time, we know no other method which has been applied to this circuit design problem and which has led to the same guaranteed result of locating exactly one solution in this huge domain, completed with a reliable error estimate." The present paper gives a novel branchandprune algorithm that obtains a unique safe box for the above system within reasonable computation times. The algorithm combines traditional interval techniques with an adaptation of discrete constraintsatisfaction techniques to continuous problems. Of particular interest is the simplicity o...
Decomposition of Arithmetic Expressions to Improve the Behavior of Interval Iteration for Nonlinear Systems
, 1991
"... Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumul ..."
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Cited by 20 (9 self)
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Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumulated and applied, straightforward linearization of the original nonlinear system sometimes leads to nonconvergent iteration. In this paper, we examine interval iterations based on an expanded system obtained from the intermediate quantities in the original system. In this system, there is no overestimation in entries of the interval Jacobi matrix, and nonlinearities can be taken into account to obtain sharp bounds. We present an example in detail, algorithms, and detailed experimental results obtained from applying our algorithms to the example.
Combining local consistency, symbolic rewriting and interval methods
 In Procs. of AISMC3, volume 1138 of LNCS
, 1996
"... Abstract. This paper is an attempt to address the processing of nonlinear numerical constraints over the Reals by combining three di erent methods: local consistency techniques, symbolic rewriting and interval methods. To formalize this combination, we de ne a generic twostep constraint processing ..."
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Cited by 14 (4 self)
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Abstract. This paper is an attempt to address the processing of nonlinear numerical constraints over the Reals by combining three di erent methods: local consistency techniques, symbolic rewriting and interval methods. To formalize this combination, we de ne a generic twostep constraint processing technique based on an extension of the Constraint Satisfaction Problem, called Extended Constraint Satisfaction Problem (ECSP). The rst step is a rewriting step, in which the initial ECSP is symbolically transformed. The second step, called approximation step, is based on a local consistency notion, called weak arcconsistency, de ned over ECSPs in terms of xed point ofcontractant monotone operators. This notion is shown to generalize previous local consistency concepts de ned over nite domains (arcconsistency) or in nite subsets of the Reals (arc Bconsistency and interval, hull and boxconsistency). A ltering algorithm, derived from AC3, is given and is shown to be correct, con uent and to terminate. This framework is illustrated by the combination of Grobner Bases computations and Interval Newton methods. The computation of Grobner Bases for subsets of the initial set of constraints is used as a rewriting step and operators based on Interval Newton methods are used together with enumeration techniques to achieve weak arcconsistency on the modi ed ECSP. Experimental results from a prototype are presented, as well as comparisons with other systems.