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16
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.
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...
Exclusion Regions for Systems of Equations
 SIAM J. NUM. ANALYSIS
, 2003
"... Branch and bound methods for nding all zeros of a nonlinear system of equations in a box frequently have the diculty that subboxes containing no solution cannot be easily eliminated if there is a nearby zero outside the box. This has the eect that near each zero, many small boxes are created by repe ..."
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Cited by 11 (7 self)
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Branch and bound methods for nding all zeros of a nonlinear system of equations in a box frequently have the diculty that subboxes containing no solution cannot be easily eliminated if there is a nearby zero outside the box. This has the eect that near each zero, many small boxes are created by repeated splitting, whose processing may dominate the total work spent on the global search. This paper discusses the reasons for the occurrence of this socalled cluster eect, and how to reduce the cluster eect by de ning exclusion regions around each zero found, that are guaranteed to contain no other zero and hence can safely be discarded. Such exclusion regions are traditionally constructed using uniqueness tests based on the Krawczyk operator or the Kantorovich theorem. These results are reviewed; moreover, re nements are proved that signi cantly enlarge the size of the exclusion region. Existence and uniqueness tests are also given.
Inclusion Isotone Extended Interval Arithmetic  A Toolbox Update
, 1996
"... In this report we deal with the correct formulation of a special extended interval arithmetic in the context of interval Newton like methods. We first demonstrate some of the problems arising from selected older definitions. Then we investigate the basic aim, concept, and properties important for de ..."
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Cited by 6 (0 self)
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In this report we deal with the correct formulation of a special extended interval arithmetic in the context of interval Newton like methods. We first demonstrate some of the problems arising from selected older definitions. Then we investigate the basic aim, concept, and properties important for defining a correct extended interval division. Finally, we give a proper way for defining the extended interval operations needed in our special context, and we prove their inclusion isotonicity. Additionally, we give some sample applications. We conclude with two updated implementations of our extended interval operations in the toolbox environments [2] and [3].
Simplification of SymbolicNumerical Interval Expressions
 in Gloor, O. (Ed.): Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation, ACM
, 1998
"... Although interval arithmetic is increasingly used in combination with computer algebra and other methods, both approaches  symbolicalgebraic and intervalarithmetic  are used separately. Implementing symbolic interval arithmetic seems not suitable due to the exponential growth in the "size" o ..."
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Cited by 5 (2 self)
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Although interval arithmetic is increasingly used in combination with computer algebra and other methods, both approaches  symbolicalgebraic and intervalarithmetic  are used separately. Implementing symbolic interval arithmetic seems not suitable due to the exponential growth in the "size" of the endpoints. In this paper we propose a methodology for "true" symbolicalgebraic manipulations on symbolicnumerical interval expressions involving interval variables instead of symbolic intervals. Due to the better algebraic properties, resembling to classical analysis, and the containment of classical interval arithmetic as a special case, we consider the algebraic extension of conventional interval arithmetic as an appropriate environment for solving interval algebraic problems. Based on the distributivity relations, a general framework for simplification of symbolicnumerical expressions involving intervals is given and some of the wider implications of the theory pertaining to inte...
GIA InC++: A Global Interval Arithmetic Library for Discontinuous Intervals
"... This document discusses theoretical and practical problems of evaluating interval arithmetic (IA) functions globally, and presents a C++ library called GIA InC++ (Global Interval Arithmetic in C++) for the task. In this implementation both algebraic and numerical IA techniques are employed. Our goal ..."
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Cited by 4 (4 self)
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This document discusses theoretical and practical problems of evaluating interval arithmetic (IA) functions globally, and presents a C++ library called GIA InC++ (Global Interval Arithmetic in C++) for the task. In this implementation both algebraic and numerical IA techniques are employed. Our goal has been to provide the user with an easy to use, portable evaluator that can be applied in practical applications without deep understanding of interval analysis. The paper can be used as a user's guide to the system. ii PREFACE The work is part of a project on applications of constraint propagation in designing and planning (19921994) funded mainly by Technology Development Centre of Finland (TEKES) and carried out by VTT Information Technology, Information Systems (formerly Laboratory for Information Processing). Trema Inc. and ViSolutions Inc. have acted as industrial partners in the project. Members of the steering committee were Dr. Tech. Juha Hynynen (chairman, ViSolutions Inc.), ...
Directed Interval Arithmetic in Mathematica: Implementation and Applications
, 1996
"... This report presents an experimental Mathematica ..."
Diagrammatic representation for interval arithmetic
 LINEAR ALGEBRA AND ITS APPLICATIONS
, 2001
"... The paper presents a diagrammatic representation of a standard interval space (the socalled MRdiagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the ISdiagram representation ..."
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Cited by 4 (0 self)
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The paper presents a diagrammatic representation of a standard interval space (the socalled MRdiagram), and shows how to represent and perform interval arithmetic and derive its various properties using the diagram. The representation is an extension and refinement of the ISdiagram representation devised earlier by the author to represent interval relations. First, the MRdiagram is defined together with appropriate graphical notions and constructions for basic interval relations and operations. Second, diagrammatic constructions for all standard arithmetic operations are presented. Several examples of the use of these constructions to aid reasoning about various simple, though nontrivial, properties of interval arithmetic are included in order to show how the representation facilitates both deeper understanding of the subject matter and reasoning about its properties.
Extended Interval Arithmetic in IEEE FloatingPoint Environment
 Interval Computations
, 1994
"... This paper describes an implementation of a general interval arithmetic extension, which comprises the following extensions of the conventional interval arithmetic: (1) extension of the set of normal intervals by improper intervals; (2) extension of the set of arithmetic operations for normal interv ..."
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Cited by 2 (1 self)
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This paper describes an implementation of a general interval arithmetic extension, which comprises the following extensions of the conventional interval arithmetic: (1) extension of the set of normal intervals by improper intervals; (2) extension of the set of arithmetic operations for normal intervals by nonstandard operations; (3) extension by infinite intervals. We give a possible realization scheme of such an universal interval arithmetic in any programming environment supporting IEEE floatingpoint arithmetic. A PASCALXSC module is reported which allows easy programming of numerical algorithms formulated in terms of conventional interval arithmetic or of any of the enlisted extended interval spaces, and provides a common base for comparison of such numerical algorithms. 1
Constrained Interval Arithmetic
"... : This paper presents an approach to solving the longstanding dependency problem in interval arithmetic. An extension to interval arithmetic, called here constrained interval arithmetic, is developed. Unlike interval arithmetic, constrained interval arithmetic has an additive inverse, a multiplicat ..."
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Cited by 1 (1 self)
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: This paper presents an approach to solving the longstanding dependency problem in interval arithmetic. An extension to interval arithmetic, called here constrained interval arithmetic, is developed. Unlike interval arithmetic, constrained interval arithmetic has an additive inverse, a multiplicative inverse and satisfies the distributive law. This means that the algebraic structure of constrained interval arithmetic is different than that of interval arithmetic. The applicability of constrained interval arithmetic is explored. 1. Introduction: It is wellknown in the interval analysis literature that interval arithmetic overestimates the resultant width of the interval when dependencies are present. This overestimation can be arbitrarily large (see, for example, (Neumaier 1990, pages 1619)). Example 1Consider y = f(x) = x(1 \Gamma x); x 2 [0; 1]: The implementation of interval arithmetic yields y = [0; 1] \Theta (1 \Gamma [0; 1]) = [0; 1] \Theta [0; 1] = [0; 1]. The actual range...