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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 ..."
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Cited by 112 (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 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 96 (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
The Extended Real Interval System
, 1998
"... Three extended real interval systems are defined and distinguished by their implementation complexity and result sharpness. The three systems are closed with respect to interval arithmetic and the enclosure of functions and relations, notwithstanding domain restrictions or the presence of singularit ..."
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Cited by 17 (1 self)
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Three extended real interval systems are defined and distinguished by their implementation complexity and result sharpness. The three systems are closed with respect to interval arithmetic and the enclosure of functions and relations, notwithstanding domain restrictions or the presence of singularities. 1 Overview Section 2 introduces the problem of defining closed interval systems. In Section 3, real and extended points and intervals are defined. In Section 4, the empty and entire intervals are used to close the extended interval system. Section ?? shows how incorrect conclusions have been reached about the result of certain interval arithmetic operatoroperand combinations. The author is grateful to Professor Arnold Neumaier for originally raising this issue. Section 9 describes how to legitimately use IEEE floatingpoint arithmetic to obtain the sharp results described in Section ??. Section 6 generalizes extended interval arithmetic to define interval enclosures of functions, with...
InC++: A local interval arithmetic library
 Research Report, (VTT, Technical Research Centre of Finland, Information Technology
, 1994
"... ..."
Interval Computations On The Spreadsheet
 Applications of Interval Computations
, 1996
"... This paper reviews work on using interval arithmetic as the basis for next generation spreadsheet programs capable of dealing with rounding errors, imprecise data, and numerical constraints. A series of ever more versatile computational models for spreadsheets are presented beginning from classical ..."
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Cited by 6 (1 self)
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This paper reviews work on using interval arithmetic as the basis for next generation spreadsheet programs capable of dealing with rounding errors, imprecise data, and numerical constraints. A series of ever more versatile computational models for spreadsheets are presented beginning from classical interval arithmetic and ending up with interval constraint satisfaction. In order to demonstrate the ideas, an actual implementation of each model as a class library is presented and its integration with a commercial spreadsheet program is explained. 1 LIMITATIONS OF SPREADSHEET COMPUTING Spreadsheet programs, such as MS Excel, Quattro Pro, Lotus 123, etc., are among the most widely used applications of computer science. Since the pioneering days of VisiCalc and others, spreadsheet programs have been enhanced immensely with new features. However, the underlying computational paradigm of evaluating arithmetical functions by using ordinary machine arithmetic has remained the same. The wor...
Computing zeros of functions using generalized interval arithmetic
 Interval Comput
, 1993
"... We consider the use of a generalized interval arithmetic in algorithms for solving nonlinear equations or systems of nonlinear equations. The algorithms can involve either derivatives or slopes. The convergence rate is improved for either form. The improvement is greater if slopes, rather than deriv ..."
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Cited by 4 (0 self)
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We consider the use of a generalized interval arithmetic in algorithms for solving nonlinear equations or systems of nonlinear equations. The algorithms can involve either derivatives or slopes. The convergence rate is improved for either form. The improvement is greater if slopes, rather than derivatives, are used. However, the slope method is applicable to only rational functions. For multidimensional problems we introduce the generalized interval arithmetic into the HansenSengupta method. Again, the convergence rate is improved. Вычисление нулей функций при помощи обобщенной интервальной арифметики Э. Р. Хансен Рассматривается использование обобщенной интервальной арифметики в алгоритмах для решения нелинейных уравнений или систем нелинейных уравнений. В эти алгоритмы могут входить либо производные, либо <наклоны>. И в том и в другом случае улучшается скорость сходимости. Но при использовании наклонов это улучшение значительнее, чем в случае производных. Однако метод наклонов применим лишь для рациональных функций. Для многомерных задач мы вводим обобщенную интервальную арифметику в метод ХансенаСенгупты. Скорость сходимости улучшается и в этом случае. c ○ E. R. Hansen, 19944 E. R. Hansen 1
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"... In this paper, a constrained optimization method for various functions such as non differentiable ones based on interval analysis is proposed. The process can be broken down into two distinct parts: the first one that approximates the space satisfying the constraints and the second one that is exclu ..."
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In this paper, a constrained optimization method for various functions such as non differentiable ones based on interval analysis is proposed. The process can be broken down into two distinct parts: the first one that approximates the space satisfying the constraints and the second one that is exclusively in charge of optimizing a given function. The first phase relies on successive bisections of the initial space while discarding the boxes that cannot verify the constraints. The last part of the algorithm is in charge of the unconstrained optimization using a Particle Swarm Optimization algorithm (commonly known as PSO).